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1.8: Relating Trigonometric Functions - Mathematics


Learning Objectives

  • State the reciprocal relationships between trig functions, and use these identities to find values of trig functions.
  • State quotient relationships between trig functions, and use quotient identities to find values of trig functions.
  • State the domain and range of each trig function.
  • State the sign of a trig function, given the quadrant in which an angle lies.
  • State the Pythagorean identities and use these identities to find values of trig functions.

Reciprocal and Pythagorean Identities

The two most basic types of trigonometric identities are the reciprocal identities and the Pythagorean identities. The reciprocal identities are simply definitions of the reciprocals of the three standard trigonometric ratios:
[ sec heta=frac{1}{cos heta} quad csc heta=frac{1}{sin heta} quad cot heta=frac{1}{ an heta}
]

Also, recall the definitions of the three standard trigonometric ratios (sine, cosine and tangent):
[ egin{array}{l}
sin heta=frac{o p p}{h y p}
cos heta=frac{a d j}{h y p}
an heta=frac{o p p}{a d y}
end{array}
]

If we look more closely at the relationships between the sine, cosine and tangent, we'll notice that (frac{sin heta}{cos heta}= an heta)
[ frac{sin heta}{cos heta}=frac{left(frac{o p p}{h y p} ight)}{left(frac{a d j}{h y p} ight)}=frac{o p p}{h y p} * frac{h y p}{a d j}=frac{o p p}{a d j}= an heta
]

Pythagorean Identities
The Pythagorean Identities are, of course, based on the Pythagorean Theorem. If we recall a diagram that was introduced in Chapter (2,) we can build these identities from the relationships in the diagram:

Using the Pythagorean Theorem in this diagram, we see that (x^{2}+y^{2}=1^{2},) so (x^{2}+y^{2}=1 .) But, also remember that, in the unit circle, (x=cos heta) and (y=sin heta)

Substituting this equality gives us the first Pythagorean Identity:
[ x^{2}+y^{2}=1
] or
[ cos ^{2} heta+sin ^{2} heta=1
] This identity is usually stated in the form:
[ sin ^{2} heta+cos ^{2} heta=1
]

If we take this identity and divide through on both sides by (cos ^{2} heta,) this will result in the first of two additional Pythagorean Identities:
[ frac{sin ^{2} heta}{cos ^{2} heta}+frac{cos ^{2} heta}{cos ^{2} heta}=frac{1}{cos ^{2} heta}
] or
[ an ^{2} heta+1=sec ^{2} heta
]

Dividing through by (sin ^{2} heta) gives us the second:
[ frac{sin ^{2} heta}{sin ^{2} heta}+frac{cos ^{2} heta}{sin ^{2} heta}=frac{1}{sin ^{2} heta}
] or
[ 1+cot ^{2} heta=csc ^{2} heta
] So, the three Pythagorean Identities we will be using are:
[ egin{array}{l}
sin ^{2} heta+cos ^{2} heta=1
an ^{2} heta+1=sec ^{2} heta
1+cot ^{2} heta=csc ^{2} heta
end{array}
]

These Pythagorean Identities are often stated in other terms, such as:
[ egin{array}{l}
sin ^{2} heta=1-cos ^{2} heta
cos ^{2} heta=1-sin ^{2} heta
an ^{2} heta=sec ^{2} heta-1
cot ^{2} heta=csc ^{2} heta-1
end{array}
]

Now that we have some basic identities to work with, let's use them to verify the equality of some more complicated statements. The process of verifying trigonometric identities involves changing one side of the given expression into the other side. since these are not really equations, we will not treat them the way we treat equations. That is to say, we won't add or subtract anything to both sides of the statement (or multiply or divide by anything on both sides either).

Another reason for not treating a trigonometric identity as an equation is that, in practice, this process typically involves just one side of the statement. In problem solving, mathematicians typically use trigonometric identities to change the appearance of a problem without changing its value. In this process, a trigonometric expression is changed into another trigonometric expression rather than showing that two trigonometric expressions are the same, which is what we will be doing.


The trigonometric identities we have discussed in this section are summarized below:

The form sin ( heta) or (cos heta) is typically used, however any letter may be used to represent the angle in question so long as it is the SAME letter in all expressions. For example, we can say that:
[ sin ^{2} heta+cos ^{2} heta=1
] or we can say that

[ sin ^{2} x+cos ^{2} x=1
] however:
[ sin ^{2} heta+cos ^{2} x eq 1
] because ( heta) and (x) could be different angles!

Quotient Identities

The definitions of the trig functions led us to the reciprocal identities, which can be seen in the Concept about that topic. They also lead us to another set of identities, the quotient identities.

Consider first the sine, cosine, and tangent functions. For angles of rotation (not necessarily in the unit circle) these functions are defined as follows:

(egin{aligned}sin heta&=dfrac{y}{r} cos heta&=dfrac{x}{r} an heta &=dfrac{y}{x}end{aligned})

Given these definitions, we can show that ( an heta =dfrac{sin heta}{cos heta}), as long as (cos heta eq 0):

(dfrac{sin heta}{cos heta} =dfrac{dfrac{y}{r}}{dfrac{x}{r}}=dfrac{y}{r} imes dfrac{r}{x}=dfrac{y}{x}= an heta).

The equation ( an heta =dfrac{sin heta}{cos heta}) is therefore an identity that we can use to find the value of the tangent function, given the value of the sine and cosine.

Let's take a look at some problems involving quotient identities.

1. Find the value of ( an heta)?

If (cos heta =dfrac{5}{13}) and (sin heta =dfrac{12}{13}), what is the value of ( an heta )?

( an heta =dfrac{12}{5})

( an heta =dfrac{sin heta}{cos heta} =dfrac{dfrac{12}{13}}{dfrac{5}{13}}=dfrac{12}{13} imes dfrac{13}{5}=dfrac{12}{5})

2. Show that (cot heta =dfrac{cos heta}{sin heta})

(cos heta sin heta =dfrac{dfrac{x}{r}}{dfrac{y}{r}}=dfrac{x}{r} imesdfrac{r}{y}=dfrac{x}{y}=cot heta)

3. What is the value of (cot heta)?

If (cos heta =dfrac{7}{25}) and (sin heta =dfrac{24}{25}), what is the value of (cot heta)?

(cot heta =dfrac{7}{24})

(cot heta =dfrac{cos heta}{sin heta}=dfrac{dfrac{7}{25}}{dfrac{24}{25}}=dfrac{7}{25} imes dfrac{25}{24}=dfrac{7}{24})

Example (PageIndex{3})

If (sin heta =dfrac{63}{65}) and (cos heta =dfrac{16}{65}), what is the value of ( an heta)?

Solution

( an heta =dfrac{63}{16}). We can see this from the relationship for the tangent function:

( an heta =dfrac{sin heta}{cos heta}=dfrac{dfrac{63}{65}}{dfrac{16}{65}}=dfrac{63}{65} imes dfrac{65}{16}=dfrac{63}{16})

Example (PageIndex{4})

If ( an heta =dfrac{40}{9}) and (cos heta =dfrac{9}{41}), what is the value of (sin heta)?

Solution

(sin heta =dfrac{40}{41}). We can see this from the relationship for the tangent function:

(egin{aligned} an heta &= dfrac{sin heta}{cos heta} sin heta &=( an heta )(cos heta ) sin heta&=dfrac{40}{9} imes dfrac{9}{41} sin heta &=dfrac{40}{41}end{aligned})

Review

Fill in each blank with a trigonometric function.

  1. If (cos heta =dfrac{1}{2}) and (cot heta =dfrac{sqrt{3}}{3}), what is the value of (sin heta )?
  2. If ( an heta =0) and (cos heta =−1), what is the value of (sin heta)?
  3. If (cot heta =−1) and (sin heta =−dfrac{sqrt{2}}{2}), what is the value of (cos heta )?

Vocabulary

TermDefinition
Quotient IdentityThe quotient identity is an identity relating the tangent of an angle to the sine of the angle divided by the cosine of the angle.

Additional Resources

Video: The Reciprocal, Quotient, and Pythagorean Identities

Cofunction Identities

A cofunction identity is a relationship between one trig function of an angle and another trig function of the complement of that angle.

In a right triangle, you can apply what are called "cofunction identities". These are called cofunction identities because the functions have common values. These identities are summarized below.


Definition of Trigonometric Functions & Basic Formulae

In a right angled triangle ABC, ∠CAB = A and ∠BCA = 90° = π/2. AC is the base, BC the altitude and AB is the hypotenuse.

We refer to the base as the adjacent side and to the altitude as the opposite side. There are six trigonometric ratios, also called trigonometric functions or circular functions. With reference to angle A, the six ratios are:

is called sine of A , and written as sinA

is called the cosine of A , and written as cosA

is called the tangent of A , and written as tanA

Obviously, $large tanA = frac $

The reciprocals of sine, cosine and tangent are called the cosecant, secant and cotangent of A respectively. We write these as cosecA , secA , cot A respectively.

Since the hypotenuse is the greatest side in a right angle triangle, sin A and cos A can never be greater than unity and cosecA and sec A can never be less than unity.

Hence , | sinA | ≤ 1, |cos A| ≤ 1, |cosec A| ≥ 1, |sec A | ≥1 , while tan A and cot A may have any numerical value lying between – ∞ to +∞

Notes:

♦ Above mentioned method relating trigonometric functions to angles and sides of a triangle is called geometric definition of trigonometric functions.

♦ All the six trigonometric functions have got a very important property in common that is periodicity.

♦ Remember that the trigonometrical ratios are real numbers and remain same so long as the angles are real.

Basic Formulae:

It is possible to express a trigonometrical ratio in terms of any one of the other ratios:

i.e. all trigonometrical functions have been expressed in terms of cotθ . This is left as an exercise for you to derive these results. Just as a hint for you, express the denominator of fraction that defines cotθ as unity (i.e. base as unity) and form a right-angled triangle to express the sides and proceed.

Illustration : Express tanθ in terms of cosθ.

$large cos heta = frac = frac<1>$ where OB is taken as unity and OA = x

Illustration : If sinθ + sin 2 θ = 1 , then prove that

cos 12 θ + 3 cos 10 θ + 3 cos 8 θ + cos 6 θ – 1 = 0.

Given that sinθ = 1 – sin 2 θ = cos 2 θ

L.H.S = cos 6 θ (cos 2 θ + 1) 3 – 1

Illustration :Prove that :

(tanθ + cotθ) 2 = tan 2 θ + cot 2 θ + 2

= sec 2 θ – 1 + cosec 2 θ – 1 + 2

(i) If sin x + cos x = m and sec x + cosec x = n prove that n(m 2 – 1) = 2 m.

(ii) If x sin 3 θ + y cos 3 θ = sinθ and x sinθ – y cosθ = 0, prove that x 2 + y 2 = 1

(iv) If a sec α + btan α = d and b sec α + a tan α = c , prove that a 2 + c 2 = b 2 + d 2


Contents

In this section, the same upper-case letter denotes a vertex of a triangle and the measure of the corresponding angle the same lower case letter denotes an edge of the triangle and its length.

Given an acute angle A = θ of a right-angled triangle, the hypotenuse c is the side that connects the two acute angles. The side b adjacent to θ is the side of the triangle that connects θ to the right angle. The third side a is said to be opposite to θ .

If the angle θ is given, then all sides of the right-angled triangle are well-defined up to a scaling factor. This means that the ratio of any two side lengths depends only on θ . Thus these six ratios define six functions of θ , which are the trigonometric functions. More precisely, the six trigonometric functions are: [4] [5]

sine sin ⁡ θ = a c = o p p o s i t e h y p o t e n u s e >= > >>> cosecant csc ⁡ θ = c a = h y p o t e n u s e o p p o s i t e >= > >>>
cosine cos ⁡ θ = b c = a d j a c e n t h y p o t e n u s e >= > >>> secant sec ⁡ θ = c b = h y p o t e n u s e a d j a c e n t >= > >>>
tangent tan ⁡ θ = a b = o p p o s i t e a d j a c e n t >= > >>> cotangent cot ⁡ θ = b a = a d j a c e n t o p p o s i t e >= > >>>

In geometric applications, the argument of a trigonometric function is generally the measure of an angle. For this purpose, any angular unit is convenient, and angles are most commonly measured in conventional units of degrees in which a right angle is 90° and a complete turn is 360° (particularly in elementary mathematics).

However, in calculus and mathematical analysis, the trigonometric functions are generally regarded more abstractly as functions of real or complex numbers, rather than angles. In fact, the functions sin and cos can be defined for all complex numbers in terms of the exponential function via power series [7] or as solutions to differential equations given particular initial values [8] (see below), without reference to any geometric notions. The other four trigonometric functions (tan, cot, sec, csc) can be defined as quotients and reciprocals of sin and cos, except where zero occurs in the denominator. It can be proved, for real arguments, that these definitions coincide with elementary geometric definitions if the argument is regarded as an angle given in radians. [7] Moreover, these definitions result in simple expressions for the derivatives and indefinite integrals for the trigonometric functions. [9] Thus, in settings beyond elementary geometry, radians are regarded as the mathematically natural unit for describing angle measures.

When radians (rad) are employed, the angle is given as the length of the arc of the unit circle subtended by it: the angle that subtends an arc of length 1 on the unit circle is 1 rad (≈ 57.3°), and a complete turn (360°) is an angle of 2 π (≈ 6.28) rad. For real number x, the notations sin x, cos x, etc. refer to the value of the trigonometric functions evaluated at an angle of x rad. If units of degrees are intended, the degree sign must be explicitly shown (e.g., sin , cos , etc.). Using this standard notation, the argument x for the trigonometric functions satisfies the relationship x = (180x/ π )°, so that, for example, sin π = sin 180° when we take x = π . In this way, the degree symbol can be regarded as a mathematical constant such that 1° = π /180 ≈ 0.0175.

The six trigonometric functions can be defined as coordinate values of points on the Euclidean plane that are related to the unit circle, which is the circle of radius one centered at the origin O of this coordinate system. While right-angled triangle definitions allow for the definition of the trigonometric functions for angles between 0 and π 2 < extstyle <2>>> radian (90°), the unit circle definitions allow the domain of trigonometric functions to be extended to all positive and negative real numbers.

The trigonometric functions cos and sin are defined, respectively, as the x- and y-coordinate values of point A . That is,

The other trigonometric functions can be found along the unit circle as

By applying the Pythagorean identity and geometric proof methods, these definitions can readily be shown to coincide with the definitions of tangent, cotangent, secant and cosecant in terms of sine and cosine, that is

hold for any angle θ and any integer k . The same is true for the four other trigonometric functions. By observing the sign and the monotonicity of the functions sine, cosine, cosecant, and secant in the four quadrants, one can show that 2 π is the smallest value for which they are periodic (i.e., 2 π is the fundamental period of these functions). However, after a rotation by an angle π , the points B and C already return to their original position, so that the tangent function and the cotangent function have a fundamental period of π . That is, the equalities

hold for any angle θ and any integer k .

The algebraic expressions for the most important angles are as follows:

Writing the numerators as square roots of consecutive non-negative integers, with a denominator of 2, provides an easy way to remember the values. [12]

Such simple expressions generally do not exist for other angles which are rational multiples of a straight angle. For an angle which, measured in degrees, is a multiple of three, the sine and the cosine may be expressed in terms of square roots, see Trigonometric constants expressed in real radicals. These values of the sine and the cosine may thus be constructed by ruler and compass.

For an angle of an integer number of degrees, the sine and the cosine may be expressed in terms of square roots and the cube root of a non-real complex number. Galois theory allows proving that, if the angle is not a multiple of 3°, non-real cube roots are unavoidable.

For an angle which, measured in degrees, is a rational number, the sine and the cosine are algebraic numbers, which may be expressed in terms of n th roots. This results from the fact that the Galois groups of the cyclotomic polynomials are cyclic.

For an angle which, measured in degrees, is not a rational number, then either the angle or both the sine and the cosine are transcendental numbers. This is a corollary of Baker's theorem, proved in 1966.

Simple algebraic values Edit

The following table summarizes the simplest algebraic values of trigonometric functions. [13] The symbol ∞ represents the point at infinity on the projectively extended real line it is not signed, because, when it appears in the table, the corresponding trigonometric function tends to + ∞ on one side, and to − ∞ on the other side, when the argument tends to the value in the table.

The modern trend in mathematics is to build geometry from calculus rather than the converse. [ citation needed ] Therefore, except at a very elementary level, trigonometric functions are defined using the methods of calculus.

Trigonometric functions are differentiable and analytic at every point where they are defined that is, everywhere for the sine and the cosine, and, for the tangent, everywhere except at π /2 + k π for every integer k .

The trigonometric function are periodic functions, and their primitive period is 2 π for the sine and the cosine, and π for the tangent, which is increasing in each open interval ( π /2 + k π , π /2 + (k + 1) π ) . At each end point of these intervals, the tangent function has a vertical asymptote.

In calculus, there are two equivalent definitions of trigonometric functions, either using power series or differential equations. These definitions are equivalent, as starting from one of them, it is easy to retrieve the other as a property. However the definition through differential equations is somehow more natural, since, for example, the choice of the coefficients of the power series may appear as quite arbitrary, and the Pythagorean identity is much easier to deduce from the differential equations.

Definition by differential equations Edit

Sine and cosine are the unique differentiable functions such that

Differentiating these equations, one gets that both sine and cosine are solutions of the differential equation

Applying the quotient rule to the definition of the tangent as the quotient of the sine by the cosine, one gets that the tangent function verifies

Power series expansion Edit

Applying the differential equations to power series with indeterminate coefficients, one may deduce recurrence relations for the coefficients of the Taylor series of the sine and cosine functions. These recurrence relations are easy to solve, and give the series expansions [14]

The radius of convergence of these series is infinite. Therefore, the sine and the cosine can be extended to entire functions (also called "sine" and "cosine"), which are (by definition) complex-valued functions that are defined and holomorphic on the whole complex plane.

Being defined as fractions of entire functions, the other trigonometric functions may be extended to meromorphic functions, that is functions that are holomorphic in the whole complex plane, except some isolated points called poles. Here, the poles are the numbers of the form ( 2 k + 1 ) π 2 < extstyle (2k+1)<2>>> for the tangent and the secant, or k π for the cotangent and the cosecant, where k is an arbitrary integer.

Recurrences relations may also be computed for the coefficients of the Taylor series of the other trigonometric functions. These series have a finite radius of convergence. Their coefficients have a combinatorial interpretation: they enumerate alternating permutations of finite sets. [15]

one has the following series expansions: [16]

Partial fraction expansion Edit

There is a series representation as partial fraction expansion where just translated reciprocal functions are summed up, such that the poles of the cotangent function and the reciprocal functions match: [17]

This identity can be proven with the Herglotz trick. [18] Combining the (–n) th with the n th term lead to absolutely convergent series:

Similarly, one can find a partial fraction expansion for the secant, cosecant and tangent functions:

Infinite product expansion Edit

The following infinite product for the sine is of great importance in complex analysis:

For the proof of this expansion, see Sine. From this, it can be deduced that

Relationship to exponential function (Euler's formula) Edit

This formula is commonly considered for real values of x , but it remains true for all complex values.

Solving this linear system in sine and cosine, one can express them in terms of the exponential function:

When x is real, this may be rewritten as

Most trigonometric identities can be proved by expressing trigonometric functions in terms of the complex exponential function by using above formulas, and then using the identity e a + b = e a e b =e^e^> for simplifying the result.

Definitions using functional equations Edit

One can also define the trigonometric functions using various functional equations.

For example, [19] the sine and the cosine form the unique pair of continuous functions that satisfy the difference formula

cos ⁡ ( x − y ) = cos ⁡ x cos ⁡ y + sin ⁡ x sin ⁡ y

In the complex plane Edit

By taking advantage of domain coloring, it is possible to graph the trigonometric functions as complex-valued functions. Various features unique to the complex functions can be seen from the graph for example, the sine and cosine functions can be seen to be unbounded as the imaginary part of z becomes larger (since the color white represents infinity), and the fact that the functions contain simple zeros or poles is apparent from the fact that the hue cycles around each zero or pole exactly once. Comparing these graphs with those of the corresponding Hyperbolic functions highlights the relationships between the two.

Many identities interrelate the trigonometric functions. This section contains the most basic ones for more identities, see List of trigonometric identities. These identities may be proved geometrically from the unit-circle definitions or the right-angled-triangle definitions (although, for the latter definitions, care must be taken for angles that are not in the interval [0, π /2] , see Proofs of trigonometric identities). For non-geometrical proofs using only tools of calculus, one may use directly the differential equations, in a way that is similar to that of the above proof of Euler's identity. One can also use Euler's identity for expressing all trigonometric functions in terms of complex exponentials and using properties of the exponential function.

Parity Edit

The cosine and the secant are even functions the other trigonometric functions are odd functions. That is:

Periods Edit

All trigonometric functions are periodic functions of period 2 π . This is the smallest period, except for the tangent and the cotangent, which have π as smallest period. This means that, for every integer k , one has

Pythagorean identity Edit

The Pythagorean identity, is the expression of the Pythagorean theorem in terms of trigonometric functions. It is

Sum and difference formulas Edit

The sum and difference formulas allow expanding the sine, the cosine, and the tangent of a sum or a difference of two angles in terms of sines and cosines and tangents of the angles themselves. These can be derived geometrically, using arguments that date to Ptolemy. One can also produce them algebraically using Euler's formula.

When the two angles are equal, the sum formulas reduce to simpler equations known as the double-angle formulae.

These identities can be used to derive the product-to-sum identities.

this is the tangent half-angle substitution, which allows reducing the computation of integrals and antiderivatives of trigonometric functions to that of rational fractions.

Derivatives and antiderivatives Edit

The derivatives of trigonometric functions result from those of sine and cosine by applying quotient rule. The values given for the antiderivatives in the following table can be verified by differentiating them. The number C is a constant of integration.

Alternatively, the derivatives of the 'co-functions' can be obtained using trigonometric identities and the chain rule:

The trigonometric functions are periodic, and hence not injective, so strictly speaking, they do not have an inverse function. However, on each interval on which a trigonometric function is monotonic, one can define an inverse function, and this defines inverse trigonometric functions as multivalued functions. To define a true inverse function, one must restrict the domain to an interval where the function is monotonic, and is thus bijective from this interval to its image by the function. The common choice for this interval, called the set of principal values, is given in the following table. As usual, the inverse trigonometric functions are denoted with the prefix "arc" before the name or its abbreviation of the function.

The notations sin −1 , cos −1 , etc. are often used for arcsin and arccos , etc. When this notation is used, inverse functions could be confused with multiplicative inverses. The notation with the "arc" prefix avoids such a confusion, though "arcsec" for arcsecant can be confused with "arcsecond".

Just like the sine and cosine, the inverse trigonometric functions can also be expressed in terms of infinite series. They can also be expressed in terms of complex logarithms.

Angles and sides of a triangle Edit

In this sections A , B , C denote the three (interior) angles of a triangle, and a , b , c denote the lengths of the respective opposite edges. They are related by various formulas, which are named by the trigonometric functions they involve.

Law of sines Edit

The law of sines states that for an arbitrary triangle with sides a , b , and c and angles opposite those sides A , B and C :

where Δ is the area of the triangle, or, equivalently,

It can be proven by dividing the triangle into two right ones and using the above definition of sine. The law of sines is useful for computing the lengths of the unknown sides in a triangle if two angles and one side are known. This is a common situation occurring in triangulation, a technique to determine unknown distances by measuring two angles and an accessible enclosed distance.

Law of cosines Edit

The law of cosines (also known as the cosine formula or cosine rule) is an extension of the Pythagorean theorem:

In this formula the angle at C is opposite to the side c . This theorem can be proven by dividing the triangle into two right ones and using the Pythagorean theorem.

The law of cosines can be used to determine a side of a triangle if two sides and the angle between them are known. It can also be used to find the cosines of an angle (and consequently the angles themselves) if the lengths of all the sides are known.

Law of tangents Edit

The following all form the law of tangents [20]

The explanation of the formulae in words would be cumbersome, but the patterns of sums and differences, for the lengths and corresponding opposite angles, are apparent in the theorem.

Law of cotangents Edit

> (the radius of the inscribed circle for the triangle)

> (the semi-perimeter for the triangle),

then the following all form the law of cotangents [20]

In words the theorem is: the cotangent of a half-angle equals the ratio of the semi-perimeter minus the opposite side to the said angle, to the inradius for the triangle.

Periodic functions Edit

The trigonometric functions are also important in physics. The sine and the cosine functions, for example, are used to describe simple harmonic motion, which models many natural phenomena, such as the movement of a mass attached to a spring and, for small angles, the pendular motion of a mass hanging by a string. The sine and cosine functions are one-dimensional projections of uniform circular motion.

Trigonometric functions also prove to be useful in the study of general periodic functions. The characteristic wave patterns of periodic functions are useful for modeling recurring phenomena such as sound or light waves. [21]

Under rather general conditions, a periodic function f(x) can be expressed as a sum of sine waves or cosine waves in a Fourier series. [22] Denoting the sine or cosine basis functions by φk , the expansion of the periodic function f(t) takes the form:

For example, the square wave can be written as the Fourier series

In the animation of a square wave at top right it can be seen that just a few terms already produce a fairly good approximation. The superposition of several terms in the expansion of a sawtooth wave are shown underneath.

While the early study of trigonometry can be traced to antiquity, the trigonometric functions as they are in use today were developed in the medieval period. The chord function was discovered by Hipparchus of Nicaea (180–125 BCE) and Ptolemy of Roman Egypt (90–165 CE). The functions of sine and versine (1 - cosine) can be traced back to the jyā and koti-jyā functions used in Gupta period Indian astronomy (Aryabhatiya, Surya Siddhanta), via translation from Sanskrit to Arabic and then from Arabic to Latin. [23] (See Aryabhata's sine table.)

All six trigonometric functions in current use were known in Islamic mathematics by the 9th century, as was the law of sines, used in solving triangles. [24] With the exception of the sine (which was adopted from Indian mathematics), the other five modern trigonometric functions were discovered by Persian and Arab mathematicians, including the cosine, tangent, cotangent, secant and cosecant. [24] Al-Khwārizmī (c. 780–850) produced tables of sines, cosines and tangents. Circa 830, Habash al-Hasib al-Marwazi discovered the cotangent, and produced tables of tangents and cotangents. [25] [26] Muhammad ibn Jābir al-Harrānī al-Battānī (853–929) discovered the reciprocal functions of secant and cosecant, and produced the first table of cosecants for each degree from 1° to 90°. [26] The trigonometric functions were later studied by mathematicians including Omar Khayyám, Bhāskara II, Nasir al-Din al-Tusi, Jamshīd al-Kāshī (14th century), Ulugh Beg (14th century), Regiomontanus (1464), Rheticus, and Rheticus' student Valentinus Otho.

Madhava of Sangamagrama (c. 1400) made early strides in the analysis of trigonometric functions in terms of infinite series. [27] (See Madhava series and Madhava's sine table.)

The terms tangent and secant were first introduced by the Danish mathematician Thomas Fincke in his book Geometria rotundi (1583). [28]

The 17th century French mathematician Albert Girard made the first published use of the abbreviations sin, cos, and tan in his book Trigonométrie. [29]

In a paper published in 1682, Leibniz proved that sin x is not an algebraic function of x . [30] Though introduced as ratios of sides of a right triangle, and thus appearing to be rational functions, Leibnitz result established that they are actually transcendental functions of their argument. The task of assimilating circular functions into algebraic expressions was accomplished by Euler in his Introduction to the Analysis of the Infinite (1748). His method was to show that the sine and cosine functions are alternating series formed from the even and odd terms respectively of the exponential series. He presented "Euler's formula", as well as near-modern abbreviations (sin., cos., tang., cot., sec., and cosec.). [23]

A few functions were common historically, but are now seldom used, such as the chord, the versine (which appeared in the earliest tables [23] ), the coversine, the haversine, [31] the exsecant and the excosecant. The list of trigonometric identities shows more relations between these functions.

  • crd(θ) = 2 sin(
  • θ / 2 )
  • versin(θ) = 1 − cos(θ) = 2 sin 2 (
  • θ / 2 )
  • coversin(θ) = 1 − sin(θ) = versin(
  • π / 2 − θ)
  • haversin(θ) =
  • 1 / 2 versin(θ) = sin 2 (
  • θ / 2 )
  • exsec(θ) = sec(θ) − 1
  • excsc(θ) = exsec(
  • π / 2 − θ) = csc(θ) − 1

The word sine derives [32] from Latin sinus, meaning "bend bay", and more specifically "the hanging fold of the upper part of a toga", "the bosom of a garment", which was chosen as the translation of what was interpreted as the Arabic word jaib, meaning "pocket" or "fold" in the twelfth-century translations of works by Al-Battani and al-Khwārizmī into Medieval Latin. [33] The choice was based on a misreading of the Arabic written form j-y-b ( جيب ), which itself originated as a transliteration from Sanskrit jīvā, which along with its synonym jyā (the standard Sanskrit term for the sine) translates to "bowstring", being in turn adopted from Ancient Greek χορδή "string". [34]

The word tangent comes from Latin tangens meaning "touching", since the line touches the circle of unit radius, whereas secant stems from Latin secans—"cutting"—since the line cuts the circle. [35]

The prefix "co-" (in "cosine", "cotangent", "cosecant") is found in Edmund Gunter's Canon triangulorum (1620), which defines the cosinus as an abbreviation for the sinus complementi (sine of the complementary angle) and proceeds to define the cotangens similarly. [36] [37]


Fast 16-bit approximation of cos(x).

This approximation never varies more than 0.69% from the floating point value you'd get by doing

Parameters

thetainput angle from 0-65535
Returns sin of theta, value between -32767 to 32767.

Definition at line 120 of file trig8.h.

Fast 8-bit approximation of cos(x).

This approximation never varies more than 2% from the floating point value you'd get by doing

Parameters

thetainput angle from 0-255
Returns sin of theta, value between 0 and 255

Definition at line 253 of file trig8.h.

Fast 16-bit approximation of sin(x).

This approximation never varies more than 0.69% from the floating point value you'd get by doing

Parameters

thetainput angle from 0-65535
Returns sin of theta, value between -32767 to 32767.

Definition at line 30 of file trig8.h.

Fast 16-bit approximation of sin(x).

This approximation never varies more than 0.69% from the floating point value you'd get by doing

Parameters

thetainput angle from 0-65535
Returns sin of theta, value between -32767 to 32767.

Definition at line 88 of file trig8.h.

Fast 8-bit approximation of sin(x).

This approximation never varies more than 2% from the floating point value you'd get by doing

Parameters

thetainput angle from 0-255
Returns sin of theta, value between 0 and 255

Definition at line 159 of file trig8.h.

Fast 8-bit approximation of sin(x).

This approximation never varies more than 2% from the floating point value you'd get by doing

Parameters

thetainput angle from 0-255
Returns sin of theta, value between 0 and 255

Definition at line 217 of file trig8.h.


Awk only provides sin(), cos() and atan2(), the three bare necessities for trigonometry. They all use radians. To calculate the other functions, we use these three trigonometric identities:

tangent arcsine arccosine
tan ⁡ θ = sin ⁡ θ cos ⁡ θ >> tan ⁡ ( arcsin ⁡ y ) = y 1 − y 2 >>>> tan ⁡ ( arccos ⁡ x ) = 1 − x 2 x >>>>

With the magic of atan2(), arcsine of y is just atan2(y, sqrt(1 - y * y)), and arccosine of x is just atan2(sqrt(1 - x * x), x). This magic handles the angles arcsin(-1), arcsin 1 and arccos 0 that have no tangent. This magic also picks the angle in the correct range, so arccos(-1/2) is 2*pi/3 and not some wrong answer like -pi/3 (though tan(2*pi/3) = tan(-pi/3) = -sqrt(3).)

atan2(y, x) actually computes the angle of the point (x, y), in the range [-pi, pi]. When x > 0, this angle is the principle arctangent of y/x, in the range (-pi/2, pi/2). The calculations for arcsine and arccosine use points on the unit circle at x 2 + y 2 = 1. To calculate arcsine in the range [-pi/2, pi/2], we take the angle of points on the half-circle x = sqrt(1 - y 2 ). To calculate arccosine in the range [0, pi], we take the angle of points on the half-circle y = sqrt(1 - x 2 ).


Ex 3.3 Class 11 Maths Question 1.
Prove that:
Solution.
L.H.S. =

Ex 3.3 Class 11 Maths Question 2.

Solution.
L.H.S. =

Ex 3.3 Class 11 Maths Question 3.

Solution.
L.H.S. =

Ex 3.3 Class 11 Maths Question 4.

Solution.
L.H.S. =

Ex 3.3 Class 11 Maths Question 5.
Find the value of:
(i) sin 75°
(ii) tan 15°
Solution.
(i) sin (75°) = sin (30° + 45°)

(ii) tan 15° = tan (45° – 30°)

Prove the following:
Ex 3.3 Class 11 Maths Question 6.

Solution.
We have,

Ex 3.3 Class 11 Maths Question 7.

Solution.
We have,

Ex 3.3 Class 11 Maths Question 8.

Solution.
We have,

Ex 3.3 Class 11 Maths Question 9.

Solution.
We have,

Ex 3.3 Class 11 Maths Question 10.
sin(n +1 )x sin(n + 2)x + cos(n +1 )x cos(n + 2)x = cosx
Solution.
We have,

Ex 3.3 Class 11 Maths Question 11.

Solution.
We have,

Ex 3.3 Class 11 Maths Question 12.
sin 2 6x – sin 2 4x= sin 2 x sin10x
Solution.

Ex 3.3 Class 11 Maths Question 13.
cos 2 2x – cos 2 6x = sin 4x sin 8x
Solution.

Ex 3.3 Class 11 Maths Question 14.
sin2x + 2 sin 4x + sin 6x = 4 cos 2 x sin 4x
Solution.
We have,

Ex 3.3 Class 11 Maths Question 15.
cot 4x (sin 5x + sin 3x) = cot x (sin 5x – sin 3x)
Solution.

Ex 3.3 Class 11 Maths Question 16.

Solution.
We have,

Ex 3.3 Class 11 Maths Question 17.

Solution.
We have,

Ex 3.3 Class 11 Maths Question 18.

Solution.

Ex 3.3 Class 11 Maths Question 19.

Solution.

Ex 3.3 Class 11 Maths Question 20.

Solution.

Ex 3.3 Class 11 Maths Question 21.

Solution.

Ex 3.3 Class 11 Maths Question 22.
cot x cot 2x – cot 2x cot 3x – cot3x cotx = 1
Solution.
We know that 3x = 2x + x.
Therefore,

Ex 3.3 Class 11 Maths Question 23.

Solution.

Ex 3.3 Class 11 Maths Question 24.
cos 4x = 1 – 8 sin 2 x cos 2 x
Solution.

Ex 3.3 Class 11 Maths Question 25.
cos 6x = 32 cos6 x – 48 cos 4 x + 18 cos 2 x -1
Solution.

We hope the NCERT Solutions for Class 11 Maths Chapter 3 Trigonometric Functions Ex 3.3 help you. If you have any query regarding NCERT Solutions for Class 11 Maths Chapter 3 Trigonometric Functions Ex 3.3, drop a comment below and we will get back to you at the earliest.


Mathematica Q&A: Plotting Trig Functions in Degrees

Got a question about Mathematica? The Wolfram Blog has answers! We’ll regularly answer selected questions from users around the web. You can submit your question directly to the Q&A Team using this form.

This week’s question comes from Brian, who is a part-time math teacher:

How do you plot trigonometric functions in degrees instead of radians?

Trigonometric functions in Mathematica such as Sin[x] and Cos[x] take x to be given in radians:

To convert from degrees to radians, multiply by π &frasl 180. This special constant is called Degree in Mathematica.

The symbol ° is a handy shorthand for Degree and is entered as Esc-d-e-g-Esc. You can also find this symbol in the Basic Math Assistant palette in the Palettes menu of Mathematica.

Using either Degree or °, you can plot trigonometric functions in degrees:

That answers the main question, but here’s a related hint.

When plotting trigonometric functions in degrees, you might also want to manually specify exactly where Mathematica draws tick marks. You can do this using the Ticks option:

(Here, Range[0, 360, 45] specifies the tick marks on the x axis, and Automatic uses the default tick marks on the y axis.)

The Ticks option is very flexible. You can specify where tick marks are drawn, what labels they should have, how long they are, and even colors and styles.

Download the Computable Document Format (CDF) file for this post to see how to get the custom tick marks used in this plot:

If you have a question you’d like to see answered in this blog, you can submit it to the Q&A Team using this form.


Formal Definitions

Consider the following right triangle:

The sides with respect to angle θ heta θ are

sin ⁡ θ = ( opposite ) ( hypotenuse ) = b c cos ⁡ θ = ( adjacent ) ( hypotenuse ) = a c tan ⁡ θ = ( opposite ) ( adjacent ) = b a . egin sin heta &= frac<( ext)><( ext)> = frac cos heta &= frac<( ext)><( ext)> = frac an heta &= frac<( ext)><( ext)> = frac. end sin θ cos θ tan θ ​ = ( hypotenuse ) ( opposite ) ​ = c b ​ = ( hypotenuse ) ( adjacent ) ​ = c a ​ = ( adjacent ) ( opposite ) ​ = a b ​ . ​

We also have the following reciprocal functions


Course Content Menu

Chapter 1 - The Six Trigonometric Functions

LessonsHomeworkQuiz
1.1 - The Rectangular Coordinate System1.11.1
1.2 - Angles, Degrees and Special Triangles1.21.2
1.3 - Trigonometric Functions1.31.3
1.4 - Introduction to the Unit Circle1.41.4
Chapter 1 Test ( 15 questions )
Chapter 2 - Trigonometry

LessonsHomeworkQuiz
2.1 - Right Triangle Trigonometry2.12.1
2.2 - Other Angles and Trigonometric Functions2.22.2
2.3 - Solving Right Triangles2.32.3
2.4 - Applications2.42.4
Chapter 2 Test ( 18 questions )
Chapter 3 - Radian Measure

LessonsHomeworkQuiz
3.1 - Reference Angle3.13.1
3.2 - Radians and Degrees3.23.2
3.3 - Circular Functions3.33.3
3.4 - Arc Length and Area of a Sector3.43.4
3.5 - Velocity3.53.5
Chapter 3 Test ( 22 questions )
Chapter 4 - Graphs of Trigonometric Functions

LessonsHomeworkQuiz
4.1 - Graphs of Basic Trigonometry Functions4.14.1
4.2 - Amplitude and Period4.24.2
4.3 - Phase Shift4.34.3
4.4 - Equations of Graphs4.44.4
4.5 - Relations & Functions4.54.5
4.6 - Inverse Trigonometric Functions4.64.6
Chapter 4 Test ( 26 questions )
Chapter 5 - Trigonometric Identities

LessonsHomeworkQuiz
5.1 - Proving Identities5.15.1
5.2 - Sum and Difference Formulas5.25.2
5.3 - Double-Angle Formulas5.35.3
5.4 - Half-Angle Formulas5.45.4
5.5 - More Identities5.55.5
Chapter 5 Test ( 22 questions )
Chapter 6 - Trigonometric Equations

LessonsHomeworkQuiz
6.1 - Trigonometric Equations6.16.1
6.2 - More Trigonometric Equations6.26.2
6.3 - Trigonometric Equations & Multiple Angles6.36.3
6.4 - Parametric Equations6.46.4
Chapter 6 Test ( 20 questions )
Chapter 7 - Triangles

LessonsHomeworkQuiz
7.1 - Law of Cosines7.17.1
7.2 - Law of Sines7.27.2
7.3 - Area of a Triangle7.37.3
7.4 - Vectors7.47.4
Chapter 7 Test ( 27 questions )
Chapter 8 - Polar Coordinates & Complex Numbers

LessonsHomeworkQuiz
8.1 - Complex Numbers8.18.1
8.2 - Trigonometric Form of a Complex Number8.28.2
8.3 - Products and Quotients in Trigonometric Form8.38.3
8.4 - Roots of a Complex Number8.48.4
8.5 - Polar Coordinates8.58.5
8.6 - Equations with Polar Coordinates and Their Graphs8.68.6
Chapter 8 Test ( 21 questions )
Trigonometry Final Exam


1.8: Relating Trigonometric Functions - Mathematics

Chapter 4 - Trigonometry and the Unit Circle <- link to CEMC Waterloo

​ 4.1 Angles and Angle Measure - CEMC Radian Measure

Choose at least 5 from Practice section (at least one of 12 and 13)

Choose at least 4 from the Apply/Extend section (one from Extend)

Choose at least 1 from the Create Connections section

Extension # 19 (engineering), 24, C5

Choose at least 5 from Practice section (at least two letters each)

Choose at least 4 from the Apply/Extend section (one from Extend)

Choose at least 1 from the Create Connections section

4.3 Trigonometric Ratios (Unit Circle Worksheet)

Choose at least 4 from Practice section (at least three letters each)

Choose at least 5 from the Apply/Extend section (one from Extend)

Choose at least 1 from the Create Connections section

​4.4 Intro to Trig Equations - CEMC - Trig Equations

Choose at least 5 from Practice section (at least three letters each)

Choose at least 4 from the Apply/Extend section (one from Extend)

Choose at least 2 from the Create Connections section

Ch4 Review - ​ Pages 215-217 &ndash 4.1 # 1-6 (at least 3)

​​Ch4 ​Trigonometry Unit Test

Chapter 5 - Trigonometric Functions and Graphs

5.1 Graphing Sine and Cosine Functions

Assigned: Pages 233-237 - # 3, 4cd, 5, 6-10, 12, 14, 15, 18 ,19, C2, C4

5.2 Transformations of Sinusoidal Functions

Assigned: Part 1 - Trig Graphs Practice

Pages 250-255 - # 5, 7, 9, 10, 13, 15, 16, 21, 23, C2, C3

Assigned: Pages 262-265 - Student Choice

5.4 Equations & Graphs of Trig Functions

Assigned: Pages 275-281 - # 1, 3, 7, 8, 10, 11, 17, 19,21, C1, C2

Trigonometry Function and Graphs Review

Ch5 Trig Functions and Graphs Test

Graphing Calculator and/or GEOGEBRA App part of test

Chapter 6 - Trigonometric Identities

6.1a Reciprocal, Quotient, and Pythagorean Identities

​​ Assigned: Practice in Study Guide # 1 - 11 1 - 6

6.1b Reciprocal, Quotient, and Pythagorean Identities

Assigned: Pages 296-298 - # 1, 5, 6 (graph calc/app), 7, 9, 12, 14, C1, C2

6.2 Sum, Difference, and Double Angle Identities

Assigned: Pages 306-308 - # 1ace, 2ace, 3, 4ace, 5-8, 10

6.3a Proofs using basic identities

6.3b Proofs using sum and difference identities

6.3c Proofs using double angle identities

Extension page 315 # 16, 17, 18, 19

6.4 Solving Trig Equations using identities

Assigned: Pages 320-321 - # 1, 2acd, 3, 5, 6, 8-12, 15, 16

Assigned: Pages 322-323 - # 1bd, 2cd, 3, 7bc, 8d, 9bc, 11, 12, 13c, 17, 19

Chapter 7/8 - Exponential & Logarithmic Functions

7.3 Solving Exponential Equations: Part 1

​​ Assigned: Pages 364-365 # 1-3, C1, C2

Practice in Study Guide # 1 - 8

Extension page 365 # 16, 17

8.1 Understanding Logarithms

​​ Assigned: Pages 380-382 # 2-7, 12-14

Practice in Study Guide # 1 - 15

Practice in Study Guide # 1 - 4

Extension page 381 # 21, 22, 24

​​ Assigned: Pages 400-403 # 1-3,7-11, 15, C2, C3

Practice in Study Guide # 1 - 13

Practice in Study Guide # 14 - 20

8.4 Solving Exponential Equations: Part 2

​​ Assigned: Pages 412-415 # 2, 7, C1

Practice in Study Guide # 1 - 12

Extension page 415 # 19, 22

8.4 Solving Logarithmic Equations

​​ Assigned: Pages 412-415 # 1ac, 4, 5, 6, 8e

Practice in Study Guide # 1 - 4

Practice in Study Guide # 5 - 16- 4

Extension page 415 # 20, 21

8.3 Law of Logarithms - Change of Base

​​ Assigned: Practice in Study Guide # 1 - 3

Practice in Study Guide # 4, 5

Practice in Study Guide # 6

Extension page 402 # 19, 20

7.1/7.2 Characteristics & Transformations of Exponential Functions

​​ Assigned: Pages 342-343 # 1-4, 5ac

Pages 351-355 # 1abc, 2, 3abc, 4d, 6abc, 7ab, C1, C2b

Practice in Study Guide # 1-3

8.1/8.2 Characteristics & Transformations of Logarithmic Functions

​​ Assigned: Pages 380-381 # 1,b, 7, 9, 10, 15, 16, 17, C1

Pages 389-391 # 1a, 2, 4ab, 5ac, 6ac, 10a Ext 15, 16a, 17

Practice in Study Guide # 1-7

7.1 Applications of Exponential Growth and Decay

​​ Assigned: Pages 342-344 # 6, 7b, 8ac, 9ad, 10acd, 11, 12

8.1-8.4 Application of Logarithmic Scales

​​ Assigned: Selected Problems # 1-11

Ch7&8 Exponents and Logarithms Test - Part A (Lessons 1 - 6)

Ch7&8 Exponents and Logarithms Test - Part B (Lessons 7 - 10)

Chapter 2 - Radical Functions

2.1 Radical Functions and Transformations

Assigned: Pages 72-77 : Practice section #1-5 (at least 2 letters each)

at least 6 from Apply/Extend (A/E) section

at least 1 question from Create Connections (CC) section

2.2 Square Root of a Function

Assigned: Pages 86-89 : at least 5 from Practice section (all parts)

at least 5 from Apply/Extend (A/E)

at least 1 from Create Connections (CC) section

2.3 Solving Radical Equations

Assigned: Pages 96-98 : Practice section #1-7 (at least 2 letters each)

at least 6 from Apply/Extend (A/E)

at least 2 questions from CC section

Assigned: Pages 99-101 : #1-6 [at least 4 (all parts)]

# 13-15 (pick 2) 16 all 17 or 18

Radical Functions Test - Feb 27/28 (may stay up to 30 minutes extra)

Chapter 9 - Rational Functions <- link to CEMC Waterloo

9.1 Transformations of Rational Functions

Assigned: Pages 442-445 : Practice section #1-6 (at least 2 letters each)

# 7, 8, 9 and at least 7 more from Apply/Extend (A/E) section

at least 1 question from Create Connections (CC) section

9.2 Analyzing Rational Functions

Assigned: Pages 451-456 : Practice section #1-6 (at least 2 letters each)

at least 9 from Apply/Extend (A/E)

at least 1 from Create Connections (CC) section

9.3 Connecting Graphs to Rational Equations

Assigned: Pages 465-467 : Practice section #1-6 (at least 2 letters each)

at least 6 from Apply/Extend (A/E)

at least 1 questions from CC section

Assigned: Pages 468-469 : #1-11 [at least 10]

Chapter 10 - Function Operations <- link to CEMC Waterloo

10.1 Sums and Differences of Functions

Assigned: Pages 483-487 : Practice section #1-8 (at least 2 letters each)

at least 6 from Apply/Extend (A/E)

at least 2 question from Create Connections (CC) section

10.2 Products and Quotients of Functions

Assigned: Pages 496-498 : Practice section #1-5 (at least 2 letters each)

at least 8 from Apply/Extend (A/E)

at least 1 question from Create Connections (CC) section

10.3 Composition of Functions

Assigned: Pages 507-509 : Practice section #1-7 (at least 2 letters each)

at least 12 from Apply/Extend (A/E)

at least 1 questions from CC section

Assigned: Pages 510-511 : 10.1 - at least 4 questions

10.2 - at least 4 questions

10.3 - at least 5 questions

Chapter 3 - Polynomial Functions <- link to CEMC Waterloo

3.1 Characteristics of Polynomial Functions

Assigned: Pages 114-117: Practice section #1-4 (at least 2 letters each)

at least 4 from Apply/Extend (A/E)

at least 1 questions from CC section

Assigned: Pages 124-125: at least 4 from Practice section (2 letters each)

at least 4 from Apply/Extend (A/E)

Assigned: Pages 133-135: at least 4 from Practice section (2 letters each)

at least 4 from Apply/Extend (A/E)

at least 1 questions from CC section

3.4 Equations and Graphs of Polynomial Functions

Assigned: Pages 147-152: at least 4 from Practice section (2 letters each)


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