# Trigonometry

Trigonometry

## Trigonometry Hi! Could someone please help me solve this question? It's asking for the sine of theta and the answer is C, but I do not know how to work it out. 10 18 13 21   ### Help pls Hi! I need help with the following question:

In a triangle ABC (right angle in B) where tan(A) = 3sec(C) solve E = (sec^2(A) –3csc(C))^(1/2).

The answer is 1, but I do not know how to solve it.

### How can I get my calculator to work correctly? I was watching a YouTube video on this subject and I was using my calculator so I could get the hang of it.

The person in the video did cos-1(0.71) and got 45.1. When I did it I got 49.7

A similar thing happened when they did sin-1(0.314). They got 18.3 and I got 20.3.

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## Angles

central to the study of trigonometry is the concept of an angle. An angle is defined as a geometric figure created by two lines drawn from the same point, known as the vertex. The lines are called the sides of an angle and their length is one defining characteristic of an angle. Another characteristic of an angle is its measurement or magnitude, which is determined by the amount of rotation, around the vertex, required to transpose one side on top of the other. If one side is rotated completely around the point, the distance traveled is known as a revolution and the path it traces is a circle.

Angle measurements are typically given in units of degrees or radians. The unit of degrees, invented by the ancient Babylonians, divides one revolution into 360 ° (degrees). Angles that are greater than 360 ° represent a magnitude greater than one revolution. Radian units, which relate angle size to the radius of the circle formed by one revolution, divide a revolution into 2 π units. For most theoretical trigonometric work, the radian is the primary unit of angle measurement.

## History of Trigonometry Outline

But the trigonometrical version is different. If you have the measurements of the two angles and the length of the side between them, then the problem is to compute the remaining angle (which is easy, just subtract the sum of the two angles from two right angles) and the remaining two sides (which is difficult). The modern solution to the last computation is by means of the law of sines. Details are at Dave's Short Trig Course, Oblique Triangles.

All trigonometrical computations require measurement of angles and computation of some trigonometrical function. The modern trigonometrical functions are sine, cosine, tangent, and their reciprocals, but in ancient Greek trigonometry, the chord, a more intuitive function, was used.

Trigonometry, of course, depends on geometry. The law of cosines, for instance, follows from a proposition of synthetic geometry, namely propositions II.12 and II.13 of the Elements. And so, problems in trigonometry have required new developments in synthetic geometry. An example is Ptolemy's theorem which gives rules for the chords of the sum and difference of angles, which correspond to the sum and difference formulas for sines and cosines.

The prime application of trigonometry in past cultures, not just ancient Greek, is to astronomy. Computation of angles in the celestial sphere requires a different kind of geometry and trigonometry than that in the plane. The geometry of the sphere was called "spherics" and formed one part of the quadrivium of study. Various authors, including Euclid, wrote books on spherics. The current name for the subject is "elliptic geometry." Trigonometry apparently arose to solve problems posed in spherics rather than problems posed in plane geometry. Thus, spherical trigonometry is as old as plane trigonometry.

### The Babylonians and angle measurement

The Babylonians were the first to give coordinates for stars. They used the ecliptic as their base circle in the celestial sphere, that is, the crystal sphere of stars. The sun travels the ecliptic, the planets travel near the ecliptic, the constellations of the zodiac are arranged around the ecliptic, and the north star, Polaris, is 90° from the ecliptic. The celestial sphere rotates around the axis through the north and south poles. The Babylonians measured the longitude in degrees counterclockwise from the vernal point as seen from the north pole, and they measured the latitude in degrees north or south from the ecliptic.

### Hipparchus of Nicaea (ca. 180 - ca. 125 B.C.E.)

Some of Hipparchus' advances in astronomy include the calculation of the mean lunar month, estimates of the sized and distances of the sun and moon, variants on the epicyclic and eccentric models of planetary motion, a catalog of 850 stars (longitude and latitude relative to the ecliptic), and the discovery of the precession of the equinoxes and a measurement of that precession.

According to Theon, Hipparchus wrote a 12-book work on chords in a circle, since lost. That would be the first known work of trigonometry. Since the work no longer exists, most everything about it is speculation. But a few things are known from various mentions of it in other sources including another of his own. It included some lengths of chords corresponding to various arcs of circles, perhaps a table of chords. Besides these few scraps of information, others can be inferred from knowledge that was taken as well-known by his successors.

### Chords as a basis of trigonometry

The chord of an angle AOB where O is the center of a circle and A and B are two points on the circle, is just the straight line AB. Chords are related to the modern sine and cosine by the formulas

where a is an angle, d the diameter, and crd an abbreviation for chord.

Some properties of chords could not have escaped Hipparchus' notice, especially in a 12-book work on the subject. For instance, a supplementary-angle formula would state that if AOB and BOC are supplementary angles, then Thales' theorem states that triangle ABC is right, so the Pythagorean theorem says the square on the chord AB plus the square on the chord BC equals the square on the diameter AC. Summarized using a modern algebraic notation

crd 2 AOB + crd 2 BOC = d 2

where d is the diameter of the circle.

Hipparchus probably constructed his table of chords using a half-angle formula and the supplementary angle formula. The half-angle formula in terms of chords is

crd 2 (t/2) = r(2r - crd (180° - t)

where r is the radius of the circle and t is an angle. Starting with crd 60° = r, Hippocrates could by means of this half-angle formula find the chords of 30°, 15°, and 7 1/2°. He could complete a table of chords in 7 1/2° steps by using crd 90°, the half-angle formula, and the supplementary angle formula.

What other relations among the chords of various angles that Hippocrates would have known remains speculation.

### Menelaus (ca. 100 C.E.)

 sin CE sin EA = sin CF sin FD sin BD sin BA
and
 sin CA sin EA = sin CD sin FD sin BF sin BE

He proved this result by first proving the plane version, then "projecting" back to the sphere. The plane version says

 CE EA = CF FD BD BA
and
 CA EA = CD FD BF BE

### Ptolemy's Theorem

Theorem. For a cyclic quadrilateral (that is, a quadrilateral inscribed in a circle), the product of the diagonals equals the sum of the products of the opposite sides.

AC BD = AB CD + AD BC

When AD is a diameter of the circle, then the theorem says

crd AOC crd BOD = crd AOB crd COD + d crd BOC.

where O is the center of the circle and d the diameter. If we take a to be angle AOB and b to be angle AOC, then we have

crd b crd (180° - a) = crd a crd (180° - b) + d crd (b - a)

which gives the difference formula

 crd (b - a) = crd b crd (180° - a) - crd a crd (180° - b) d

With a different interpretation of a and b, the sum formula results:

 crd (b + a) = crd b crd (180° - a) + crd a crd (180° - b) d

These, of course, correspond to the sum and difference formulas for sines.

Armed with his theorem, Ptolemy could complete his table of chords from 1/2° to 180° in increments of 1/2°.

### Trigonometry

The primary source of information in this outline is Thomas Heath's A History of Greek Mathematics, Clarendon Press, Oxford, 1921, currently reprinted by Dover, New York, 1981.

## Trigonometry Table

The trigonometric table is made up of trigonometric ratios that are interrelated to each other &ndash sine, cosine, tangent, cosecant, secant, cotangent. These ratios, in short, are written as sin, cos, tan, cosec, sec, cot, and are taken for standard angle values. You can refer to the trigonometric table chart to know more about these ratios. ### Trigonometry Formulae

The basic trigonometry formulae are:

• sin &theta = Opposite Side/Hypotenuse
• cos &theta = Adjacent Side/Hypotenuse
• tan &theta = Opposite Side/Adjacent Side

The complete list of trigonometric formulae involving trigonometry ratios and trigonometry identities is listed for easy access. Here's a list of all the trigonometric formulas for you to learn and revise.

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## Free Trigonometry Diagnostic Tests

High school Trigonometry classes introduce students to various trigonometric identities, properties, and functions in detail. Students typically take Trigonometry after completing previous coursework in Algebra and Geometry, but before taking Pre-Calculus and Calculus. Information students learn in Trigonometry helps them succeed in later higher-level mathematics courses, as well as in science courses like Physics, where trigonometric functions are used to model certain physical phenomena.

Like Pre-Algebra, Algebra I, and Algebra II classes, Trigonometry classes focus on functions and graphs. Trigonometry in particular investigates trigonometric functions, and in the process teaches students how to graph sine, cosine, secant, cosecant, tangent, cotangent, arcsin, arccos, and arctan functions, as well as how to perform phase shifts and calculate their periods and amplitudes. Trigonometric operations are also discussed, and students also learn about trigonometric equations, including how to understand, set up, and factor trig equations, how to solve individual trigonometric equations, as well as systems of trigonometric equations, how to find trig roots, and how to use the quadratic formula on trigonometric equations.

Trigonometric identities are also discussed in Trigonometry classes students learn about the sum and product identities, as well as identities of inverse operations, squared trigonometric functions, halved angles, and doubled angles. Students also learn to work with identities with angle sums, complementary and supplementary identities, pythagorean identities, and basic and definitional identities.

Another major part of Trigonometry is learning to analyze specific kinds of special triangles. Students learn to determine angles and side lengths in 30-60-90 and 45-45-90 right triangles using the law of sines and the law of cosines, as well as how to identify similar triangles and determine proportions using proportionality.

Trigonometry also teaches students about the unit circles and radians, focusing on how to convert degrees into radians and vice versa. Complementary, supplementary, and coterminal angles are all discussed. This focus on angles in the unit circle is also applied to the coordinate plane when angles in different quadrants are examined.

As may now be apparent, many students find themselves very apprehensive about taking, and keeping up with, a Trigonometry course. Resources like Varsity Tutors&rsquo free Trigonometry Practice Tests can help them channel any nervousness they feel about the course into a process of active review that will benefit them. Each Trigonometry Practice Test features a dozen multiple-choice Trigonometry questions, and each question comes with a full step-by-step explanation to help students who miss it learn the concepts being tested. Questions are organized in Practice Tests, which draw from various topics taught in Trigonometry questions are also organized by concept. So, if a student wants to focus on only answering questions about using the law of sines, questions organized by concept makes this possible. Using Varsity Tutors&rsquo free Trigonometry Practice Tests, students can practice material they find difficult and reduce apprehension they may feel about Trigonometry.

## Trigonometry

This course will teach you all of the fundamentals of trigonometry, starting from square one: the basic idea of similar right triangles. In the first sequences in this course, you'll learn the definitions of the most common trigonometric functions from both a geometric and algebraic perspective.

In this course, you'll master trigonometry by solving challenging problems and interacting with animated graphs, not by relying on rote memorization. Additionally, you'll learn how to apply trigonometry in the contexts of measuring and manipulating sound waves and creating complex, artistic designs using polar graphing.

### Introduction to Trigonometry

#### Exploring the Sine

Understand the sine function and how it gets used.

#### Graphing the Sine

What does the graph of the sine function look like?

#### Thinking in Polar

Use distance and angles to graph with circles and reveal new patterns.

### The Functions of Trigonometry

Learn the six basic trig functions in an intuitive way.

Get acquainted with two ways to measure an angle.

#### Optional: Greek Letter Primer

Use this tutorial to get comfortable with Greek letters, which are common in trigonometry.

#### Sine and Cosine

Explore the fundamental properties of the two fundamental trigonometry functions.

#### The Unit Circle

Create a reference tool to help find the exact value of trig functions.

### Measuring

Use trigonometry functions to solve for sides and angles.

#### Solving Right Triangles

Apply what you've learned about sine, cosine, and tangent to find missing sides of triangles.

#### Solving for Angles

Invert trigonometry functions to solve for angles instead of sides.

#### Law of Sines

Calculate side lengths and angle measures on any triangle, not just right triangles.

## Trigonometry Upon successful completion of Acellus Trigonometry, students will have attained a mastery of the trig foundational skills necessary for success in higher mathematics. Students will have mastered the unit circle, memorizing the coordinates of various key angles to quickly determine the lengths of the sides of common right triangles. Students will know how to use the sine, cosine, tangent and their reciprocal and inverse functions to determine unknown sides and angles of right triangles. They know what the graphs of these functions look like and how to translate them. They will know how to calculate arc length and sector area. Students will be confident using various trig formulas, such as the Law of Sines and the Law of Cosines, as well as the area formula for triangles. They also are familiar with and know how to use the trig identities. Students are familiar working with vectors and know how to calculate magnitude and direction from horizontal and vertical components and vice versa. They also know how to add vectors both geometrically and algebraically. Students know how to solve trig equations. Throughout this course, students gain experience using trigonometry to solve problems based on real-world situations. This course was developed by the International Academy of Science. Learn More

### Scope and Sequence

Following this unit students are presented with the Mid-Term Review and Exam.

Unit 7 – Powers, Roots, and Complex Numbers This unit discusses graphs of sine, cosine, tangent, secant, cosecant, and cotangent. Also covered are amplitude, period, horizontal and vertical translations, and a review of graphing concepts. Unit 8 – Identities This unit reviews identities and discusses cofunction and negative angle identities, and simplifying expressions. Unit 9 – Solving Trignometric Equations This unit covers combining like terms, square roots, factoring, and quadratics. Unit 10 – More Identities This unit includes sum and difference formulas for sine, cosine, and tangent, as well as double-angle formulas, and half-angle formulas. Unit 11 – Problem Solving This unit discusses problem solving in trigonometry.

Following this unit students are presented with the Final Review and Exam.