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In this section you will:
  • Use right triangles to evaluate trigonometric functions.
  • Find function values for 30° ( π 6 ) , 45° ( π 4 ) , and 60° ( π 3 ) .
  • Use equal cofunctions of complementary angles.
  • Use the definitions of trigonometric functions of any angle.
  • Use right-triangle trigonometry to solve applied problems.

Mt. Everest, which straddles the border between China and Nepal, is the tallest mountain in the world. Measuring its height is no easy task and, in fact, the actual measurement has been a source of controversy for hundreds of years. The measurement process involves the use of triangles and a branch of mathematics known as trigonometry. In this section, we will define a new group of functions known as trigonometric functions, and find out how they can be used to measure heights, such as those of the tallest mountains.

Using right triangles to evaluate trigonometric functions

[link] shows a right triangle with a vertical side of length y and a horizontal side has length x . Notice that the triangle is inscribed in a circle of radius 1. Such a circle, with a center at the origin and a radius of 1, is known as a unit circle    .

Graph of quarter circle with radius of 1. Inscribed triangle with an angle of t. Point of (x,y) is at intersection of terminal side of angle and edge of circle.

We can define the trigonometric functions in terms an angle t and the lengths of the sides of the triangle. The adjacent side    is the side closest to the angle, x . (Adjacent means “next to.”) The opposite side    is the side across from the angle, y . The hypotenuse    is the side of the triangle opposite the right angle, 1. These sides are labeled in [link] .

A right triangle with hypotenuse, opposite, and adjacent sides labeled.
The sides of a right triangle in relation to angle t

Given a right triangle with an acute angle of t , the first three trigonometric functions are listed.

Sine sin  t = opposite hypotenuse
Cosine cos  t = adjacent hypotenuse
Tangent tan  t = opposite adjacent

A common mnemonic for remembering these relationships is SohCahToa, formed from the first letters of “ S ine is o pposite over h ypotenuse, C osine is a djacent over h ypotenuse, T angent is o pposite over a djacent.”

For the triangle shown in [link] , we have the following.

sin  t = y 1 cos  t = x 1 tan  t = y x

Given the side lengths of a right triangle and one of the acute angles, find the sine, cosine, and tangent of that angle.

  1. Find the sine as the ratio of the opposite side to the hypotenuse.
  2. Find the cosine as the ratio of the adjacent side to the hypotenuse.
  3. Find the tangent as the ratio of the opposite side to the adjacent side.

Evaluating a trigonometric function of a right triangle

Given the triangle shown in [link] , find the value of cos α .

A right triangle with side lengths of 8, 15, and 17. Angle alpha also labeled which is opposite to the side labeled 8.

The side adjacent to the angle is 15, and the hypotenuse of the triangle is 17.

cos ( α ) = adjacent hypotenuse = 15 17
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Given the triangle shown in [link] , find the value of sin t .

A right triangle with sides of 7, 24, and 25. Also labeled is angle t which is opposite the side labeled 7.

7 25

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Reciprocal functions

In addition to sine, cosine, and tangent, there are three more functions. These too are defined in terms of the sides of the triangle.

Secant sec  t = hypotenuse adjacent
Cosecant csc  t = hypotenuse opposite
Cotangent cot  t = adjacent opposite

Take another look at these definitions. These functions are the reciprocals of the first three functions.

sin  t = 1 csc  t csc  t = 1 sin  t cos  t = 1 sec  t sec  t = 1 cos  t tan  t = 1 cot  t cot  t = 1 tan  t

Questions & Answers

A laser rangefinder is locked on a comet approaching Earth. The distance g(x), in kilometers, of the comet after x days, for x in the interval 0 to 30 days, is given by g(x)=250,000csc(π30x). Graph g(x) on the interval [0, 35]. Evaluate g(5)  and interpret the information. What is the minimum distance between the comet and Earth? When does this occur? To which constant in the equation does this correspond? Find and discuss the meaning of any vertical asymptotes.
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Prove that (sinA/1-cosA - 1-cosA/sinA) (cosA/1-sinA - 1-sinA/cosA) = 4
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Source:  OpenStax, Algebra and trigonometry. OpenStax CNX. Nov 14, 2016 Download for free at https://legacy.cnx.org/content/col11758/1.6
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