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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

root under 3-root under 2 by 5 y square
Himanshu Reply
The sum of the first n terms of a certain series is 2^n-1, Show that , this series is Geometric and Find the formula of the n^th
amani Reply
cosA\1+sinA=secA-tanA
Aasik Reply
why two x + seven is equal to nineteen.
Kingsley Reply
The numbers cannot be combined with the x
Othman
2x + 7 =19
humberto
2x +7=19. 2x=19 - 7 2x=12 x=6
Yvonne
because x is 6
SAIDI
what is the best practice that will address the issue on this topic? anyone who can help me. i'm working on my action research.
Melanie Reply
simplify each radical by removing as many factors as possible (a) √75
Jason Reply
how is infinity bidder from undefined?
Karl Reply
what is the value of x in 4x-2+3
Vishal Reply
give the complete question
Shanky
4x=3-2 4x=1 x=1+4 x=5 5x
Olaiya
hi can you give another equation I'd like to solve it
Daniel
what is the value of x in 4x-2+3
Olaiya
if 4x-2+3 = 0 then 4x = 2-3 4x = -1 x = -(1÷4) is the answer.
Jacob
4x-2+3 4x=-3+2 4×=-1 4×/4=-1/4
LUTHO
then x=-1/4
LUTHO
4x-2+3 4x=-3+2 4x=-1 4x÷4=-1÷4 x=-1÷4
LUTHO
A research student is working with a culture of bacteria that doubles in size every twenty minutes. The initial population count was  1350  bacteria. Rounding to five significant digits, write an exponential equation representing this situation. To the nearest whole number, what is the population size after  3  hours?
David Reply
v=lbh calculate the volume if i.l=5cm, b=2cm ,h=3cm
Haidar Reply
Need help with math
Peya
can you help me on this topic of Geometry if l help you
litshani
( cosec Q _ cot Q ) whole spuare = 1_cosQ / 1+cosQ
Aarav Reply
A guy wire for a suspension bridge runs from the ground diagonally to the top of the closest pylon to make a triangle. We can use the Pythagorean Theorem to find the length of guy wire needed. The square of the distance between the wire on the ground and the pylon on the ground is 90,000 feet. The square of the height of the pylon is 160,000 feet. So, the length of the guy wire can be found by evaluating √(90000+160000). What is the length of the guy wire?
Maxwell Reply
the indicated sum of a sequence is known as
Arku Reply
how do I attempted a trig number as a starter
Tumwe Reply
cos 18 ____ sin 72 evaluate
Het Reply
Practice Key Terms 6

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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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