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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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Got questions? Get instant answers now!

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

given 4cot thither +3=0and 0°<thither <180° use a sketch to determine the value of the following a)cos thither
Snalo Reply
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Mark Reply
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Miranda
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Miranda Drice
jai
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jai
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Miranda
I am living in india
jai
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Miranda
what is the formula for calculating algebraic
Propessor Reply
I think the formula for calculating algebraic is the statement of the equality of two expression stimulate by a set of addition, multiplication, soustraction, division, raising to a power and extraction of Root. U believe by having those in the equation you will be in measure to calculate it
Miranda
state and prove Cayley hamilton therom
sita Reply
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Propessor
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Miranda
the Cayley hamilton Theorem state if A is a square matrix and if f(x) is its characterics polynomial then f(x)=0 in another ways evey square matrix is a root of its chatacteristics polynomial.
Miranda
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jai
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Propessor
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jai
What is algebra
Pearl Reply
algebra is a branch of the mathematics to calculate expressions follow.
Miranda
Miranda Drice would you mind teaching me mathematics? I think you are really good at math. I'm not good at it. In fact I hate it. 😅😅😅
Jeffrey
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Jeffrey
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Miranda
but seriously, Im really bad at math. And I hate it. But you see, I downloaded this app two months ago hoping to master it.
Jeffrey
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Miranda
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Miranda
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Jeffrey
I'm going to 11grade
Miranda
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Miranda
gotta practice, holmie
Steve
if you never use it you won't be able to appreciate it
Steve
I don't know why. But Im trying to like it.
Jeffrey
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Jeffrey
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Miranda
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Nelson Reply
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Miranda
I dont know. lol
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Miranda
ask nelson. lol
Jeffrey
answer and questions in exercise 11.2 sums
Yp Reply
how do u calculate inequality of irrational number?
Alaba
give me an example
Chris
and I will walk you through it
Chris
cos (-z)= cos z .
Swadesh
cos(- z)=cos z
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what is a algebra
Jallah Reply
(x+x)3=?
Narad
6x
Obed
what is the identity of 1-cos²5x equal to?
liyemaikhaya Reply
__john __05
Kishu
Hi
Abdel
hi
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Nokwanda
C'est comment
Abdel
Hi
Amanda
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SORIE
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SORIE
I speak French
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SORIE
it's 12
what is the function of sine with respect of cosine , graphically
Karl Reply
tangent bruh
Steve
cosx.cos2x.cos4x.cos8x
Aashish Reply
sinx sin2x is linearly dependent
cr Reply
what is a reciprocal
Ajibola Reply
The reciprocal of a number is 1 divided by a number. eg the reciprocal of 10 is 1/10 which is 0.1
Shemmy
 Reciprocal is a pair of numbers that, when multiplied together, equal to 1. Example; the reciprocal of 3 is ⅓, because 3 multiplied by ⅓ is equal to 1
Jeza
each term in a sequence below is five times the previous term what is the eighth term in the sequence
Funmilola Reply
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Kenny 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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