# 7.2 Right triangle trigonometry

 Page 1 / 12
In this section you will:
• Use right triangles to evaluate trigonometric functions.
• Find function values for $\text{\hspace{0.17em}}30°\left(\frac{\pi }{6}\right),45°\left(\frac{\pi }{4}\right),$ and $\text{\hspace{0.17em}}60°\left(\frac{\pi }{3}\right).$
• 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 $\text{\hspace{0.17em}}y\text{\hspace{0.17em}}$ and a horizontal side has length $\text{\hspace{0.17em}}x.\text{\hspace{0.17em}}$ 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    .

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] .

Given a right triangle with an acute angle of $\text{\hspace{0.17em}}t,$ the first three trigonometric functions are listed.

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.

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 $\text{\hspace{0.17em}}\mathrm{cos}\text{\hspace{0.17em}}\alpha .$

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

$\begin{array}{ccc}\hfill \mathrm{cos}\left(\alpha \right)& =& \frac{\text{adjacent}}{\text{hypotenuse}}\hfill \\ & =& \frac{15}{17}\hfill \end{array}$

Given the triangle shown in [link] , find the value of $\text{\hspace{0.17em}}\text{sin}\text{\hspace{0.17em}}t.$

$\frac{7}{25}$

## 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.

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

what is the period of cos?
Patrick
if tan alpha + beta is equal to sin x + Y then prove that X square + Y square - 2 I got hyperbole 2 Beta + 1 is equal to zero
sin^4+sin^2=1, prove that tan^2-tan^4+1=0
what is the formula used for this question? "Jamal wants to save \$54,000 for a down payment on a home. How much will he need to invest in an account with 8.2% APR, compounding daily, in order to reach his goal in 5 years?"
i don't need help solving it I just need a memory jogger please.
Kuz
A = P(1 + r/n) ^rt
Dale
how to solve an expression when equal to zero
its a very simple
Kavita
gave your expression then i solve
Kavita
Hy guys, I have a problem when it comes on solving equations and expressions, can you help me 😭😭
Thuli
Tomorrow its an revision on factorising and Simplifying...
Thuli
ok sent the quiz
kurash
send
Kavita
Hi
Masum
What is the value of log-1
Masum
the value of log1=0
Kavita
Log(-1)
Masum
What is the value of i^i
Masum
log -1 is 1.36
kurash
No
Masum
no I m right
Kavita
No sister.
Masum
no I m right
Kavita
tan20°×tan30°×tan45°×tan50°×tan60°×tan70°
jaldi batao
Joju
Find the value of x between 0degree and 360 degree which satisfy the equation 3sinx =tanx
what is sine?
what is the standard form of 1
1×10^0
Akugry
Evalute exponential functions
30
Shani
The sides of a triangle are three consecutive natural number numbers and it's largest angle is twice the smallest one. determine the sides of a triangle
Will be with you shortly
Inkoom
3, 4, 5 principle from geo? sounds like a 90 and 2 45's to me that my answer
Neese
Gaurav
prove that [a+b, b+c, c+a]= 2[a b c]
can't prove
Akugry
i can prove [a+b+b+c+c+a]=2[a+b+c]
this is simple
Akugry
hi
Stormzy
x exposant 4 + 4 x exposant 3 + 8 exposant 2 + 4 x + 1 = 0
x exposent4+4x exposent3+8x exposent2+4x+1=0
HERVE
How can I solve for a domain and a codomains in a given function?
ranges
EDWIN
Thank you I mean range sir.
Oliver
proof for set theory
don't you know?
Inkoom