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7.2 Right Triangle Trigonometry

In this section, we will study the following topics:

Evaluating trig functions of acute angles using right triangles

Use Fundamental Identities

Use the Complimentary Angle Theorem

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Take a look at the right triangle, with an acute angle, , in the figure below.

Notice how the three sides are labeled in reference to .

The sides of a right triangle

Side adjacent to

S

ide

op

po

site

Hypotenuse

We will be reviewing special ratios of these sides of the right triangle, with respect to angle, .

These ratios are better known as our six basic trig functions.

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Six Trigonometric Functions

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5

To remember the definitions of Sine, Cosine and Tangent, we use the acronym :

“SOH CAH TOA”

Definitions of the Six Trigonometric Functions

O A O

H H AS C T

Find the value of each of the six trigonometric functions of the angle .

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Find the exact value of the six trig functions of :

Example

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Hint: First find the length of the hypotenuse using the Pythagorean Theorem.

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Example (cont)

5

9

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So the six trig functions are:

sin

cos

tan

opp

hyp

adj

hyp

opp

adj

csc

sec

cot

hyp

opp

hyp

adj

adj

opp

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Given that is an acute angle and , find the exact value of the six trig functions of .

Example

12cos

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10 3 10Given sin and cos ,

10 10find the value of each of the four remaining trigonometric functions of .

This is known as a Pythagorean Identity.

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Divide each side by cos2 to derive 2nd Pythagorean Identity.

2 2sin cos 1

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Divide each side by sin2 to derive 3rd Pythagorean Identity.

2 2sin cos 1

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Find the exact value of each expression. Do not use a calculator.

cos1 3( ) cos 35 ( ) cotcsc 35 3sin

3

a b

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tan 75( ) ( ) cos38 sin 52

cot15a b

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End of Section 7.2