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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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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
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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
5
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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
13
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10 3 10Given sin and cos ,
10 10find the value of each of the four remaining trigonometric functions of .
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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