4.2: Exact Values for Inverse Sine, Cosine, and Tangent
You are working with a triangular brace in shop class. The brace is a right triangle, and the length of one side of the bracket is
Can you find the angle between the legs of the brace?
By the time you finish reading this Concept, you'll be able to answer this question.
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James Sousa Example: Determine Trig Function Values Using Reference Triangles
Guidance
Inverse trig functions can be useful in a variety of math problems for finding angles that you need to know. In many cases, such as angles involving multiples of
Recall the unit circle and the critical values. With the inverse trigonometric functions, you can find the angle value (in either radians or degrees) when given the ratio and function. Make sure that you find all solutions within the given interval.
Example A
Find the exact value of the expression without a calculator, in
Solution: This is a value from the special right triangles and the unit circle.
Recall that
Example B
Find the exact value of the expression without a calculator, in
Solution: This is a value from the special right triangles and the unit circle.
Example C
Find the exact value of the expression without a calculator, in
Solution: This is a value from the special right triangles and the unit circle.
Vocabulary
Trigonometric Inverse: The trigonometric inverse is a function that undoes a trig function to give the original argument of the function. It can also be used to find an angle from the ratio of two sides.
Guided Practice
1. Find the exact value of the inverse function of
2. Find the exact value of the inverse function of
3. Find the exact value of the inverse function of
Solutions:
1.
2.
3.
Concept Problem Solution
Using your knowledge of the values of trig functions for angles, you can work backward to find the angle that the brace makes:
Practice
Find the exact value of each expression without a calculator, in

sin−1(2√2) 
cos−1(12) 
sin−1(1) 
cos−1(−3√2) 
tan−1(−3√3) 
tan−1(−1) 
sin−1(3√2) 
cos−1(2√2) 
csc−1(2√) 
sec−1(−2) 
cot−1(3√3) 
sec−1(23√2)  \begin{align*}\csc^{1} \left ( \frac{2\sqrt{3}}{2} \right )\end{align*}
 \begin{align*}\cot^{1} \left ( \sqrt{3} \right )\end{align*}
 \begin{align*}\cot^{1} \left ( 1 \right )\end{align*}
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Term  Definition 

Trigonometric Inverse  A trigonometric inverse is a function that undoes a trigonometric function to give the original argument of the function. It can also be used to find a missing angle of a triangle from the ratio of two sides of the triangle. 
Image Attributions
Here you'll learn to find the angle for common values of inverse trigonometric functions.