Lesson Overview:

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Inverse trig functions can either be read as sin−1(x) or arcsin(x). In this lesson we will mostly use sin−1(x).  They are used to find the unknown measure of a right triangle when two side lengths are known. They are the inverse of regular trig functions.

a. Things to Remember

b. Inverse Function Graphs

c. Evaluating Trig Functions

d. Evaluating Composite Inverse Trig Functions

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Part A: Things to Remember  

 - Not all functions have an inverse, only one-to-one functions

 - To check if a “function” is actually a function, use the Vertical line test

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 - To check if a function is a one-to-one function, use the Horizontal line test

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 - Sine, cosine, and tangent functions are not one-to-one, so we must restrict the domain.

Part B: Inverse Function Graphs

 

 

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Inverse Sine Function:

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If we restrict the domain of f(x)=sin(x) to [−π/2,π/2] The range is [−1,1] .

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Inverse Cosine Function:

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The domain of the inverse cosine function is [−1,1] and the range is [0,π]

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Inverse Tan Function:

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The domain of the inverse tangent function is (−∞,∞) and the range is (−π/2,π/2) . 

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Part C: Evaluating Inverse Trig Functions 

 

 

 

 

 

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When evaluating inverse trig functions, first look to see if the variable within the parenthesis (x) is

positive or negative. If positive, your answer is within the 1st quadrant (π/n). If it is an inverse

cosine function, and the x is negative, your answer is within the 2nd quadrant (n-1)π/n. If it is an

inverse sine or tangent function with a negative x, notice your formula in the 4th quadrant is

(– π/n) and not (2n-1) π /n.

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When (x) is positive, answer it as though you

are answering with the left hand trick from 4.1.

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When Deciding if it is True or False...

use the range. If the number within the parenthesis falls in between the range, it is true.

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Function              Domain               Range

sin − 1 ( x )        [ − 1 , 1 ]          [ − π 2 , π 2 ]

cos − 1 ( x )       [ − 1 , 1 ]            [ 0 , π ]

tan − 1 ( x )       ( − ∞ , ∞ )          ( − π 2 , π 2 )                                                   

f(x) = sin ( x )      (-∞ , + ∞)             [-1 , 1]

f(x) = cos ( x )     (-∞ , + ∞)             [-1 , 1]

f(x) = tan ( x )     All real numbers  (-in , + ∞)

                           except π/2 + n*π

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Part D: Evaluating Composite Inverse Trig Functions

When you evaluate composite, or compound, inverse functions, you combine what you learned in this lesson with lesson 4.1. You use Q.R.S to evaluate the inside function, then use your answer as (x) to solve for the next function.

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The Unit Circle on the SAT

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Since the square root of 2 over 2 is equal to π/4, we know

due to the unit circle that π/4= 45 degrees. Even though it

is inverse sine, the degree is still positive.

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Trig Functions in the Real World

Example 1:

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Example 2:

The base of a ladder is placed 3 feet away from a 10 -foot-high wall, so that the top of the ladder meets the top of the wall. What is the measure of the angle formed by the ladder and the ground?

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Conclusion:

In mathematics, the inverse trigonometric functions are the inverse functions of the trigonometric functions. Specifically, they are the inverses of the sine, cosine, tangent, cotangent, secant, and cosecant functions, and are used to obtain an angle from any of the angle's trigonometric ratios.