Holt McDougal Algebra Inverses of Trigonometric Functions toolbox Pg. 953 (2-10;16-24;30-31, 41 why4)

Holt McDougal Algebra Inverses of Trigonometric Functions How do you evaluate inverse trigonometric functions? How do you use trigonometric equations and inverse trigonometric functions to solve problems? Essential Questions

Holt McDougal Algebra Inverses of Trigonometric Functions You have evaluated trigonometric functions for a given angle. You can also find the measure of angles given the value of a trigonometric function by using an inverse trigonometric relation.

Holt McDougal Algebra Inverses of Trigonometric Functions The expression sin -1 is read as “the inverse sine.” In this notation, -1 indicates the inverse of the sine function, NOT the reciprocal of the sine function. Reading Math

Holt McDougal Algebra Inverses of Trigonometric Functions The inverses of the trigonometric functions are not functions themselves because there are many values of θ for a particular value of a. For example, suppose that you want to find cos -1. Based on the unit circle, angles that measure and radians have a cosine of. So do all angles that are coterminal with these angles.

Holt McDougal Algebra Inverses of Trigonometric Functions Example 1: Finding Trigonometric Inverses Find all possible values of cos -1. Step 1 Find the values between 0 and 2 radians for which cos θ is equal to. Use the x-coordinates of points on the unit circle.

Holt McDougal Algebra Inverses of Trigonometric Functions Example 1 Continued Step 2 Find the angles that are coterminal with angles measuring and radians. Add integer multiples of 2  radians, where n is an integer Find all possible values of cos -1.

Holt McDougal Algebra Inverses of Trigonometric Functions Because more than one value of θ produces the same output value for a given trigonometric function, it is necessary to restrict the domain of each trigonometric function in order to define the inverse trigonometric functions.

Holt McDougal Algebra Inverses of Trigonometric Functions Trigonometric functions with restricted domains are indicated with a capital letter. The domains of the Sine, Cosine, and Tangent functions are restricted as follows. Sinθ = sinθ for {θ| } θ is restricted to Quadrants I and IV. Cosθ = cosθ for {θ| } θ is restricted to Quadrants I and II. Tanθ = tanθ for {θ| } θ is restricted to Quadrants I and IV.

Holt McDougal Algebra Inverses of Trigonometric Functions These functions can be used to define the inverse trigonometric functions. For each value of a in the domain of the inverse trigonometric functions, there is only one value of θ. Therefore, even though tan -1 has many values, Tan -1 1 has only one value.

Holt McDougal Algebra Inverses of Trigonometric Functions

Holt McDougal Algebra Inverses of Trigonometric Functions The inverse trigonometric functions are also called the arcsine, arccosine, and arctangent functions. Reading Math

Holt McDougal Algebra Inverses of Trigonometric Functions Example 2A: Evaluating Inverse Trigonometric Functions Evaluate each inverse trigonometric function. Give your answer in both radians and degrees.  Find value of θ for or whose Cosine. Use x-coordinates of points on the unit circle.

Holt McDougal Algebra Inverses of Trigonometric Functions Example 2B: Evaluating Inverse Trigonometric Functions Evaluate each inverse trigonometric function. Give your answer in both radians and degrees. The domain of the inverse sine function is {a|1 = – 1 ≤ a ≤ 1}. Because is outside this domain. Sin -1 is undefined.

Holt McDougal Algebra Inverses of Trigonometric Functions Example 3: Safety Application A painter needs to lean a 30 ft ladder against a wall. Safety guidelines recommend that the distance between the base of the ladder and the wall should be of the length of the ladder. To the nearest degree, what acute angle should the ladder make with the ground?

Holt McDougal Algebra Inverses of Trigonometric Functions Example 3 Continued θ 7.5 Step 1 Draw a diagram. The base of the ladder should be (30) = 7.5 ft from the wall. The angle between the ladder and the ground θ is the measure of an acute angle of a right triangle.

Holt McDougal Algebra Inverses of Trigonometric Functions Example 3 Continued Step 2 Find the value of θ. Use the cosine ratio. Substitute 7.5 for adj. and 30 for hyp. Then simplify. The angle between the ladder and the ground should be about 76°

Holt McDougal Algebra Inverses of Trigonometric Functions Example 4A: Solving Trigonometric Equations Solve each equation to the nearest tenth. Use the given restrictions. sin θ = 0.4, for – 90° ≤ θ ≤ 90° The restrictions on θ are the same as those for the inverse sine function.  = Sin -1 (0.4) ≈ 23.6° Use the inverse sine function on your calculator.

Holt McDougal Algebra Inverses of Trigonometric Functions Example 4B: Solving Trigonometric Equations Solve each equation to the nearest tenth. Use the given restrictions. sin θ = 0.4, for 90° ≤ θ ≤ 270° The terminal side of θ is restricted to Quadrants ll and lll. Since sin θ > 0, find the angle in Quadrant ll that has the same sine value as 23.6°. θ ≈ 180° – 23.6° ≈ 156.4° θ has a reference angle of 23.6°, and 90° < θ < 180°.