Section 7.5 Inverse Circular Functions Chapter 7 Section 7.5 Inverse Circular Functions

sin-1x = arcsin(x) = The angle between -/2 and /2 whose sine is x. Inverse Trigonometric Functions In this section we focus on how to get the value for the angle (either in degrees or radians) if you know the value for the sine, cosine or tangent. This process is formalized in what are called the inverse trigonometric functions. For some numbers the values of these functions can be found by hand, but for most a calculator is needed. Inverse Sine Function This function is commonly denoted in one of two ways: sin-1x or arcsin(x) The meaning of this function is as follows for any number x between -1 and 1: sin-1x = arcsin(x) = The angle between -/2 and /2 whose sine is x. Think of the “answer” to the inverse sine function as an angle  that has a certain relationship with x. The angle  will have the following property: IF  = sin-1x = arcsin(x) THEN sin = x For example, sin-1(½) = arcsin(½) is the number =/6, because sin(/6) = ½. The tricky part here is that the “answer” to the inverse sine function must be between the angles -/2 and /2 (-90 and 90 if you are using degrees).

1. x the input must be between -1 and 1 The restrictions you have on the values  and x for the relation  = sin-1x = arcsin(x) are: 1. x the input must be between -1 and 1 2. The angle  the output is between -/2 and /2 Both of these come from the values of sine on the unit circle. (Remember the sine is the distance up and down on the unit circle.) 1  Notice that x can not get bigger than 1 or less than -1 and hit the unit circle with a horizontal line. The value of sin-1x is undefined if x is bigger than 1 or less than -1. The horizontal green line hits the circle in 2 places once on the right half and on the left half. Since the angles on the left half are not between -/2 and /2 we do not use them for the inverse sine. To find the values for the inverse sine for certain numbers we can do using this idea of drawing the unit circle and going down by the appropriate amount. The numbers that you can find the values for the inverse sine by hand are:

Find: Find: Find: For numbers other than the ones that represent sides of either a 45-45-90, a 30-60-90 triangle or 0 or ±1 you will need to use a calculator. Example: Find sin-1(.75) with a calculator sin-1(.75) = .848062… (if calculator is in radian mode) sin-1(.75) = 48.5904… (if calculator is in degree mode)

cos-1x = arccos(x) = The angle between 0 and  whose cosine is x. Inverse Cosine Function This function is commonly denoted in one of two ways: cos-1x or arccos(x) The meaning of this function is as follows for any number x between -1 and 1: cos-1x = arccos(x) = The angle between 0 and  whose cosine is x. Think of the “answer” to the inverse cosine function as an angle  that has a certain relationship with x. The angle  will have the following property: IF  = cos-1x = arccos(x) THEN cos = x For example, cos-1(½) = arccos(½) is the number =/3, because cos(/3) = ½. Unlike the inverse sine the “answer” to the inverse cosine function must be between the angles 0 and  (0 and 180 if you are using degrees). 1  The restrictions you have on the values  and x for the relation  = cos-1x = arccos(x) are: 1. x the input must be between -1 and 1 2. The angle  the output is between 0 and  Both of these come from the values of cosine on the unit circle. (Remember the cosine is the distance left and right on the unit circle.)

IF  = tan-1x = arctan(x) THEN tan = x Find: For numbers other than the ones that represent sides of either a 45-45-90, a 30-60-90 triangle or 0 or ±1 you will need to use a calculator. Example: Find cos-1(.375) with a calculator cos-1(.375) = 1.1864… (if calculator is in radian mode) cos-1(.375) = 67.9757…(if calculator is in degree mode) Inverse Tangent Function This function is commonly denoted in one of two ways: tan-1x or arctan(x) The meaning of this function is as follows for any number x between -1 and 1: tan-1x = arctan(x) = The angle between -/2 and /2 whose tangent is x. Think of the “answer” to the inverse cosine function as an angle  that has a certain relationship with x. The angle  will have the following property: IF  = tan-1x = arctan(x) THEN tan = x For example, tan-1(1) = arctan(1) is the number  =/4, because tan(/4) = 1. Just like the inverse sine the “answer” to the inverse tangent function must be between the angles -/2 and /2 (-90 and 90 if you are using degrees).

The values of all the trig functions for  ! Algebraic and Numeric Properties of Inverse Trigonometric Functions The inverse trigonometric functions have relationships with all the other trigonometric functions. Rather than listing out some more trigonometric identities (I know you must be getting tired of that at this point) I want to show you a method for getting them. Construction of Corresponding Right Triangle What you do is to construct a triangle that conforms to the inverse trig relation you have. This is something we did before when we were doing right triangle trig. Suppose  = cos-1(⅔) = arcos(⅔). We make a right triangle with one angle  and label the sides so that cos = ⅔. We then solve for the remaining side. The values of all the trig functions for  !

The construction of the corresponding right triangle will even work for algebraic expressions. For this it is often useful to think of the number x as x/1. Example: Change cos(tan-1x) to an expression with x. 1. Set  = tan-1(x) and construct the corresponding triangle by labeling the sides. 2. Solve for the remaining side. This can also be done for the other trigonometric functions. There are several identities that are very useful.

𝑦=arcsin 𝑥 𝑦=arccos 𝑥 𝑦=arctan 𝑥