Warm-up: 1) Make a quick sketch of each relation 2) Consider the pair of relations above as inverse functions. What is wrong with this consideration? The top function will not pass the horizontal line test so it does not really have an inverse because it is not 1-1. We must set up restrictions in the domain (x  0) in order to have an inverse function relationship! HW: Inverse Trig Functions

4.7 Inverse Trig Functions Objective: Evaluate inverse Trig functions

A restriction on the domain is necessary to create an inverse function for each trig function. Consider the sine function. The sine function does not pass the horizontal line test. We need to restrict the domain as we did with the warm-up. How would YOU restrict the domain?

The interval from -/2 to /2 considers all the outputs of the sine function from –1 to 1. This interval includes the origin. Quadrant I angles generate the positive ratios. Quadrant IV angles generate the negative ratios.

Plotting the special angles from the unit circle.  

The new table generates the graph of the inverse. The domain of the restricted section of the sine function is So the range of sin-1 or arcsin is To graph the inverse function, we interchange the input and output values. The range of the restricted section of sine is [-1 ,1] So the domain of the arcsin is [-1, 1].

Note how each point on the original graph gets “reflected” onto the graph of the inverse. etc. You will see the inverse notation as both:

In the tradition of inverse functions we have: Unless instructed to use degrees, you should assume inverse trig functions will generate outputs of real numbers (in radians). For the trig function the input is the angle and the output is the ratio. For the inverse trig function the input is the ratio and the output is the angle.

The other inverse trig functions are generated by using similar restrictions on the domain. Consider the cosine function: What do you think would be a good domain restriction for the cosine?

The restricted interval for the cosine is in the red frame The restricted interval for the cosine is in the red frame. This interval includes all outputs from –1 to 1 and all inputs in the first and second quadrants. Since the domain and range for the section are the domain and range for the inverse cosine are

Like the sine function, the domain of the region of the tangent that generates the arctan is y=arctan(x) y=tan(x)

Cotangent, Cosecant, and Secant require similar restrictions on their domains in order to generate an inverse. y = cscx y = cotx y = secx

Box indicates domain restriction to generate inverse functions Restricted Domain: Restricted Range: Restricted Domain: Restricted Range: Restricted Domain: Restricted Range: Restricted Domain: Restricted Range: Restricted Domain: Restricted Range: Restricted Domain: Restricted Range: Box indicates domain restriction to generate inverse functions

arcsin(x) arccos(x) arctan(x) arccot(x) arccsc(x) arcsec(x) Domain Range

Summary of parameters: arcsin(x) arccos(x) arctan(x) arccot(x) arccsc(x) arcsec(x) Domain Range When x < 0, y = arcsin(x) will be in which quadrant? y < 0 in IV When x < 0, y = arccos(x) will be in which quadrant? y > 0 in II When x < 0, y = arctan(x) will be in which quadrant? y < 0 in IV

When finding the inverse trig function of a ratio we use only half the unit circle to find the angle. cos-1(x) cot-1(x) sec-1(x) sin-1(x) tan-1(x) csc-1(x) Due to domain and range restrictions we use negative angles when appropriate.  

Special right triangles can be very helpful with respect to the basics. 1 2 Use the special triangles above to answer the following.

Lets try a few more. 1 2

Example: y x -1

Negative inputs for the arccos can be a little tricky using special triangles. x -1 2 1 2 From the triangle you can see that arccos(1/2) = 60 degrees. But negative inputs for the arccos generate angles in Quadrant II so we have to use 60 degrees as a reference angle in the second quadrant.

You should be able to do inverse trig calculations without a calculator when special angles from the special triangles are involved. You should also be able to do inverse trig calculations without a calculator for quadrantal angles. Quadrantal angles are the angles between the quadrants—angles like To solve arccos(-1) for example, you could draw a quick sketch of the cosine section: observe that arccos(-1) =

Example: Find the exact value of each expression without using a calculator. When your answer is an angle, express it in radians. cos-1(x) cot-1(x) sec-1(x) sin-1(x) tan-1(x) csc-1(x)

Sneedlegrit: Find the exact value of each expression. HW: Inverse Trig Functions