T2.1 e To Find the Inverse Functions for sin Ө, cos Ө, tan Ө cot Ө, sec Ө, & csc Ө “It’s an obstacle illusion” –Alan-Edward Warren, Sr Got ID?

2 Active Learning Assignment Questions?

3 LESSON: To find the inverse for sin Ө, cos Ө, and tan Ө. This is written as Sin -1, Cos -1, and Tan -1. It can also be written as Arcsin, Arccos, and Arctan. These two terms are INTERCHANGEABLE!!!!!!!! If sin Ө = ratio, and we know the angle, we can find the ratio. (Put your calculator back in degrees.) Ex: sin 30° = x Just use your calculator (or build the triangle to find it). 0.5 = x 1 2

4 BUT, what if we know the ratio, but want to solve for the angle (in degrees)? Start with: How do we get Ө by itself? We take the sine inverse of both sides! Since Sin -1 inverses sin, this is all we have left. Now, we use the calculator (see 2 nd function above sin). (Pronounced sine inverse) Put a RATIO into the inverse function. Get an ANGLE out!

5 Inverse functions must pass both the vertical line test and the horizontal line test because when we take the inverse, we switch the domain and range (the ratio becomes the questions (x) and the angle becomes the answer (y)) Ө Ө Ө Ө Ө Ө ratio

6 Positive and Negative Quadrants for Inverse Functions I IV III II All positive in QI = 30° = -30° = 60° = 120° 90° -90° (Reciprocals go together, too.) 180° 0° * * * * *

7 Radian vs. Degree. What is the difference between these two? You cannot tell by the expression. You must receive instructions as to which one is needed. Once that is established, just change your calculator to the appropriate setting. Let’s try to find it in radians. = In radians, to four decimal places. in degrees in radians Same as

8 Try these: Degrees, 2 dec. pl. Radians, 4 dec. pl °124.75° Either Degrees or Radians Try: Ө ratio Error!

9 It’s the same process, except for changing the function to it’s reciprocal. Given, find in degrees: To solve for Ө, we use the reciprocal property for csc Ө Flip that equation! Inverse both sides Simplify Solve, in degrees, two decimal places. What about Sec -1 x, Csc -1 x, and Cot -1 x ? The same!

Try: Degrees, 2 dec. pl. Radians, 4 dec. pl. 10 *

11 Active Learning Assignment: P 289: Written Exercises 1 – 4 Find each in degrees (2 dec. pl.). (Make sure your calculator is in degrees) 1. cot Ө = csc Ө = 3 3. sec Ө = ( ) 4.sec Ө = csc Ө = cot Ө = 0.75 TEST on T2.1a, T2.1c, T2.1d, T2.1e & T2.1f on Friday, (3 x 5 card, only) AND (Answers to these six are on the next page.)

12 Answers for 1 – 6 on previous page: In Degrees: ° ° ° ° ° °