Understanding Inverse Trigonometric Functions

1. 8-2: Inverse Trigonometric Functions (Day 1) Essential Question: What are the restricted domains for the sin, cos, and tan functions?

2. 8-2: Inverse Trig Functions • Because the sine, cosine, and tangent functions repeat forever, it helps if we restrict the domain we’re looking at and limit the number of possible solutions • This means we won’t have to worry about adding 2k, k, etc. • The restricted sine function is a sine function whose domain is restricted to [-/2,/2] • This covers everything from the minimum (-1) to the maximum (1) of a standard sine function • All your answers should be within thisdomain.

3. 8-2: Inverse Trig Functions • As we’ve used before, the calculator has a button (sin-1) to calculate the inverse sine function. • Special Angles • Use the charts you’ve copied before, or • Use degree mode, then convert your answer into radians • In radian mode, divide your answer by π, and convert to a fraction • Example: sin-1 ½ • There were two solutions based on the chart we drew: /6 and 5/6. Only /6 is in the range of [-/2,/2], which makes it our answer. • Calculator (degree): sin-1 (½) = 30˚ * 2/360 = /6 • Calculator (radian): sin-1 (½) = .5236 / π = 1/6 = /6 • Everything else • Use the calculator (radian mode) • Example: sin-1 (-0.795) = -.9190

4. 8-2: Inverse Trig Functions • The restricted cosine function is a cosine function whose domain is restricted to [0, ] • This covers everything from the maximum (1) to the minimum (-1) of a standard sine function • All your answers should be within this domain. • Problems are solved the same way as the restricted sine function • Example #1: cos-1 ½ • Example #2: cos-1 (-0.63) /3 2.2523

5. 8-2: Inverse Trig Functions • The restricted tangent function is a tangent function whose domain is restricted to [-/2,/2] • All your answers should be within this domain. • Problems are solved the same way as the restricted sine/cosine functions • Example #1: tan-1 1 • Example #2: tan-1 136 /4 1.5634

6. 8-2: Inverse Trig Functions • Assignment • Page 536 – 537 • 1 – 23 (odds)

7. 8-2: Inverse Trigonometric Functions (Day 2) Essential Question: What are the restricted domains for the sin, cos, and tan functions?

8. 8-2: Inverse Trig Functions • Two-part functions • Example #1: Find cos-1(sin /6) without using a calculator • Solution: Work inside out • sin /6 = ½ • cos-1 (½) = /3 • Your turn: Find cos-1(cos 5/4) • cos 5/4 = • cos-1 () = 3/4

9. 8-2: Inverse Trig Functions • When you have inverse trig functions combined with regular trig functions, you can use right triangles to find exact values • Example: Find the exact value of cos(tan-1 ) • Solution steps: • Draw a triangle. • Use SOH-CAH-TOA to establish the ratios for two sides. • Use the Pythagorean theorem to figure out the 3rd side • Apply the outside ratio • tan = opposite/adjacent 

10. 8-2: Inverse Trig Functions • The same technique allows us to write combined functions as an algebraic expression • Example: Write sin(cos-1 v) as an algebraic expression in terms of v • Solution steps: • Draw a triangle. Write the “v” as a fraction (v/1) and label sides • Use the Pythagorean theorem to figure out the 3rd side • Apply the outside ratio • cos = adjacent/hypotenuse 

11. 8-2: Inverse Trig Functions • Assignment • Page 537 • 27 – 45 (odds)