Essential Question: What are the restricted domains for the sin, cos, and tan functions?

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 this domain.

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 a) Use the charts you’ve copied before, or b) Use degree mode, then convert your answer into radians c) 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) =

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 

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  / 4 

8-2: Inverse Trig Functions Assignment Page 536 – – 23 (odds)

Essential Question: What are the restricted domains for the sin, cos, and tan functions?

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

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 3 rd side Apply the outside ratio tan = opposite / adjacent 

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 3 rd side Apply the outside ratio cos = adjacent / hypotenuse 

8-2: Inverse Trig Functions Assignment Page – 45 (odds)