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

8-2: Inverse Trig FunctionsBecause 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 solutionsThis means we won’t have to worry about adding 2k, 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 functionAll your answers should be within this

domain. x

1

–1

y

2

4

2

4

8-2: Inverse Trig FunctionsAs 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, orb) Use degree mode, then convert your answer into radiansc) 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

8-2: Inverse Trig FunctionsThe 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 functionAll your answers should be within this domain.Problems are solved the same way as the

restricted sine function

Example #1: cos-1 ½Example #2: cos-1 (-0.63) x

1

–1

y

/3

2.2523 2

4

4

3

4

8-2: Inverse Trig FunctionsThe 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 1Example #2: tan-1 136

/4

1.5634

x

1

–1

y

2

4

2

4

8-2: Inverse Trig FunctionsAssignment

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

8-2: Inverse Trig FunctionsTwo-part functions

Example #1: Find cos-1(sin /6) without using a calculatorSolution: Work inside out

sin /6 = ½ cos-1 (½) = /3

Your turn: Find cos-1(cos 5/4) cos 5/4 = cos-1 ( ) =

2

2

2

2 3/4

8-2: Inverse Trig FunctionsWhen you have inverse trig functions combined with regular

trig functions, you can use right triangles to find exact valuesExample: 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

5

2

522 2

2

2 5

9

3

c

c

c

2

3

2

3

8-2: Inverse Trig FunctionsThe 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 vSolution 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

2 2 2

2 2

2

1

1

1

v b

b v

b v

v

1

21 v

21 v

8-2: Inverse Trig FunctionsAssignment

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