1

Trig/PrecalcChapter 4.7 Inverse trig functions

ObjectivesEvaluate and graph the inverse

sine functionEvaluate and graph the remaining

five inverse trig functionsEvaluate and graph the

composition of trig functions

2

The basic sine function fails the horizontal line test. It is not one-to-one so we can’t find an inverse function unless we restrict the domain. Highlight the curve –π/2 < x < π/2

On the interval [-π/2, π/2] for sin x: the domain is [-π/2, π/2] and the range is [-1, 1]

We switch x and y to get inverse functions So for f(x) = sin-1 x the domain is [-1, 1] and range is [-π/2, π/2]

π 2ππ/2-π/2

y = sin(x)

Therefore

3

Graphing the Inverse

When we get rid of all the duplicate numbers we get this curve

Next we rotate it across the y=x line producing this curve

-10 -5 5 10

6

4

2

-2

-4

-6

-10

-55

10

6 4 2 -2 -4 -6

First we draw the sin curve

This gives us:Domain : [-1 , 1]

Range: 2, 2

4

4

2

-2

-4

-5 5

Inverse sine function y = sin-1 x or y = arcsin x

The sine function gives us ratios representing opposite over hypotenuse in all 4 quadrants.

The inverse sine gives us the angle or arc length on the unit circle that has the given ratio.

Remember the phrase “arcsine of x is the angle or arc whose sine is x”.

π/2

-π/2

1

5

Evaluating Inverse Sine

If possible, find the exact value.a. arcsin(-1/2) = ____

We need to find the angle in the range [-π/2, π/2] such that sin y = -1/2

What angle has a sin of ½? _______What quadrant would it be negative and within

the range of arcsin? ____Therefore the angle would be ______

6

IV6

6

6

Evaluating Inverse Sine cont.

b. sin-1( ) = ____ We need to find the angle in the range [-π/2, π/2] such that

sin y =

What angle has a sin of ? _______What quadrant would it be positive and within the range of

arcsin? ____Therefore the angle would be ______

c. sin-1(2) = _________ Sin domain is [-1, 1], therefore No solution

3

32

32

32

√3 2

1

I

3

3

No Solution

7

Graphs of Inverse Trigonometric Functions

The basic idea of the arc function is the same whether it is arcsin, arccos, or arctan

8

Inverse Functions Domains and Ranges y = arcsin x

Domain: [-1, 1] Range:

y = arccos x Domain: [ -1, 1] Range:

y = arctan x Domain: (-∞, ∞) Range:

,2 2

0,

,2 2

y = Arcsin (x)

y = Arccos (x)

y = Arctan (x)

9

Evaluating Inverse Cosine

If possible, find the exact value.a. arccos(√(2)/2) = ____ We need to find the angle in the range

[0, π] such that cos y = √(2)/2

What angle has a cos of √(2)/2 ? _______What quadrant would it be positive and within the range of arccos? ____Therefore the angle would be ______

b. cos-1(-1) = __ What angle has a cos of -1 ? _______

10

Warnings and Cautions!Inverse trig functions are equal to the arc trig

function. Ex: sin-1 θ = arcsin θ

Inverse trig functions are NOT equal to the reciprocal of the trig function. Ex: sin-1 θ ≠ 1/sin θ

There are NO calculator keys for: sec-1 x, csc-1 x, or cot-1 x

And csc-1 x ≠ 1/csc x sec-1 x ≠ 1/sec x cot-1 x ≠ 1/cot x

11

Evaluating Inverse functions with calculators ([E] 25 & 34)If possible, approximate to 2 decimal places.19. arccos(0.28) = ____

22. arctan(15) = _____

26. cos-1(0.26) = ____

34. tan-1(-95/7) = ____Use radian mode unless degrees are asked for.

12

Guided practice Example of [E] 28 & 30Use an inverse trig functionto write θ as a function of x. 28. Cos θ = 4/x so θ = cos-1(4/x) where x > 0

30. tan θ = (x – 1)/(x2 – 1) θ = tan-1(x – 1)/(x2 – 1) where x – 1 > 0 , x > 1

“θ as a function of x” means to write an equation of the form θ equal to an expression with x in it.

4

x

1

10

x

13

Composition of trig functions

Find the exact value, sketch a triangle. cos(tan-1 (2)) = _____

This means tan θ = 2 so…draw the triangle Label the adjacent and opposite sides

Find the hypo. using Pyth. Theorem

So the

θ

2

1

√5

2 5cos5

14

Example

Write an algebraic expression that is equivalent to the given expression.

cos(arctan(1/x))

u

x

1

2

22

1cos11

x x xuxx

1) Draw and label the triangle

---(let u be the unknown angle)

2) Use the Pyth. Theo. to compute the hypo

3) Find the cot of u

2 1x

You Try! Evaluate: -4/3

0 rad.

csc[arccos(-2/3)] (Hint: Draw a triangle)

Rewrite as an algebraic expression:

3arcsin2

3arcsin sin2

3tan arccos5

arccos tan 2

3

2

3 5 5

2

2

11

vv

Word problem involving sin or cos function: P type 1

pcalc643

ALEKS

An object moves in simple harmonic motion with amplitude 12 cm and period 0.1 seconds. At time t = 0 seconds , its displacement d from rest is 12 in a negative direction, and initially it moves in a negative direction.

Give the equation modeling the displacement d as a function of time t.

Undo HelpClear

Next >> Explain

Word problem involving sin or cos function: P type 2

pcalc643

ALEKS

The depth of the water in a bay varies throughout the day with the tides. Suppose that we can model the depth of the water with the following function. h(t) = 13 + 6.5 sin 0.25t

In this equation, h(t) is the depth of the water in feet, and t is the time in hours.

Find the following. If necessary, round to the nearest hundredth.

Frequency of h: cycles per hourPeriod of h: hoursMinimum depth of the water: feet Undo HelpClear

Next >> Explain