Ch 4.7: Inverse Trig Functions

Inverse of Sine

• Inverse: Must pass horizontal line test, so we must limit the domain of sine to make it one-to-one– Interval: , then y = sin(x) has an inverse– Written: y = arcsin(x) or y = sin-1(x)– Remember: y = sin-1(x) iff x = sin(y)

2 2x

y = sin(x)

x

y

2

2

Ex: Find the exact value for1

arcsin2

Ask yourself, where on the unit circle does sin = ?

Remember, you must be between

1

2

2 2x

:6

ANSWER

y = Arcsin (x)

Inverse of Cosine

• Inverse: Must pass horizontal line test, so we must limit the domain of sine to make it one-to-one– Interval: , then y = cos(x) has an inverse– Written: y = arccos(x) or y = cos-1(x)– Remember: y = cos-1(x) iff x = cos(y)

0 x

Ex: Find the exact value for1

arccos2

Ask yourself, where on the unit circle does cos = ?

Remember, you must be between

1

2

0 x :

3ANSWER

0

y = cos(x)

x

y

Inverse of Tangent

• Inverse: Must pass horizontal line test, so we must limit the domain of sine to make it one-to-one– Interval: , then y = tan(x) has an inverse– Written: y = arctan(x) or y = tan-1(x)– Remember: y = tan-1(x) iff x = tan(y)

2 2x

Ex: Find the exact value for arctan 1

Ask yourself, where on the unit circle does tan = 1?

Remember, you must be between 2 2

x

:

4ANSWER

2

2

y = tan(x)

x

y

y = Arctan (x)

Approximating Values

• By definition, inverses are supposed to be in radians• Check to see whether the number is in radians or

degrees• To do inverse trig functions, hit “2nd” then the trig

function• Round 4 places• Some will not work!

Ex: Arcsin(0.2447) = Ex: sin-1(2) =

0.2472

Inverse Prop.: Recall f(f-1(x))=x & f-1(f(x))=x

For -1 x 1 and sin(sin-1(x)) = x & sin-1(sin(y)) = y

For -1 x 1 and

cos(cos-1(x)) = x & cos-1(cos(y)) = y

For x is a real number and tan(tan-1(x)) = x & tan-1(tan(y)) = y

**Pay attention to make sure the values fall within the parameters of the inverse!**

2 2y

0 y

2 2y

1 1: sin sin

3Ex

1sin sin 1 1x x if x

!TRUE

1

3

: tan arctan 5Ex

1tan tan is realx x if x

!TRUE

5

5:arcsin sin

3Ex

arcsin(sin( ))

2 2y y if y

! So work each piece at timeFalse

5 3sin

3 2

3arcsin

2

3

More complex problems• Determine the quadrant• Draw a triangle, label the parts• Using the triangle, answer the problem

1 3: cos sin

5Ex

arcsin :2 2

x

Thus, either Quadrant I or IV. Since -3/5, you are in IV!!

cosA

H

4

5

1 3: sin tan

2Ex

1tan :2 2

x

Thus, either Quadrant I or IV. Since 3/2, you are in I!!

13sin

O

H

3

13 3 13

13

Most complex problems• Follow the same rules from previous slide, but

now you will have variables in your answer

1 4: cot cosEx

x

2 16x

2 2 2

:

4

Pythagorean Theorem

opposite x 2 2 16opposite x

2 16opposite x

cota

o

2

4

16x

2

2

16

16

x

x

2

2

4 16

16

x

x