4.7Inverse Trigonometric

Functions

Definition of the Inverse Trig. Functions

1. arcsin x quadrants I and IV22

ππ≤≤− y

2. arccos x quadrants I and II π≤≤y0

3. arctan x quadrants I and IV22

ππ<<− y

Find the values of :

⎟⎠

⎞⎜⎝

⎛−2

1arcsin I or IV

-12

3

630

π−°− or

2

3sin 1−

I or IV 32

1360

πor°

2

2arccos I or II

1

21

445

πor°

( )1arccos −

-1

10

πor°180

Use your calculator to approximate the value.Calculator in radians.

arctan( -8.45 )

arcsin( 0.2247 ) arccos( 2 )

-1.453001

.226635

error

Remark: the domain of the inverse functionsof cosine and sine are [-1,1].

5

xUse an inverse trig.function to write as a function of x.

€

θ

€

θ

€

cosθ =x

5

€

arccos cosθ( ) = arccosx

5

⎛

⎝ ⎜

⎞

⎠ ⎟

€

θ =arccosx

5

⎛

⎝ ⎜

⎞

⎠ ⎟

Find the exact value.

tan[arctan(-5)]-5

1

What is the tan of this triangle? -5

⎟⎠

⎞⎜⎝

⎛3

5sinarcsin

π3

5πis coterminal to what angle that is in the domain of arcsin?

3

π−=⎟⎟

⎠

⎞⎜⎜⎝

⎛⎟⎠

⎞⎜⎝

⎛−3

sinarcsinπ

3

π−

Find the exact value.

⎟⎠

⎞⎜⎝

⎛3

2arccostan

2

35

2

5==

adjopp

⎥⎦

⎤⎢⎣

⎡⎟⎠

⎞⎜⎝

⎛−5

3arcsincos

-35

4

5

4==

hypadj

Find sin(arccos 3x)3

10 ≤≤x

3x

1? =

1 = (3x)2 + b2

bx =− 291

sin(arccos 3x)1

91 2x

hyp

opp −==

cot(arccos 3x) 291

3

x

x

−=