4.5 (Day 1) Inverse Sine & Cosine

Remember: the inverse of a function is found by switching the x & y values (reflect over line y = x)

Domains become ranges…. ranges become domains

We want the inverse sine & inverse cosine to be functions(pass vertical line test) so we need to restrict their domains –can’t be all real numbers

To denote we want the inverses to be functions, we use capital letters Sin–1x and Cos–1x

OR Arcsin x and Arccos x

y = Sin–1 x

D = [–1, 1]

Want range to include (–) & (+) valuesChoose QI for (+) valuesWhich quad is closest to QI that contains (–) values?

y = Cos–1 x

D = [–1, 1](the range for sin θ)

(the range for cos θ)

III

III IV

(+)(+)

(–) (–)

,2 2

R

close

III

III IV

(–)

(–) (+)

0,R

close

(+)

These re

strict

ions tell u

s where

we draw

the r

eferen

ce θ &

△

*What type of answer is required

(1) sin x cos x

(2) Sin–1 xCos–1 x

trig function of an angle is a ratiotrig (angle) = ratio

a) Sin–1 0.3240

0.3300

b) Arcsin 0.5681

0.6042

Ex 1) Evaluate to 4 decimal places (radian mode)

inverse function of a ratio is an angletrig–1 (ratio) = angle Find the θ and draw it’s picture in the correct quadrant!

c) Cos–1 (–0.56)

2.1652

short △ QIII

Ex 2) Evaluate. Find exact value if possible.

a) 1

1

Sin sin4

2Sin

2 4

b)

22ratio

θ2

21

1

Cos cos2

Cos (0)2

ratio 0

c) 7Arcsin sin

6

1Arcsin

2 6

12ratio

θ

tall △ QIV

c) 5

Arccos cos3

1Arcsin

2 3

12ratio

θ

picture

Ex 3) Determine the exact value.

a) 1 5sin Cos

9

2 14sin

9

θ θ

56 2 14

picture

b) 3cos Arcsin

5

4cos

5

θ θ

9

5

(a ratio!)

–35

4

(a ratio!)

Optics: Light is refracted when it travels from air to water. i is the angle of incidence (in air) and r is the angle of refraction (in water). Equation is:

Ex 4) If a light ray makes a 30° angle with the vertical in air, determine the angle with the vertical in water.

*Degree mode*

sin 4

sin 3

i

r

12

1

sin30 4 4

sin 3 sin 33 3

4sin sin2 83

Sin8

22

r r

r r

r

r

Homework

#405 Pg 220 #1–13 all, 15–18, 22, 25, 28–31