3.4 Circular Functions

x2 + y2 = 1

is a circle centered at the origin with radius 1

call it “The Unit Circle”

(1, 0)

Ex 1) For the radian measure , find the value of sine & cosine.

cos 02

sin 12

2

2

Any point on this circle can be defined in terms of sine & cosine(x, y) (cos θ, sin θ)

(0, 1)

We can draw a “reference” triangle by tracing the x-value first & then the y-value to get to a point. Use Pythagorean Theorem to find hypotenuse.

Ex 2) The terminal side of an angle θ in standard position passes through (3, 7). Draw reference triangle & find exact value of cos θ and sin θ.

cosadj x

hyp r sin

opp y

hyp r

2 2 2

2

2

7 3

49 9

58

58

r

r

r

r

(3, 7)

3

7

3 3 58cos

5858

x

r

7 7 58sin

5858

y

r

(x, y)

x

yr

θ

θ

r

Ex 3) Find the exact values of cos θ and sin θ for θ in standard position with the given point on its terminal side.

–1

21,

5Q

2

5

22 2 4 29 29

( 1) 15 25 25 5

r

295

1 5 5 29cos 1

2929

25

295

2 5 2 29sin

5 2929

θr

Reminder:

3cos

2

III

III IV

(+, +)(–, +)

(–, –) (+, –)

sinθ = 0sinθ = 0

cosθ = 0

cosθ = 0

Ex 4) State whether each value is positive, negative, or zero.

a) cos 75° b) sin (–100°) c)

positive negative zero

Ex 5) An angle θ is in standard position with its terminal side

in the 2nd quadrant. Find the exact value of cos θ if 8

sin10

–6

810

why negative?

Pythag says:x2 + 82 = 102

x2 = 36 x = ±6

so…

8sin

10

y

r

x

cosx

r

6cos

10

Homework

#304 Pg 145 #1–49 odd