Aim: How do we find the exact values of trig functions?

Do Now: Evaluate the following trig ratios

a) sin 45

b) sin 60

c) sin 135

HW: p.380 # 34,36,38,42

p.391 # 8,12,14,16,24,26

y

xO-1

1

Draw 135 on the standard position

Form a triangle in quadrant II.

B

We will use the ΔAOB to find the trig ratios for angle 135

135

A

The triangle is an isosceles right triangle and AOB is 45

How do we proceed?

First of all, we need to find the reference angle.

Reference angle: An acute angle that is formed by the x-axis and the terminal side of an angle in standard position.

135OA

If which is in the quadrant II, therefore, the reference angle is formed by and the negative x- axis. Then the reference angle is 45

y

x

1

-1

2

Use the triangle in quadrant II, we can find the trig ratios of 135

135

Reference angle 45

1tan,2

1cos,

2

1sin

Principal angle

y

x225

Reference angle 45 in quadrant III

y

x315

Reference angle 45 in quadrant IV

y

x

45

If the angle is in quadrant I, then the principal angle and reference angle are the same

The rules to find the reference angle for any angle within 360

Quadrant I: Principal angle & reference angle are the same

Quadrant II: angle is (180 – θ)

Quadrant III: angle is (θ – 180)

Quadrant IV: angle is

Where θ represents principal angle

(180 – θ)

Based on the rules, 30, 150, 210 and 330 all have the same reference angle.

30: the reference angle is still 30 in quadrant I

150: the reference angle is 180 – 150 = 30 in quadrant II

210: the reference angle is 210 – 180 = 30 in quadrant III

330: the reference angle is 360 – 330 = 30 in quadrant IV

To find the trig ratios for any angle from 0 to 360, we first find the reference angle then use the rules of ASTC to determine the signs.

Find the value of a) sin 225 b) cos 315

225 is in quadrant III and the reference angle is 45 sin 225 is just like sin 45 in quad III

2

2

2

1225sin

315 is in quadrant IV and the reference angle is 45 cos 315 is just like cos 45 in quad IV

2

2

2

1315cos

Find the exact values of the following trig functions

a)sin 240

b)sin 225

c) cos 135

d) sin -330

2

3

2

2

2

2

2

1

e) cos 120

f) cos 225

g) cos 315

h) cos -60

2

1

2

2

2

2

2

1

i) sin 420

j) tan 315

k)tan 210

l)tan -120

2

3

1

3

3

3