A man is standing at the top of a cliff, 25 meters above the water below. He is looking out at a boat in the distance. The angle of depression is 30˚.

1) How far away from the bottom of the cliff is the boat?

2) How far away is the boat from the man?

Warm-Up: February 28, 2014

Homework Questions?

Trigonometric Functions of Any

AngleSection 4.4

We are no longer restricting ourselves to points on the unit circle.

Points can now be any distance “r” away from the origin

Points are still always on the terminal side of θ. Initial side is still the positive x-axis.

What’s New?

22 yxr

Name Abbreviation Relation to (x, y) and r

Restriction

Sine sin y/r

Cosine cos x/r

Tangent tan y/x x≠0

Cosecant csc r/y y≠0

Secant sec r/x x≠0

Cotangent

cot x/y y≠0

The Six Trigonometric Functions

Let P=(-5, -12) be a point on the terminal side of θ. Find each of the six trigonometric functions of θ.

Example 1

Let P=(4, -3) be a point on the terminal side of θ. Find each of the six trigonometric functions of θ.

You-Try #1

All Students Take Calculus

AS

T C

If tan θ >0 and cos θ < 0, in which quadrant does θ lie?

You-Try #3

Given that and find the value of the rest of the trigonometric functions

Example 44csc 0tan

Given that and find the value of the rest of the trigonometric functions

You-Try #4

3

1tan 0cos

Page 474 #1-33 odd

Assignment

Given that and find the value of the rest of the trigonometric functions

Warm-Up: January 29, 2014

0csc 12

5tan

Homework Questions?

We like dealing with acute angles (0˚<θ<90˚). We can often find trig functions of non-acute

angles by using an acute reference angle Let θ be a non-acute angle in standard

position that lies in a quadrant (not on an axis)

Its reference angle is the positive acute angle θ’ formed by the terminal side of θ and the x-axis

Reference Angles

Quadrant I:

Quadrant II:

Quadrant III:

Quadrant IV:

Finding the Reference Angle

360'

180'

180'

'

Find the reference angle θ’ for each of the following angles:

Example 5

5.2)4

3)

225)

330)

d

c

b

a

Find the reference angle θ’ for each of the following angles:

You-Try #5

6)6

7)

120)

160)

d

c

b

a

1) Find the associated reference angle, θ’2) Find the trig function value of θ’3) Use the quadrant which θ lies in to choose

the appropriate sign to the value found in step 2

Using Reference Angles to Evaluate Trigonometric Functions

Find the exact value of each of the following:

Example 6

6

17sec)

225sin)

240csc)4

7tan)

d

c

b

a

Find the exact value of each of the following:

You-Try #6

4cot)

120tan)

135cos)6

5sin)

d

c

b

a

Trig Values Table

Trig Value Table Practice (1 side) Page 474 #35-65 Odd No calculators!

Assignment