Ch 6.3: Trigonometric functions of any angle

In this section, we will

1. extend the domains of the trig functions (as well as the needfor it) using the unit circle

2. look at some properties of the unit circle

3. evaluate trig. values using reference angles

4. investigate the relationships between trig. functions

Why extend the right triangle definition?

1) 0 < θ since it’s an interior angle of a right triangle.I No sin(0◦) or cos(−50◦) can be defined through the right

triangle definition.

2) θ < 90 since we have a right triangle and the sum of theinterior angles in a triangle is 180.

I In fact, if θ = 90, then we have two right angles. ⇒ Notriangle at all!!

I If θ > 90, Same problem!!

Conclusion: With right triangle definition, we can define sine,cosine, and tangent, but ONLY for θ ∈ (0, 90).⇒ We need to extend our definition (and will use the unit circle todo so..)

The Unit Circle definition - for sine, cosine, and tangent

DefinitionLet (x , y) be a point on the unit circle and θ is the angle betweenthe positive x-axis (initial side) and the ray from 0 which goesthrough (x , y) (terminal side). Then,

x = cos θ , y = sin θ ,y

x= tan θ.

ExamplesUse the unit circle definition to evaluate sin θ and cos θ for thefollowing angles: θ = 0◦, 90◦, 180◦, 270◦, 360◦

Let’s check!

Let θ ∈ (0, 90) and (x , y) be corresponding point on the unit circle.

New definition says cos θ = x and sin θ = y .Q: If θ ∈ (0, 90), can we say the same using the old definition?(Do we even have a right triangle?)

Properties of the unit circle (UC)

Since any point P(x , y) on the unit circle (UC) has the followingrelation:

x2 + y2 = 1

if we’re given one of x, y in (x , y) on the UC, then we know theother coordinate.Example) If y = 1

2 , and (x , y) in 2nd Q, what is x?

Properties of the unit circle (UC) - Continued

If point P(x , y) on UC, then due to UC being symmetric to both xand y axis (and also to the origin):

(x ,−y), (−x , y), and (−x ,−y)

are all on the UC as well.Example) If x = −3

5 and y = −45 , a) verify that (x , y) is on the UC,

and b) find three other points on the UC?

Reference Angle

DefinitionFor a non-quadrantal angle θ in standard position, the acute angleθr formed by the terminal side and the nearest x-axis is called thereference angle.

Examples - Reference angles

Find the reference angle for a) θ = 315◦, b) θ = 1900◦, c)θ = −120◦.

More Examples

Find the reference angle for each rotation given.a. θ = 5π

6 b. θ = 4π3 c. θ = −7π

4 .

Signs of the Trig functionsKnowing that cos θ = x and sin θ = y , (and tan θ = y

x )

Ex1) Let sinα > 0 and tanα < 0. Which quadrant is α in?

Ex2) cosβ < 0 and cotβ > 0.

Examples - Evaluating trig. values using reference anglesEvaluate the following:

1. cos 330◦

2. sin(7π6 )

What if (x , y) is NOT on the unit circle?

I sin θ

I cos θ

I tan θ

I csc θ

I sec θ

I cot θ

Definition

DefinitionGiven (x , y) with r =

√x2 + y2 and the corresponding θ,

sin θ =y

r, cos θ =

x

r, tan θ =

y

x

csc θ =1

sin θ=

r

y, sec θ =

1

cos θ=

r

x, cot θ =

1

tan θ=

x

y

(Note: denominators must not be 0.)Q: For which (quadrantal) angles are these functions not defined?

ExamplesFind the values of the 6 trig functions, given the following.

1. (7, 24)2. (−3,−1)

More example

Given sin θ = 5/13 and cos θ < 0, find the values of other ratios.

Fundamental Trigonometric Identities

Reciprocal Identities:

csc θ =1

sin θ, sec θ =

1

cos θ, cot θ =

1

tan θ

Ratio Identities:

tan θ =sin θ

cos θ=

sec θ

csc θ, cot θ =

cos θ

sin θ

Cofunction Identities : From the unit circle definition

Theoremcos θ = sin(π2 − θ), csc θ = sec(π2 − θ), cot θ = tan(π2 − θ)

Examples

Write the following in terms of its cofunction.

1. sin−60◦

2. cos π3

3. cot 120◦

4. sec π2

Homework for Ch 6.3: pg. 497

4, 18, 19, 44, 51, 65, 70, 71, 91, 92, 94