Ch 6.3: Trigonometric functions of any angle

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 need

for 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 < ¦È 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) ¦È < 90 since we have a right triangle and the sum of the

interior angles in a triangle is 180.

I

I

In fact, if ¦È = 90, then we have two right angles. ? No

triangle at all!!

If ¦È > 90, Same problem!!

Conclusion: With right triangle definition, we can define sine,

cosine, and tangent, but ONLY for ¦È ¡Ê (0, 90).

? We need to extend our definition (and will use the unit circle to

do so..)

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

Definition

Let (x, y ) be a point on the unit circle and ¦È is the angle between

the positive x-axis (initial side) and the ray from 0 which goes

through (x, y ) (terminal side). Then,

x = cos ¦È ,

y = sin ¦È ,

y

= tan ¦È.

x

Examples

Use the unit circle definition to evaluate sin ¦È and cos ¦È for the

following angles: ¦È = 0? , 90? , 180? , 270? , 360?

Let¡¯s check!

Let ¦È ¡Ê (0, 90) and (x, y ) be corresponding point on the unit circle.

New definition says cos ¦È = x and sin ¦È = y .

Q: If ¦È ¡Ê (0, 90), can we say the same using the old definition?

(Do we even have a right triangle?)

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