Lesson 4.2

A circle with center at (0, 0) and radius 1 is called a unit circle.

The equation of this circle would be

122 yx

(1,0)

(0,1)

(0,-1)

(-1,0)

Let's pick a point on the circle. We'll choose a point where the x is 1/2. If the x is 1/2, what is the y value?

(1,0)

(0,1)

(0,-1)

(-1,0)

x = 1/2

122 yx

12

1 22

y

4

32 y

2

3y

2

3,

2

1

2

3,

2

1

(1,0)

(0,1)

(0,-1)

(-1,0)

Now, draw a right triangle from the origin to the point when x = ½. Use this triangle to find the values of sin, cos, and tan.

2

1

2

31

sin

cos2

1

121

tan 3

2123

2

3,

2

12

3

123

For any point on the unit circle:

Sine is the y value

Cosine is the x value.

Tangent is

(1,0)

(0,1)

(0,-1)

(-1,0)

2

3,

2

1sin

cos

tan

2

3,

2

1

2

2

2

2

Find the sin, cos, and tan of the angle

2

2,

2

2

1

22

22

How many degrees are in each division?

45°

45°

2

2,

2

290°

1,0

0°

135°

2

2,

2

2

180° 0,1

225°

270°315°

2

2,

2

2

2

2,

2

2

1,0

225sin2

2

0,1

How about radians?

45°

2

2,

2

290°

1,0

0°

135°

2

2,

2

2

180° 0,1

225°

270°315°

2

2,

2

2

2

2,

2

2

1,0

4

7sin

2

2

0,1

4

2

4

3

4

5

2

34

7

How many degrees are in each division?

30°

30°

2

1,

2

3

90°

1,0

0°

120°

180° 0,1

210°

270°

330°

1,0

330cos2

3

0,1

60°150°

240°300°

2

3,

2

1

2

3,

2

1

2

3,

2

1

2

1,

2

3

2

1,

2

3

2

1,

2

3

2

3,

2

1240sin2

3

How about radians?

30°

30°

2

1,

2

3

90°

1,0

0°

120°

180° 0,1

210°

270°

330°

1,0

0,1

60°150°

240°300°

2

3,

2

1

2

3,

2

1

2

3,

2

1

2

1,

2

3

2

1,

2

3

2

1,

2

3

2

3,

2

1

6

2

3,

2

1

Let’s think about the function f() = sin

Domain: All real numbers.

Range: -1 sin 1

(1, 0)

(0, 1)

(-1, 0)

(0, -1)

Let’s think about the function f() = cos

Domain: All real numbers

Range: -1 cos 1

(1, 0)

(0, 1)

(-1, 0)

(0, -1)

Let’s think about the function f() = tan

Domain:

Range: All real numbers

x ≠ n( /2), where n is an odd integer

Trig functions:

2

3,

2

1

Look at the unit circle and determine sin 420°.

So sin 420° = sin 60°.

Sine is periodic with a period of 360° or 2.

Reciprocal functions have the same period.

PERIODIC PROPERTIES

sin( + 2) = sin cosec( + 2) = cosec

cos( + 2) = cos sec( + 2) = sec

tan( + ) = tan cot( + ) = cot

4

9tan

This would have the same value as

4tan

1

Subtract the period until you reach an angle measure you know.

Positive vs. Negative & Even vs. Odd

?3

cos isWhat

?3

cos isWhat

Remember negative angle means to go clockwise

2

1

2

1

2

3,

2

1

Even if: f(-x) = f(x) Odd if: f(-x) = -f(x)

coscos Cosine is an even function.

?3

sin isWhat

?3

sin isWhat

2

3

2

3

2

3,

2

1

sinsin

?3

tanisWhat

?3

tanisWhat

2

3,

2

1

3

3

Sine is an odd function.

EVEN-ODD PROPERTIESsin(- ) = - sin (odd) cosec(- ) = - cosec (odd)

cos(- ) = cos (even) sec(- ) = sec (even)

tan(- ) = - tan (odd) cot(- ) = - cot (odd)

Problem Set 4.2