section 4.3 the unit circle. (0, 1) (0, -1) (1, 0)(-1, 0) 0, 0˚ 2π, 360˚ π2π2, 90˚ π, 180˚...

Section 4.3 The Unit Circle

(0, 1)

(0, -1)

(1, 0)(-1, 0)

0, 0˚ 2π, 360˚

π2

, 90˚

π, 180˚

3π 2

, 270˚

Quadrantal Angles

“All Students Take Calculus” tells you where each trig function is positive.

AS

T C

, 180 0, 02, 360

3

2

2

Let’s begin with an easy family… 4

2

2

2

2

1

45

2

2,

2

2

What are the coordinates?

4

Now, reflect the triangle to the second quadrant…

What are the coordinates?

, 180 0, 02, 360

3

2

2

2

2

2

2

1

45

2

2,

2

2

4

Now, reflect the triangle to the third quadrant…

1

-2

2

2

2

-2

2,

2

2 3

4

What are the coordinates?

Now, reflect the triangle to the fourth quadrant…

, 180 0, 02, 360

3

2

2

2

2

2

2

1

45

2

2,

2

2

4

1

-2

2

2

2

-2

2,

2

2 3

4

-2

2, -

2

2 5

4

What are the coordinates?

, 180 0, 02, 360

3

2

2

2

2

2

2

1

45

2

2,

2

2

4

1

-2

2

2

2

-2

2,

2

2 3

4

-2

2, -

2

2 5

4

2

2, -

2

2 7

4

EX. Evaluate the six trig functions at θ = .

**REMEMBER!!! cos θ is represented by the x-coordinates and sin θ is represented by the y-coordinates.**

4

EX. Evaluate the six trig functions at θ = .3

4

Homework: worksheet

30

1 1

2

3

2

6

3

2,1

2

Now, reflect the triangle to the second quadrant.

Complete the family… 6

30

1 1

2

3

2

6

3

2,1

2

Now, reflect the triangle to the third quadrant.

5

6

1

2

-3

2

-3

2,1

2

30

1 1

2

3

2

6

3

2,1

2

Now, reflect the triangle to the fourth quadrant.

5

61

2

-3

2

-3

2,1

2

-3

2, -

1

2

What are the coordinates?

7

6

What are the coordinates?

30

11

2

3

2

6

3

2,1

2 5

6

1

2

-3

2

-3

2,1

2

-3

2, -

1

2 7

6

3

2, -

1

2 11

6

EX. Evaluate the six trig functions at θ = .

6

EX. Evaluate the six trig functions at θ = .

Homework: worksheet

7

6

3

3

2

1

2

1

2,

3

2

3

1

, 180 0, 02, 360

3

2

2

Complete the family…

Now, reflect the triangle to the second quadrant.

60˚

1

, 180 0, 02, 360

3

2

2

60

3

2

1

2

1

2,

3

2

3

1

-1

2

What are the coordinates?

-1

2,

3

2 2

3

3

2

Now, reflect the triangle to the third quadrant.

1

, 180 0, 02, 360

3

2

2

60

3

2

1

2

1

2,

3

2

3

13

2

-1

2

-1

2,

3

2 2

3

What are the coordinates?

-1

2, -

3

2 4

3

Now, reflect the triangle to the fourth quadrant.

1

, 180 0, 02, 360

3

2

2

60

3

2

1

2

1

2,

3

2

3

13

2

-1

2

-1

2,

3

2 2

3

-1

2, -

3

2 4

3

1

2, -

3

2

What are the coordinates?

5

3

Shortcut to remember ordered pairs in quadrant 1

1 32 23 1

1) Parenthesis2) Fractions with denominator of 23)

4) Square roots

EX. Evaluate the six trig functions at θ = .

3

EX. Evaluate the six trig functions at θ = .

Homework: worksheet

2

3