04/19/23 10.3: Extending the Trig Ratios

10.3: Extending the Trig Ratios

Expectation:G1.3.3: Determine the exact values of

sine, cosine, and tangent for 0°, 30°, 45°, 60°, and their integer multiples and apply in various

contexts.

If the angles ∠X and ∠Y each measure between 0° and 90°, and if

sin X = cos Y, what is the sum of the measures of the angles ∠X and ∠Y?

A.30

B.45

C.60

D.90

E.135

04/19/23 10.3: Extending the Trig Ratios

04/19/23 10.3: Extending the Trig Ratios

Angle of Rotation

An angle is an angle of rotation iff:

a. its vertex is the origin

b. one side is the positive x-axis

c. the other side is a rotation of the first side centered at the

origin.

04/19/23 10.3: Extending the Trig Ratios

Angles of Rotation

θ

04/19/23 10.3: Extending the Trig Ratios

Angles of Rotation

θ

04/19/23 10.3: Extending the Trig Ratios

Angles of Rotation

θ

04/19/23 10.3: Extending the Trig Ratios

Angles of Rotation

θ

04/19/23 10.3: Extending the Trig Ratios

Unit Circle

Defn: A circle is a unit circle iff:

a. its center is the origin (0,0).

b. its radius is 1.

04/19/23 10.3: Extending the Trig Ratios

Unit Circle: x2 + y2 = 1

(1,0)

(0,-1)

(0,1)

(-1,0)

04/19/23 10.3: Extending the Trig Ratios

Who Cares?

We can use unit circles and trig to find coordinates of points on a unit circle.

04/19/23 10.3: Extending the Trig Ratios

(1,0)

(0,-1)

(0,1)

(-1,0) 30°

A

What are the coordinates of A?

04/19/23 10.3: Extending the Trig Ratios

What are the coordinates of B?

(1,0)

(0,-1)

(0,1)

(-1,0) 45°

B

04/19/23 10.3: Extending the Trig Ratios

What are the coordinates of C?

(1,0)

(0,-1)

(0,1)

(-1,0)

C

60°

04/19/23 10.3: Extending the Trig Ratios

What are the coordinates of D?

(1,0)

(0,-1)

(0,1)

(-1,0)

60°

D

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What is the angle of rotation for the hypotenuse below?

(1,0)

(0,-1)

(0,1)

(-1,0)

A (.866, .5)

04/19/23 10.3: Extending the Trig Ratios

???????????

What is the cos 30?

What is the sin 30?

Compare sin 30, cos 30 and the (x,y) coordinates of A.

04/19/23 10.3: Extending the Trig Ratios

What is the angle of rotation for the hypotenuse below?

(1,0)

(0,-1)

(0,1)

(-1,0)

B(-.707, .707)

04/19/23 10.3: Extending the Trig Ratios

???????????

What is the cos 135?

What is the sin 135?

Compare sin 135, cos 135 and the (x,y) coordinates of B.

04/19/23 10.3: Extending the Trig Ratios

What is the angle of rotation for the hypotenuse below?

(1,0)

(0,-1)

(0,1)

(-1,0)

C(-.5, -.866)

04/19/23 10.3: Extending the Trig Ratios

???????????

What is the cos 240?

What is the sin 240?

Compare sin 240, cos 240 and the (x,y) coordinates of C.

04/19/23 10.3: Extending the Trig Ratios

What is the angle of rotation for the hypotenuse below?

(1,0)

(0,-1)

(0,1)

(-1,0)

D (.5, -.866)

04/19/23 10.3: Extending the Trig Ratios

???????????

What is the cos 300?

What is the sin 300?

Compare sin 300, cos 300 and the (x,y) coordinates of D.

04/19/23 10.3: Extending the Trig Ratios

Sine and Cosine on a Unit Circle

Defn: Let θ be a rotation angle. Then sin θ is the y-coordinate of the image of P(1,0) rotated θ about the origin and cos θ is the x-coordinate.

P’= (cos θ, sin θ)

04/19/23 10.3: Extending the Trig Ratios

What are sin (-60) and cos (-60)?

What are the sin 440 and cos 440?

04/19/23 10.3: Extending the Trig Ratios

Negative Angles

sin (-θ) = - sin θ

cos (- θ) = cos (θ)

04/19/23 10.3: Extending the Trig Ratios

Angles Larger than 360°

If θ > 360, then:

sin θ = sin (θ - 360n)

cos θ = cos (θ - 360n)

where n is a whole number.

04/19/23 10.3: Extending the Trig Ratios

Verify trig identity number 1:

tan θ = sin θcos θ

04/19/23 10.3: Extending the Trig Ratios

Verify trig identity number 2:

sin2 θ + cos2 θ = 1

04/19/23 10.3: Extending the Trig Ratios

Graphing Sine and Cosine

For θ = 0, 30, 45, 60, 90, 120, 135, 150, 180, 210, 225, 240, 270, 300, 315, 330 and 360, determine sin θ and cos θ.

It may be helpful to organize your data into a chart.

Graph your data.

04/19/23 10.3: Extending the Trig Ratios

-1

-0.5

0

0.5

1

0 90 180 270 360

sin

cos

04/19/23 10.3: Extending the Trig Ratios

A satellite orbits a planet at 1° per hour. Let the radius of the orbit equal 1 and determine the ordered pair coordinates of the satellite after 497 hours. Assume it starts at (1,0).

04/19/23 10.3: Extending the Trig Ratios

Give 2 angles, θ, between 0 and 360 that have cos θ = .7071.

04/19/23 10.3: Extending the Trig Ratios

Assignment

pages 652-653,

# 13-55 (odds)