10.3: Extending the Trig Ratios
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Transcript of 10.3: Extending the Trig Ratios
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
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)
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)