13.2 – angles and the unit circle
DESCRIPTION
13.2 – Angles and the Unit Circle . Angles and the Unit Circle. For each measure, draw an angle with its vertex at the origin of the coordinate plane. Use the positive x -axis as one ray of the angle. 1.90°2.45°3.30° 4.150°5.135°6.120°. Angles and the Unit Circle. - PowerPoint PPT PresentationTRANSCRIPT
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13.2 – Angles and the Unit Circle
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Angles and the Unit CircleFor each measure, draw an angle with its vertex at the origin of the coordinate plane. Use the positive x-axis as one ray of the angle.
1. 90° 2. 45° 3. 30°
4. 150° 5. 135° 6. 120°
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Angles and the Unit Circle
Solutions1. 2. 3.
4. 5. 6.
For each measure, draw an angle with its vertex at the origin of the coordinate plane. Use the positive x-axis as one ray of the angle.
1. 90° 2. 45° 3. 30°
4. 150° 5. 135° 6. 120°
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The Unit Circle
1-1
1
-1
The Unit Circle-Radius is always one unit-Center is always at the origin-Points on the unit circle relate to the periodic function
30
Let’s pick a point on the unit circle. The positive angle always goes counter-clockwise from the x-axis.
The x-coordinate of this has a value of the cosine of the angle. The y-coordinate has a value of the sine of the angle.
In order to determine the sine and cosine we need a right triangle.
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The Unit Circle
1-1
1
-1
The angle can also be negative. If the angle is negative, it is drawn clockwise from the x axis.
- 45
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Angles and the Unit CircleFind the measure of the angle.
Since 90 + 60 = 150, the measure of the angle is 150°.
The angle measures 60° more than a right angle of 90°.
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Angles and the Unit CircleSketch each angle in standard position.
a. 48° b. 310° c. –170°
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Let’s Try SomeDraw each angle of the unit circle.
a.45o
b.-280 o
c.-560 o
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The Unit CircleThe Unit Circle
Definition: A circle centered at the origin with a radius of exactly one unit.
|-------1-------|(0 , 0) (1,0)(-1,0)
(0, 1)
(0, -1)
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What are the angle measurements What are the angle measurements of each of the four angles we just of each of the four angles we just
found?found?
180°
90°
270°
0°360°2π
π/2
π
3π/2
0
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The Unit Circle
1-1
1
-1
Let’s look at an example
30
The x-coordinate of this has a value of the cosine of the angle. The y-coordinate has a value of the sine of the angle.
In order to determine the sine and cosine we need a right triangle.
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The Unit Circle
1-1
1
-1
30
Create a right triangle, using the following rules:1.The radius of the circle is the hypotenuse.2.One leg of the triangle MUST be on the x axis. 3.The second leg is parallel to the y axis.
30
601
Remember the ratios of a 30-60-90 triangle-
2
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The Unit Circle
1-1
1
-1
30
30
601
2
X- coordinate
Y- coordinate
P
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Angles and the Unit CircleFind the cosine and sine of 135°.
Use a 45°-45°-90° triangle to find sin 135°.
From the figure, the x-coordinate of point A
is – , so cos 135° = – , or about –0.71. 22
22
opposite leg = adjacent leg
0.71 Simplify.
= Substitute. 22
The coordinates of the point at which the terminal side of a 135° angle intersects are about (–0.71, 0.71), so cos 13 –0.71 and sin 135° 0.71.
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Angles and the Unit CircleFind the exact values of cos (–150°) and sin (–150°).
Step 1: Sketch an angle of –150° in standard position. Sketch a unit circle.
x-coordinate = cos (–150°)y-coordinate = sin (–150°)
Step 2: Sketch a right triangle. Place the hypotenuse on the terminal side of the angle. Place one leg on the x-axis. (The other leg will be parallel to the y-axis.)
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Angles and the Unit Circle(continued)
The triangle contains angles of 30°, 60°, and 90°.
Step 3: Find the length of each side of the triangle.
hypotenuse = 1 The hypotenuse is a radius of the unit circle.
shorter leg = The shorter leg is half the hypotenuse.12
12
32longer leg = 3 = The longer leg is 3 times the shorter leg.
32
12
Since the point lies in Quadrant III, both coordinates are negative. The longer leg lies along the x-axis, so
cos (–150°) = – , and sin (–150°) = – .
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Let’s Try SomeDraw each Unit Circle. Then find the cosine and sine of each angle.
a.45o
b.120o
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45° Reference Angles - Coordinates45° Reference Angles - CoordinatesRemember that the unit circle is overlayed on a coordinate plane (that’s how
we got the original coordinates for the 90°, 180°, etc.)
Use the side lengths we labeled on the QI triangle to determine coordinates.
45°135°
315°
225°
( , )
( , )
( , )
( , ) 2
2
22
22
22
22
22
22
22
22
22
π/43π/4
5π/4 7π/4
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30-60-90 Green Triangle30-60-90 Green TriangleHolding the triangle with the single fold down and double fold to the left, label each
side on the triangle.
Unfold the triangle (so it looks like a butterfly) and glue it to the white circle with the triangle you just labeled in quadrant I, on top of the blue butterfly.
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60° Reference Angles - Coordinates60° Reference Angles - Coordinates
Use the side lengths we labeled on the QI triangle to determine coordinates.
60°120°
300°
240°
( , )
( , )
( , )
( , ) 2
3
23
21
23
21
23
21
21
21
23
π/32π/3
4π/3 5π/3
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30-60-90 Yellow Triangle30-60-90 Yellow TriangleHolding the triangle with the single fold down and double fold to the left, label each
side on the triangle.
Unfold the triangle (so it looks like a butterfly) and glue it to the white circle with the triangle you just labeled in quadrant I, on top of the green butterfly.
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30° Reference Angles 30° Reference Angles We know that the quadrant one angle formed by the triangle is 30°.
That means each other triangle is showing a reference angle of 30°. What about in radians?
Label the remaining three angles.
30°150°
330°210
°
π/6
7π/6
5π/6
11π/6
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30° Reference Angles - Coordinates30° Reference Angles - Coordinates
Use the side lengths we labeled on the QI triangle to determine coordinates.
30°150°
330°210
°
( , )
( , )
( , )
( , ) 2
1
21
23
21
23
21
23
23
23
21
π/6
7π/6
5π/6
11π/6
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Final ProductFinal Product
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The Unit CircleThe Unit Circle