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13.2 – Angles and the Unit Circle

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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° ° °

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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° ° ° Solutions 1. 2. 3. 4. 5. 6.

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The Unit Circle The Unit Circle Radius is always one unit

Center is always at the origin Points on the unit circle relate to the periodic function Let’s pick a point on the unit circle. The positive angle always goes counter-clockwise from the x-axis. 1 30 -1 1 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. -1 In order to determine the sine and cosine we need a right triangle.

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The Unit Circle The angle can also be negative. If the angle is negative, it is drawn clockwise from the x axis. 1 -1 – 45 1 -1

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Angles and the Unit Circle

Find the measure of the angle. The angle measures 60° more than a right angle of 90°. Since = 150, the measure of the angle is 150°.

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Angles and the Unit Circle

Sketch each angle in standard position. a. 48° b. 310° c. –170°

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Let’s Try Some Draw each angle of the unit circle. 45o -280 o -560 o

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The Unit Circle Definition: A circle centered at the origin with a radius of exactly one unit. (0, 1) | | (-1,0) (0 , 0) (1,0) (0, -1) 9

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What are the angle measurements of each of the four angles we just found?

π/2 90° 0° 2π 180° 360° π 270° 3π/2 10

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The Unit Circle Let’s look at an example

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. 1 In order to determine the sine and cosine we need a right triangle. 30 -1 1 -1

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The Unit Circle 1 Create a right triangle, using the following rules:

The radius of the circle is the hypotenuse. One leg of the triangle MUST be on the x axis. The second leg is parallel to the y axis. 30 -1 1 Remember the ratios of a triangle- 2 60 -1 1 30

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The Unit Circle 2 60 1 1 30 P X- coordinate 30 -1 1 Y- coordinate -1

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Angles and the Unit Circle

Find the cosine and sine of 135°. From the figure, the x-coordinate of point A is – , so cos 135° = – , or about –0.71. 2 Use a 45°-45°-90° triangle to find sin 135°. opposite leg = adjacent leg 0.71 Simplify. = Substitute. 2 The coordinates of the point at which the terminal side of a 135° angle intersects are about (–0.71, 0.71), so cos –0.71 and sin 135°

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Angles and the Unit Circle

Find 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. 1 2 1 2 3 longer leg = = The longer leg is times the shorter leg. 3 2 1 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 Some Draw each Unit Circle. Then find the cosine and sine of each angle. 45o 120o

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45° Reference Angles – Coordinates

Remember 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. ( , ) 3π/4 ( , ) 135° 45° π/4 7π/4 5π/4 225° 315° ( , ) ( , ) 18

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Green Triangle Holding 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. 19

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60° Reference Angles – Coordinates

Use the side lengths we labeled on the QI triangle to determine coordinates. 2π/3 π/3 60° 120° ( , ) ( , ) 5π/3 4π/3 ( , ) ( , ) 240° 300° 20

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Yellow Triangle Holding 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. 21

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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. π/6 150° 30° 5π/6 11π/6 330° 210° 7π/6 22

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30° Reference Angles – Coordinates

Use the side lengths we labeled on the QI triangle to determine coordinates. ( , ) ( , ) 150° 30° π/6 5π/6 7π/6 11π/6 330° 210° ( , ) ( , ) 23

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Final Product 24

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The Unit Circle 25

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