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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°5.135°6.120°

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Angles and the Unit Circle Solutions 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 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 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 Circle Find the measure of the angle. Since = 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 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. a.45 o b.-280 o c.-560 o

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

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What are the angle measurements of each of the four angles we just found? 180° 90° 270° 0° 360° 2π2π π /2 π 3 π /2 0

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The Unit Circle 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 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 Remember the ratios of a triangle- 2

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The Unit Circle X- coordinate Y- coordinate P

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Angles and the Unit Circle Find 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 – 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 13 –0.71 and sin 135° 0.71.

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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 = 1The hypotenuse is a radius of the unit circle. shorter leg = The shorter leg is half the hypotenuse longer leg = 3 = The longer leg is 3 times the shorter leg 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. a.45 o b.120 o

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

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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.

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

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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.

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

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

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Final Product

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The Unit Circle

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