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Chapter 13 Sec 2 Angles and Degree Measure

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2 of 21 Algebra 2 Chapter 13 Sections 2 & 3 An angle in standard position has its vertex at the origin and initial side on the positive x–axis. An angle in standard position has its vertex at the origin and initial side on the positive x–axis. initial side terminal side Standard Position

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3 of 21 Algebra 2 Chapter 13 Sections 2 & 3 Angles that have a counterclockwise rotation have a positive measure. Angles that have a counterclockwise rotation have a positive measure. 130 0º0º 90º 180º 270º Positively Counterclockwise

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4 of 21 Algebra 2 Chapter 13 Sections 2 & 3 Angles that have a clockwise rotation have a negative measure. Angles that have a clockwise rotation have a negative measure. – 130 0º0º – 90º – 180º – 270º Clockwise means negative

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5 of 21 Algebra 2 Chapter 13 Sections 2 & 3 Now let’s look at angle measures 30, 150, 210, and ° (1, 0) (–1, 0) 30 º 30° 150° 210° 330° They all form a 30° angle with the x-axis, so they should all have the same sine, cosine, and tangent values…only the signs will change! The angle to the nearest x-axis is called the reference angle. All angles with the same reference angle will have the same trig values except for sign changes. Unit Circle

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6 of 21 Algebra 2 Chapter 13 Sections 2 & 3 Unit Circle A unit circle is a circle with radius 1. A unit circle is a circle with radius 1. If we have an angle between 0 o and 90 o in standard position. Let P(x, y) be the point of intersection. If a perpendicular segment is drawn we create a right triangle, where y is opposite θ and x is adjacent to θ. If we have an angle between 0 o and 90 o in standard position. Let P(x, y) be the point of intersection. If a perpendicular segment is drawn we create a right triangle, where y is opposite θ and x is adjacent to θ. Right triangles can be formed for angles greater than 90 o, simply use the reference angle. Right triangles can be formed for angles greater than 90 o, simply use the reference angle.

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7 of 21 Algebra 2 Chapter 13 Sections 2 & 3 Radian… still A point P(x, y) is on the unit circle if and only if its distance from the origin is 1. A point P(x, y) is on the unit circle if and only if its distance from the origin is 1. The radian measure of an angle is the length of the corresponding arc on the unit circle. The radian measure of an angle is the length of the corresponding arc on the unit circle. Since Since P(x, y) s α

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8 of 21 Algebra 2 Chapter 13 Sections 2 & 3 Degree/Radian Conversion

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9 of 21 Algebra 2 Chapter 13 Sections 2 & 3 Example 1 a. Change 115 o to radian measure in terms of π.. b. Change radian to degree measure.

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10 of 21 Algebra 2 Chapter 13 Sections 2 & 3 30° and 45° Radians You will need to know these conversions. You will need to know these conversions.

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11 of 21 Algebra 2 Chapter 13 Sections 2 & 3 Coterminal angles are angles that have the same initial and terminal side, but differ by the number of rotations. Coterminal angles are angles that have the same initial and terminal side, but differ by the number of rotations. Since one rotation equals 360 , the measures of coterminal angles differ by multiples of 360 . Since one rotation equals 360 , the measures of coterminal angles differ by multiples of 360 . 300 – 360 = 300 60 = – 60 420 Coterminal Angles

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12 of 21 Algebra 2 Chapter 13 Sections 2 & 3 Example 2 Find one positive and one negative coterminal angle. a. 45 o 45 o o = 405 o and 45 o – 360 o = –315 o b. 225 o 225 o o = 585 o and 225 o – 360 o = –135 o

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Chapter 13 Sec 3 Trigonometric Functions

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14 of 21 Algebra 2 Chapter 13 Sections 2 & 3 Radius other than 1. Suppose we have a hypotenuse with a length other than 1. For our example we’ll use r as the length. Suppose we have a hypotenuse with a length other than 1. For our example we’ll use r as the length. In standard position r extends from the Origin to point P(x, y). In standard position r extends from the Origin to point P(x, y).

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15 of 21 Algebra 2 Chapter 13 Sections 2 & 3 Quadrantal Angle If a terminal side of an angle coincides with one of the axes, the angle is called a quadrantal angle. See below for examples: A full rotation around the circle is 360 o. Measures more than 360 o represent multiple rotations.

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16 of 21 Algebra 2 Chapter 13 Sections 2 & 3 To find the values of trig functions of angles greater than 90 , you will need to know how to find the measures of the reference angle. If θ in nonquadrantal, its reference angle is formed by the terminal side of the given angle and the x-axis. Reference Angles

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17 of 21 Algebra 2 Chapter 13 Sections 2 & 3 Example 1 Find the reference angle for each angle. a. 312 o Since 312 o is between 270 o and 360 o the terminal side is in fourth quad. Therefore, 360 o – 312 o = 48 o. b. –195 o the coterminal angle is 360 o – 195 o = 165 o this put us in the second quadrant so… 180 o – 165 o = 15 o

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18 of 21 Algebra 2 Chapter 13 Sections 2 & 3 0°/360° (1, 0) 90°(0, 1) 270°(0, –1) 180° (–1, 0) S tudents S ine values are positive (csc, too) A ll A ll values are positive T ake T angent values are positive (cot, too) C alculus C osine values are positive (sec, too) Determining Sign

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19 of 21 Algebra 2 Chapter 13 Sections 2 & 3 Example 2 θ Find the values of the six trigonometric functions for angle θ in standard position if a point with coordinates (–15, 20) lies on the terminal side.

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20 of 21 Algebra 2 Chapter 13 Sections 2 & 3 Example 3 θ θ Find the values of the six trigonometric functions Suppose θ is an angle in standard position whose terminal side lies in the Quadrant III. If find the remaining five trigonometric functions of θ.

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21 of 21 Algebra 2 Chapter 13 Sections 2 & 3 Daily Assignment Chapter 13 Sections 2 & 3 Study Guide Pg 177 #4 – 7 Pg 178 – 180 Odd

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