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Standard-Position-of-an-Angle-FULL.ppt
- 1. Trigonometry The study oftriangles and the relationships between their sides and angles
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- 3. Angle in StandardPosition It is easy to see the direction of an angle if it is presented in reference to a rectangular coordinate system. Let the origin of the coordinate system be at the vertex of the angle and let the positive half of the x-axis form the initial side of the angle. The terminal side shall be in any of the four quadrants of the coordinate system. Angle drawn in this manner are called angles in standard position.
- 4. Angle in Standard Position Anangle in standard position has for its initial side the positive half of the x-axis in a rectangular coordinate system.
- 5. Let’s look atan angle in standard position, where the initial side is ALWAYS on the positive x-axis and the vertex is at the origin. The terminal side can be anywhere and defines the angle. A positive angle is described by starting at the initial side and rotating counterclockwise to the terminal side (angle ). terminal side initial side vertex A negative angle is described by rotating clockwise (angle ).
- 6. Depending upon thedegree measure of the angle, the terminal side can land in one of the four quadrants. Angles can be larger than 360º by simply wrapping around the quadrants again. (450º, 540º, 630º, 720º, etc.) I II III IV I II III IV -90 -180 -270 -360 I II III 90 180 270 360
- 7. Name the quadrantof the terminal side. 1) 140o 7) 80o 2) 315o 8) -475o 3) -168o 9) -25o 4) 475o 10) 1030o 5) -340o 11) -1030o 6) 670o 12) -225o
- 8. Angles and are coterminal since they share the same sides. Coterminal Angles are angles that share the same terminal side, but have different angle measures. There are also several other angles that are coterminal to .
- 9. To find acoterminal angle: add or subtract 360º (or any multiple of 360o) to the given angle . Both are coterminal angles to Example: 35 + 360 = 395º = 35º 35 – 360 = -325º Find a negative and positive coterminal angle to -425o
- 10. Find one positiveand one negative coterminal angle for each angle below. 1) 140o 7) 80o 2) 315o 8) -475o 3) -168o 9) -25o 4) 475o 10) 1030o 5) -340o 11) -1030o 6) 670o 12) -225o
- 11. Botanical Name Narcissus 'Trigonometry' Plant CommonName Trigonometry Daffodil The flowers of a Trigonometry Daffodil are of almost geometric precision with their repeating patterns. Repeating patterns occur in sound, light, tides, time, and nature. To analyse these repeating, cyclical patterns, we need to study the cyclical functions branch of trigonometry.
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- 13. Angles in StandardPosition Initial arm Vertex Terminal arm x y To study circular functions, we must consider angles of rotation.
- 14. If the terminalarm moves counter- clockwise, angle A is positive. A x y If the terminal side moves clockwise, angle A is negative. A x y Positive or Negative Rotation Angle
- 15. 30 60 120 150 210 240 300 330 90 180 270 0 Benchmark Angles SpecialAngles Degrees 45 135 225 315 360 Math 30-1 15
- 16. Sketch each rotationangle in standard position. State the quadrant in which the terminal arm lies. 400° - 170° -1020° 1280° Math 30-1 16
- 17. Coterminal angles areangles in standard position that share the same terminal arm. They also share the same reference angle. 50° Rotation Angle 50° Terminal arm is in quadrant I Positive Coterminal Angles Counterclockwise 50° + (360°)(1) = Negative Coterminal Angles Clockwise -310° 770° -670° 410° 50° + (360°)(2) = 50° + (360°)(-1) = 50° + (360°)(-2) =
- 18. Coterminal Angles inGeneral Form By adding or subtracting multiples of one full rotation, you can write an infinite number of angles that are coterminal with any given angle. θ ± (360°)n, where n is any natural number Why must n be a natural number?
- 19. Sketching Angles andListing Coterminal Angles Sketch the following angles in standard position. Identify all coterminal angles within the domain -720° < θ < 720° . Express each angle in general form. a) 1500 b) -2400 c) 5700 Positive Negative General Form 5100 -2100 1200 -6000 2100 -1500 150 360 , n n N 240 360 , n n N - 570 360 , n n N Positive Negative General Form Positive Negative General Form , -5700 , 4800 -5100
- 20. Radian Measure: Trigand Calculus The radian measure of an angle is the ratio of arc length of a sector to the radius of the circle. a r number of radians = arc length radius When arc length = radius, the angle measures one radian. How many radians do you think there are in one circle?
- 21. Construct arcs onthe circle that are equal in length to the radius. Radian Measure 2 6.283185307... radians C 2r arc length 2(1) http://www.geogebra.org/en/upload/files/ppsb/radian.ht ml One full revolution is Math 30-1 21
- 22. Radian Measure One radianis the measure of the central angle subtended in a circle by an arc of equal length to the radius. 2r r r = a r O r r s = r 1 radian = 1 revolution of 360 Therefore, 2π rad = 3600. Or, π rad = 1800. r 2 rads Angle measures without units are considere d to be in radians. Math 30-1 22
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- 25. Sketching Angles andListing Coterminal Ang Sketch the following angles in standard position. Identify all coterminal angles within the domain -4π < θ < 4π . Express each angle in general form. a) b) c) Positive Negative General Form 5 2 , 6 n n N 4 2 , 3 n n N - 10.47 2 , n n N Positive Negative General Form Positive Negative General Form 5 6 4 3 - 10.47 17 6 7 6 - 19 , 6 - 2 3 8 , 3 10 3 - 4.19 2.1 - , 8.38 -
- 26. Angles and CoterminalAngles Degrees and Radians Page 175 1, 6, 7, 8, 9, 11a, c, d, e, h