An angle in the Cartesian plane is in standard position if its vertex lies at the origin and its initial arm lies on the positive x-axis.
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1 Learning Goals 1. To understand what standard position represents. 2. To understand what a principal and related acute angle are. 3. To understand that positive angles are measured by a counter-clockwise rotation. 4. To understand what the CAST rule tells you. 5. To understand that you will be calculating two angles. 6. To understand how to use the related acute to solve for the principal angle. An angle in the Cartesian plane is in standard position if its vertex lies at the origin and its initial arm lies on the positive x-axis. 1
2 Principal Angle (θ - the counter-clockwise angle between the initial arm and the terminal arm of an angle in standard position. Its value is between 0 and 360. Related Acute Angle (β) - the acute angle between the terminal arm of an angle in standard position and the x-axis when the terminal arm lies in quadrants 2, 3, or 4. An angle in standard position is determined by a counterclockwise rotation and is always positive. An angle determined by a clockwise rotation is always negative. If the terminal arm of an angle in standard position lies in quadrants 2, 3, or 4, there exists a related acute angle (β AND a principal angle θ. 2
3 If the terminal arm of the principal angle lies in quadrant 2 then the related acute angle is calculated as β = θ. If the terminal arm of the principal angle lies in quadrant 3 then the related acute angle is calculated as β = θ
4 If the terminal arm of the principal angle lies in quadrant 4 then the related acute angle is calculated as β = θ. Any point can be represented as a point on a circle of radius r, and has the formula x 2 + y 2 = r 2. Use the image below to help you write an expression for each of the primary trigonometric functions in terms of x, y, and r for each quadrant. 4
5 Note: For any principal angle (θ greater than 90, the values of the primary trig ratios are either the same as, or the negatives of, the ratios for the related acute angle (β. The CAST rule is an easy way to remember which primary trigonometric ratios of the principal angle (θ are positive in which quadrant. Given that all trigonometric ratios are each positive and negative in two quadrants, each ratio will result in two possible principal angles. 5
6 Sine When the ratio of sin θ is positive, the principal angle is located in either quadrant or The principal angle is either: θ 1 = θ or θ 2 = β, when 0 < θ < 180. When the ratio of sin θ is negative, the principal angle is located in either quadrant or The principal angle is either: θ 3 = β or θ 4 = β, when 180 < θ < 360. **The trig inverse of most negative ratios will yield a negative angle. The related acute angle (β is equal to the absolute (positive) value of this calculated (negative) angle.** 6
7 Cosine When the ratio of cos θ is positive, the angle is either in quadrant or The principal angle is either: θ 1 = θ or θ 2 = β, when 0 < θ < 90 or 270 < θ < 360. When the ratio of cos θ is negative, the principal angle is located in either quadrant or The principal angle is either: θ 3 = β or θ 4 = β, when 90 < θ < 270. Tangent When the ratio of tan θ is positive, the angle is either in quadrant or The principal angle is either: θ 1 = θ or θ 2 = β, when 0 < θ < 90 or 180 < θ < 270. When the ratio of tan θ is negative, the angle is either in quadrant or The principal angle is either: θ 3 = β or θ 4 = β, when 90 < θ < 180 or 270 < θ <
8 Example One Determine all possible principal angles for θ where 0 θ 360. a) tan θ = 0.67 b sin θ = 0.8 April 08, 2015 Example Two Angle θ is a principal angle that lies in quadrant 2 such that 0 θ 360. Given each trigonometric ratio, i. determine the exact values of x, y, and r ii. sketch angle θ in standard position iii. determine the principal angle θ and the related acute angle β to the nearest degree 8
9 Example Three Given that sin θ =, where 0 θ 360, a) in which quadrant could the terminal arm of θ lie? b) Determine all possible primary trigonometric ratios for θ. c) Evaluate all possible values of θ to the nearest degree. Complete: p #2, 4-6, 8, 12. 9
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