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Algebra 2A Unit 10B Notes: Coordinate Plane Trigonometry Name________________________ ______ Date_______ ______ Day 4: Angles of Rotation (10.2) • With right triangle trig, we examined trig functions for angles in a right triangle – angles less than 90°. • We can evaluate trig functions for other types of angles – angles 90° or larger and even negative angles. • To do this, we are going to look at angles on a coordinate plane. Angles in Standard Position: _______________________________________________________________ • ¼ rotation: ________ • ½ rotation: ________ • ¾ rotation: ________ • 1 full rotation: _______ Example #1: Draw an angle with the given measure in standard position. A. 320° B. –110° C. 990° Coterminal Angles: _______________________________________________________________________ How to find coterminal angles: Example #2: Find the measures of a positive angle and a negative angle that are coterminal with each given angle. A. θ = 65° B. θ = 410° C. θ = –150° Reference Angles: _________________________________________________________________________ How to find reference angles: Example #3: Find the measure of the reference angle for each given angle. A. θ = 135° B. θ = –105° C. θ = 325° Using Trig Functions for ANY Angle: • Create a “reference triangle” with the reference angle Example #4: P(–3, 6) is a point on the terminal side of in standard position. Find the exact value of the six trigonometric functions for . y x Practice Problems Draw an angle with the given measure in standard position. 1. –120° 2. 240° 3. –585° Find the measures of a positive angle and a negative angle that are coterminal with each angle. 4. θ = 515° 5. θ = −93° 6. θ = 162° ________________________ ________________________ ________________________ ________________________ ________________________ ________________________ Find the measures of the reference angle for each given angle. 7. θ = 211° ________________________ 10. θ = 119° ________________________ 8. θ = −55° ________________________ 11. θ = −160° ________________________ 9. θ = 555° ________________________ 12. θ = 435° ________________________ P is a point on the terminal side of in standard position. Find the exact value of the six trigonometric functions for 13. P(−5, 5) 14. P(12, –9) 15. P(−7, −5) ______________________ ______________________ ______________________ ______________________ ______________________ ______________________ ______________________ ______________________ ______________________ Day 5: Radians & The Unit Circle (10.3) Radian: Example #1: Convert each angle from degrees to radians A. 60° B. –240° C. 160° Example #2: Convert each angle from radians to degrees A. 4 3 B. 2 Unit Circle: • Yesterday, we used points on a coordinate plane to find values of trigonometric functions for any angle measure. • Today we will focus on trigonometric values for special angles. • We will use a Unit Circle, or a circle with a radius of 1, to model these special angles. On the Unit Circle… Cos = __________ Sin = __________ Tan = __________ C. 5 6 Filling in the Unit Circle: Using the Unit Circle: Practice: Find the exact value of each trigonometric function using the unit circle. 1. sin 30° ___________________________ 4. cos 330° ___________________________ 7. cos 2π 3 ___________________________ 10. sin 315° ___________________________ 13. cos 5π 3 ___________________________ 16. cos –270° ___________________________ 2. cos 45° ___________________________ 5. sin 150° ___________________________ 8. tan 5π 4 ___________________________ 11. cos 225° ___________________________ 14. sin − π 6 ___________________________ 17. sin –225° ___________________________ 3. tan 60° ___________________________ 6. tan 210° ___________________________ 9. tan 5π 6 ___________________________ 12. tan 270° ___________________________ 15. tan π ___________________________ 18. sin 2π ___________________________ Day 6: Graphs of Sine and Cosine (11.1) Warm Up: Using your Unit Circle, fill in the table below using the decimal approximation for each value. Round your values to the nearest hundredth. Graph: f(θ) = sinθ (θ , sinθ) Graph: f(θ) = cosθ (θ , cosθ) Properties of Parent Sine and Cosine Graphs: • Periodic Function: f(θ) = sinθ f(θ) = cosθ Domain Range Period Exploring Tranformations: Using your graphing calculator (in radian mode) and what you know about translations and stretches/compressions, describe the transformations to the graphs of the sine equations below: **Use ZoomTrig to get an appropriate viewing window** 1. f(θ) = 2sinθ 2. f(θ) = sin2θ 3. f(θ) = sinθ + 2 4. f(θ) = sin(θ + 2) 5. f(θ) = –sinθ Predicting Equations: Write an equation that describes the given transformation of the parent function f(θ) = sinθ, then use your graphing calculator to check your prediction. 6. Vertical translation down 1 7. Vertical stretch by a factor of 4 8. Horizontal translation right 3 9. Horizontal stretch by a factor of 2 10. Reflection over the x-axis and a vertical stretch by a factor of 3