Warm_Up #1

1. Warm_Up #1 Determine all six simplified trigonometric functions of ө. 13-2 - Angles of Rotation

2. PAGE 933 3-25 odd 13-2 - Angles of Rotation

3. Determine all six simplified trigonometric functions of ө. Warm_Up #1 • sin ө= cos ө = tan ө = • csc ө = sec ө = cot ө= 10/29/2014 3:28 PM 13.2 - Angles of Rotation 13-2 - Angles of Rotation 3

4. Angles of Rotations Section13.2 13-2 - Angles of Rotation

5. Definitions Angle of rotationis formed by two rays with a common endpoint (called the vertex). One ray is called theinitialside. The other ray is called the terminal side. The measure of the angle is determined by the amount and direction of rotation from the initial side to the terminal side. y terminal side • The angle measure is positive if the rotation is counterclockwise, and negative if the rotation is clockwise. • A full revolution (counterclockwise) corresponds to 360°. x vertex initial side 13-2 - Angles of Rotation

6. Example 1 Draw 210° with the given measure in standard position. Then determine in which quadrant the terminal side lies. • Please label the axis when drawing angles • Must draw the angle and its arrow (to indicate both the direction) receive full credit 210° Terminal side is in Quadrant III 13-2 - Angles of Rotation

7. Example 2 Draw –45°with the given measure in standard position. Then determine in which quadrant the terminal side lies. –45° Terminal side is in Quadrant IV 13-2 - Angles of Rotation

8. Example 3 Draw 510° with the given measure in standard position. Then determine in which quadrant the terminal side lies. • Get the actual angle 510° - 360° = 150° • So the terminal side makes 1 complete revolution and continues another 150°. 150° 510° Terminal side is in Quadrant II • 510° and 150° are called coterminal (their terminal sides coincide). • An angle coterminal with a given angle can be found by adding or subtracting multiples of 360°. 13-2 - Angles of Rotation

9. Coterminal Angles Coterminalangles are angles in standard position with the same terminal side • To determine the coterminalangles, add and/or subtract 360° • CoterminalAngles can be negative Find the measures of a positive and negative angles that are coterminal with ө=40° Magic Number: 360° 40° 13-2 - Angles of Rotation

10. Your Turn Find the measures of a positive and negative angles that are coterminal with ө =380° 13-2 - Angles of Rotation

11. Warm_Up #2 Draw the angle of: 1) 740° 2) –100 ° Find the measure of the positive and negative coterminal angle for: 3) 120° 4) –150° 13-2 - Angles of Rotation

12. 5 4 3 Review Find all trig ratios of ө • Sine ө= Cosine ө = Tangent ө = • SIN COS TAN • Cosecant ө = Secant ө = Cotangent ө= • CSC SEC COT 13-2 - Angles of Rotation

13. 5 4 3 Review Find all trig ratios of ө • Sine ө= Cosine ө = Tangent ө = • SIN COS TAN • Cosecant ө = Secant ө = Cotangent ө= • CSC SEC COT 13-2 - Angles of Rotation

14. Equation in Standard Position For ө be an angle in standard position with any point (x, y)… • Sine ө= Cosine ө = Tangent ө = • SIN COS TAN • Cosecant ө = Secant ө = Cotangent ө= • CSC SEC COT 13-2 - Angles of Rotation

15. Equation in Standard Position For ө be an angle in standard position with any point (x, y)… 13-2 - Angles of Rotation

16. Equation in Standard Position For ө be an angle in standard position with any point (x, y)… S A When ALL trig functions are positive When SIN is positive Quadrant II (– , +) Quadrant I (+, +) C T When TAN is positive When COS is positive Quadrant III (–, –) Quadrant IV (+, –) “All Students Take Calculus” 13-2 - Angles of Rotation

17. Steps Start the point at the origin Identify and plot the point onto the coordinate plane Determine the missing side using the radius equation Use Trigonometric Functions to solve 13-2 - Angles of Rotation

18. Example 4 Let (3, 4) be a point on the terminal side of ө. Determine the value of the six trigonometric functions for ө. 4 3 13-2 - Angles of Rotation

19. Example 4 Let (3, 4) be a point on the terminal side of ө. Determine the value of the six trigonometric functions for ө. 13-2 - Angles of Rotation

20. Example 4 Let (3, 4) be a point on the terminal side of ө. Determine the value of the six trigonometric functions for ө. • Sine ө= Cosine ө = Tangent ө = • SIN COS TAN • Cosecant ө = Secant ө = Cotangent ө= • CSC SEC COT 13-2 - Angles of Rotation

21. Example 5 Let (–3, 4) be a point on the terminal side of ө. Determine the value of the six trigonometric functions for ө. • Sine ө= Cosine ө = Tangent ө = • SIN COS TAN • Cosecant ө = Secant ө = Cotangent ө= • CSC SEC COT 13-2 - Angles of Rotation

22. Your Turn Let (1, –1) be a point on the terminal side of ө. Determine the value of the six trigonometric functions for ө. • Sine ө= Cosine ө = Tangent ө = • SIN COS TAN • Cosecant ө = Secant ө = Cotangent ө= • CSC SEC COT 13-2 - Angles of Rotation

23. Reference Angles • Reference angles is a positive acute angle formed by the terminal side of ө and the x-axis. They are viewed as linear pairs. • No reference trigonometric values of measure are greater than or equal to 90° or less than or equal to 0° 13-2 - Angles of Rotation

24. Reference Angles

25. 135° Example 5 Given θ = 135°, determine the reference angle for each given angle. • θ = 135° 45° The measure of the reference angle is 45°. 13-2 - Angles of Rotation

26. –105° Example 6 Given θ = –105°, determine the reference angle for each given angle. • θ = –105° 75° The measure of the reference angle is 75°. 13-2 - Angles of Rotation

27. Your Turn Find the measure of the reference angle for each given angle. θ = 212° θ = 300° The measure of the reference angle is 32°. The measure of the reference angle is 60°. 13-2 - Angles of Rotation

28. Assignment Finish Worksheet and Pg 939 (43-49 odd) 13-2 - Angles of Rotation