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Name ________________________________________ Date __________________ Class__________________ Reteach LESSON 13-3 The Unit Circle Radians are a real number measure of rotation. To convert between radians and degrees, use the following identity. π radians = 180° To convert from radians to degrees, solve the identity for 1 radian. 180° 1 radian = π radians To convert from degrees to radians, solve the identity for 1 degree. π radians 1 degree = 180° Convert 60° to radians. ⎛ π radians ⎞ π ⎟ = radians 60° = 60° ⎜ ⎜ 180° 3 ⎟ 3 ⎝ ⎠ Convert Use dimensional analysis to help. Notice that the degrees cancel so the remaining unit is radians. 5π radians to degrees. 4 ⎛ 5π 5π radians = ⎜ 4 ⎝ 4 ⎞ ⎛ 45 180° radians ⎟ ⎜ ⎠ ⎝ π radians ⎞ ⎟ = 225° ⎠ The radians cancel so the remaining unit is degrees. Convert each measure from degrees to radians. 1. −45° 2. 150° ⎛ π radians ⎞ −45° = −45° ⎜ ⎟ ⎝ 180° ⎠ ⎛ π radians ⎞ 150° = 150° ⎜ ⎟ ⎝ 180° ⎠ _____________________________________ _________________________________ 4. −120° 3. 210° _____________________________________ _________________________________ Convert each measure from radians to degrees. 5. 4π radians 3 6. − 4π ⎛ 4π ⎞ ⎛ 180° ⎞ radians = ⎜ radians ⎟ ⎜ ⎟ 3 ⎝ 3 ⎠ ⎝ π radians ⎠ − _____________________________________ 7. π 6 radians 8. _____________________________________ 3π radians 2 3π ⎛ 3π ⎞ ⎛ 180° ⎞ radians = ⎜ − radians ⎟ ⎜ ⎟ 2 ⎝ 2 ⎠ ⎝ π radians ⎠ _________________________________ 5π radians 3 _________________________________ Original content Copyright © by Holt McDougal. Additions and changes to the original content are the responsibility of the instructor. 13-22 Holt Algebra 2 Name ________________________________________ Date __________________ Class__________________ LESSON 13-3 Reteach The Unit Circle (continued) Use a reference angle to find the exact value of the sine, cosine, and tangent of an angle in any quadrant. Find the value of the trigonometric functions of 150°. Step 1 Sketch the angle. Find the measure of the reference angle. 180° − 150° = 30° Step 2 Draw the triangle with the reference angle. Label the sides with their lengths. Step 3 Find the sine, cosine, and tangent of 30°. 1 sin 30° = 2 cos 30° = tan 30° = Step 4 3 2 1 3 The diagram shows the quadrants in which each trigonometric function is positive. Adjust the signs for 150°. sin 150° = 1 2 cos 150° = − tan 150° = − 3 2 1 3 Complete to find the exact value of the sine, cosine, and tangent of 315°. 9. Find the measure of the reference angle. ___________________________________ 10. Find the sine, cosine, and tangent of the reference angle. _______________________________ _______________________________ _______________________________ 11. Adjust the signs to find the sine, cosine, and tangent of 315°. _______________________________ _______________________________ _______________________________ Original content Copyright © by Holt McDougal. Additions and changes to the original content are the responsibility of the instructor. 13-23 Holt Algebra 2 9. 5π radians 3 10. − π 18 7. radians 11. 320° 9. ⎛1 3⎞ 12. a. ⎜ , ⎜ 2 2 ⎟⎟ ⎝ ⎠ 3 b. 2 1 2 14. 1 15. 0 16. − 1 2 18. 3 13. 17. − 3 2 35π radians 18 π 12. − 7π radians 12 1 3 3 13. − ; ;− 2 2 3 14. − 2 2 ; ; −1 2 2 15. − 3 1 ;− ; 3 2 2 16. 1 3 3 ;− ;− 2 2 3 17. − 2 2 ;− ;1 2 2 18. 3 1 ;− ;− 3 2 2 19. 628 ft 19. 43π radians 36 2 2 ; ;1 2 2 1. 75° 2. 3. −290° 4. −π radians 5. 300° 6. 210° 23. − 8. 54° 25. 138 ft 7. 20π radians 9 9. 7π radians 36 11. 1 13. − 2 2 10. − 10. 234° 37π radians 30 11. Practice B radians 15 8. 63° 1 3 3 21. − ; − ;− 2 2 3 3 1 ; ;− 3 2 2 20. 0; −1; 0 22. − 2 2 ; ; −1 2 2 2 2 ;− ; −1 2 2 24. Reteach 1 2 1. − π 4 radians 2. 5π radians 6 12. − 3 3 3. 14. − 2 2 5. 240° 6. −270° 7. 30° 8. 300° 15. 3 1 3 3 ;− 16. ; − 2 2 3 17. 2 2 ;− ; −1 2 2 18. 20. − 2 2 ;− ;1 21. − 2 2 22. 2073 mi 2 2 2 cos 45° = 2 tan 45° = 1 2 2 2 cos 315° = 2 tan 315° = −1 11. sin 315° = − Practice C 5π radians 2 2. 3. 50° 4. − 5. 315° 6. −330° 2π radians 3 10. sin 45° = 3 1 ;− ; 3 2 2 1. −270° 4. − 9. 45° 3 1 ; ; 3 2 2 1 3 3 19. − ; ;− 2 2 3 7π radians 6 10π radians 9 Challenge 1. 6080 ft 2. 1,600,921 mi; 66,705 mi/h Original content Copyright © by Holt McDougal. Additions and changes to the original content are the responsibility of the instructor. A57 Holt Algebra 2