NAME _____________________________________________ DATE ____________________________ PERIOD _____________ 5-4 Practice Sum and Difference Identities Find the exact value of each trigonometric expression. 5𝜋 1. cos 12 2. sin (–165°) 4. csc 915° 5. tan (− 7. cos (–15°) 8. tan 15° 9. cos (− 11. sin 20° cos 10° + cos 20° sin 10° 12. 10. cos 11𝜋 12 3. tan 345° 7𝜋 ) 12 6. sec 𝜋 12 7𝜋 ) 12 π 5𝜋 + tan 9 36 π 5𝜋 1 − tan tan 9 36 tan Simplify each expression. 13. cos 3𝜋 2 cos π – sin 3𝜋 2 sin π 15. cos 70° cos 20° – sin 70° sin 20° tan 30° − tan 𝑥 14. 1 + tan 30° tan 𝑥 𝜋 𝜋 16. sin 12 cos y – sin y cos 12 Write each trigonometric expression as an algebraic expression. 1 2 17. sin (arccos x + arcsin x) 18. cos (arccos − arcsin 𝑥) 19. cos (arcsin x + arccos x) 20. sin (arctan 3 – arcsin x) NAME _____________________________________________ DATE ____________________________ PERIOD _____________ Verify each cofunction or reduction identity. 21. sin (360° + θ) = sin θ 22. cos (180° – θ) = –cos θ Find the solutions to each expression on the interval [0, 2π). 5π 5π 23. cos ( 4 + 𝑥) + sin ( 4 − 𝑥) = 0 𝜋 𝜋 25. cos ( 4 − 𝑥) – sin ( 4 − 𝑥) = –1 3𝜋 3𝜋 27. cos ( 2 + 𝑥) + sin ( 2 − 𝑥) = 0 2π 2π 24. sin ( 3 − 𝑥) + sin ( 3 + 𝑥) = 0 26. sin (π + x) + sin (π + x) = 1 28. tan (π – x) + tan (π – x) = –2 29. Sound waves can be modeled by the equations of the form 𝑦1 = 20 sin (3x + θ). A wave traveling in the opposite direction can be modeled by 𝑦2 = 20 sin (3x – θ). Show that 𝑦1 + 𝑦2 = 40 sin 3x cos θ. 30. ENGINEERING Two highways branch off each other at an angle of 75°. An engineer uses tan 75° to determine the height of an exit ramp at a particular point. Find the exact value of tan 75°. 31. ELECTRICITY The current I in amperes in an alternating current at time t in seconds can be found with the formula 7𝜋 I = 30 sin (50𝜋𝑡 − 3 ). Rewrite the formula in terms of one or more functions of 50πt. 32. TEMPERATURE A city’s average daily high temperature can be modeled by y = 14.33 sin (0.56x – 2.44) + 60.79, where x = 1 corresponds to January. Rewrite the formula using the Sine Difference Identity.