www.mathportal.org Math Formulas: Trigonometry Identities Right-Triangle Definitions 1. sin α = Opposite Hypotenuse 2. cos α = Adjacent Hypotenuse 3. tan α = Opposite Adjacent 4. csc α = 1 Hypotenuse = sin α Opposite 5. sec α = 1 Hypotenuse = cos α Adjacent 6. cot α = 1 Adjacent = tan α Opposite Reduction Formulas 7. sin(−x) = − sin(x) 8. cos(−x) = cos(x) 9. sin 10. cos 11. sin 12. cos π 2 π 2 π 2 π 2 − x = cos(x) − x = sin(x) + x = cos(x) + x = − sin(x) 13. sin(π − x) = sin(x) 14. cos(π − x) = − cos(x) 15. sin(π + x) = − sin(x) 16. cos(π + x) = − cos(x) Basic Identities 17. sin2 x + cos2 x = 1 18. tan2 x + 1 = 1 cos2 x 19. cot2 x + 1 = 1 sin2 x Sum and Difference Formulas 1 www.mathportal.org 20. sin(α + β) = sin α · cos β + sin β · cos α 21. sin(α − β) = sin α · cos β − sin β · cos α 22. cos(α + β) = cos α · cos β − sin α · cos β 23. cos(α − β) = cos α · cos β + sin α · cos β 24. tan(α + β) = tan α + tan β 1 − tan α · tan β 25. tan(α − β) = tan α − tan β 1 + tan α · tan β Double Angle and Half Angle Formulas 26. sin(2 α) = 2 · sin α · cos α 27. cos(2 α) = cos2 α − sin2 α 2 tan α 1 − tan2 α r 1 − cos α =± 2 r 1 + cos α =± 2 28. tan(2 α) = 29. sin α 2 30. cos α 2 31. tan 32. α 1 − cos α sin α = = 2 sin α 1 − cos α r α 1 + cos α tan = ± 2 1 − cos α Other Useful Trig Formulas Law of sines 33. sin α sin β sin γ = = α β γ Law of cosines a2 = b2 + c2 − 2 · b · c · cos α 34. b2 = a2 + c2 − 2 · a · c · cos β c2 = a2 + b2 − 2 · a · b · cos γ Area of triangle 35. A= 1 a b sin γ 2 2