Section 5.1 Fundamental Identities

1. Chapter 5Trigonometric Identities Section 5.1 Fundamental Identities Section 5.2 Verifying Identities Section 5.3 Cos Sum and Difference Section 5.4 Sin & Tan Sum and Dif Section 5.5 Double-Angle Identities Section 5.6 Half-Angle Identities

2. Section 5.1 Fundamental Identities • Review of basic Identities • Negative-Angle Identities • Fundamental Identities

3. y r x r y x sin θ = cos θ = tan θ = Hypotenuse = r opposite side = y θ A adjacent side = x

4. r y r x x y csc θ = sec θ = cot θ = B Hypotenuse = r opposite side = y θ A C adjacent side = x

5. The Reciprocal Identities sin £ = csc £ = cos £ = sec £ = tan £ = cot £ = 1 csc £ 1 sin £ 1 sec £ 1 cos £ 1 cot £ 1 tan £

6. The quotient Identities tan £ = = cot £ = = sin £ cos £ y x x y cos £ sin £

7. The Negative-Angle Identities sin(-£) = - sin £ cos(-£) = cos £ tan(-£) = - tan £

8. x2 + y2 = r2 • or • cos2θ + sin2θ = 1 r2 r2 r2 r θ y x This is our first Pythagorean identity

9. r θ y x Pythagorean identities • cos2θ + sin2θ 1 • or • 1 + tan2θ = sec2θ • or • tan2θ + 1 = sec2θ = cos2θ cos2θ cos2θ

10. r θ y x Pythagorean identities • cos2θ + sin2θ 1 • or • cot2θ + 1 = csc2θ • or • 1 + cot2θ = csc2θ = sin2θ sin2θ sin2θ

11. Section 5.2 Verifying Identities • Verify Identities by Working with One Side • Verify Identities by Working with Two Sides

12. Hints for Verifying Identities • Learn the fundamental identities and their equivalent forms. • Simplify using sin and cos. • Keep in mind the basic algebra applies to trig functions. • You can always go down to x, y, and r

13. Section 5.3 Cos Sum & Difference • Difference Identity for Cosine • Sum Identity for Cosine • Co-function Identities • Applying the Sum and Difference Identities

14. Cosine of the Sum or Difference cos(A + B) = cos A cos B – sin A sin B cos(A - B) = cos A cos B + sin A sin B

15. Co-function Identities sin (90à - £à) = cos £à cos (90à - £à) = sin £à tan (90à - £à) = cot £à csc (90à - £à) = sec £à sec (90à - £à) = csc £à cot (90à - £à) = tan £à

16. Section 5.4 Sine and TangentSum and Difference Identities • Sum Identity for Sine • Difference Identity for Sine • Applying the Sum and Difference Identities for Sine

17. Sine of the Sum or Difference sin(A + B) = sin A cos B + cos A sin B sin(A - B) = sin A cos B - cos A sin B

18. Tangent of the Sum or Difference tan (A + B) = tan (A - B) = tan A + tan B 1 – tan A tan B tan A - tan B 1 + tan A tan B

19. Section 5.5 Double-Angle Identities • Double-Angle Identities • Verifying Identities with Double Angels • Applying Double-Angle Identities

20. Double-Angle Identity Cosine cos(2A) = cos(A+A) = cos A cos A – sin A sin A = cos2 A – sin2 A or cos(2A) = cos2 A – sin2 A = (1 - sin2 A) – sin2 A = 1 - 2sin2 A or 2cos2 A - 1

21. Double-Angle Identity Sine sin(2A) = sin(A+A) = sin A cos A + cos A sin A = 2sin A cos A

22. Double-Angle Identity Tangent tan 2A = tan (A + A) = = tan A + tan A 1 – tan A tan A 2 tan A 1 – tan2A

23. Section 5.6 Half-Angle Identities • Half-Angel Identities • Using the Half-Angle Identities

24. Half-Angle Identity Sine cos 2A = 1 - 2sin2 A -cos 2A -cos 2A 0 = 1 - 2sin2 A – cos 2A - 2sin2 A -2sin2 A -2sin2 A = 1 – cos 2A sin2 A = (cos 2A – 1) 2

25. Half-Angle Identity Sine (cont.) sin A = sin = ‘ñ ‘ñ 1 – cos 2A 2 1 – cos A 2 A 2

26. Half-Angle Identity Cosine cos 2A = 2cos2 A - 1 +1 +1 cos 2A + 1 = 2cos2 A 2cos2 A = 1 + cos 2A cos2 A = (1 + cos 2A) 2

27. Half –Angle Identity Cosine (cont.) cos A = cos = ‘ñ ‘ñ 1 + cos 2A 2 1 + cos A 2 A 2

28. Half-Angle Identity Tangent tan = = tan = ñ ‘ñ 1 + cos A 2 A 2 1 – cos A 2 sin A 2 A 2 cos ‘ñ A 2 1 – cos A 1 + cos A

29. Half-Angle Identity Tangent (cont) tan = = tan = = A 2 A 2 A 2 sin 2sin cos A 2 A 2 cos A 2 2cos2 ( ) A 2 sin 2 sin A A 2 ( ) A 2 1 + 2cos 1 + cos A

30. Half-Angle Identity Tangent (cont) Using the other formula we get: tan = 1 - cos A A 2 sin A