Trigonometric Identities II Double Angles. By Mr Porter

Summary of Definitions From the Mathematics Course. Hypotenuse Adjacent Opposite θ α Reciprocal Relationships Complementary Relationships Negative Angle

Students should know these identities!. Pythagorean Identities of Trigonometry. Sum and Difference of Angles in Trigonometry. For any angle θ Students should know these identities!.

Students need to know these identities!. Double Angle Identities of Trigonometry. To derive the following double angle identities, use the sum identities with α = β substitution. Note: the cos αhas other forms: Students need to know these identities!.

Example 1: If cos A = 3/5, and A is acute, find the exact value of: It is a good idea to write down the trigonometric identities Example 1: If cos A = 3/5, and A is acute, find the exact value of: a) cos 2A b) sin 2A. First, construct a right angle triangle and find all missing sides using Pythagoras’ Theorem. A 3 5 4 From the definition of sin 2A: sin 2A = 2sin A cos A This allows the calculation of the exact values of sin A and tan A. sin A = 4/5 from our triangle. From the definition of cos 2A: cos 2A = cos2 A – sin2 A sin A = 4/5 from our triangle above. We could have used cos 2A = 2 cos2 A – 1

Example 2: If tan A = 4/3, and cos B = 12/13, where 0° < B < A < 90°, find the exact value of: a) sin 2A b) tan 2B. First, construct a right angle triangles and find all missing sides using Pythagoras’ Theorem. A 3 5 4 B 12 13 There are 2 definition of tan 2B we could use. tan B = 5/12 from our triangle. This allows the calculation of the exact values of sin θ, cos θ and tan θ. From the definition of sin 2A: sin 2A = 2 sin A cos A sin A = 4/5 and cos A = 3/5 from our triangle above. It is a good idea to write down the trigonometric identities

What Quadrant is ‘A’ in? = 2nd Quad Example 3: If sin A = 3/4, where 90° < A < 180°, find the exact value of: a) cos 2A b) tan 2A. First, construct a right angle triangle and find all missing sides using Pythagoras’ Theorem. A -√7 4 3 Using the definition of tan 2A : A is in 2nd Quad tan A = -3/√7 from our triangle. This allows the calculation of the exact values of sin θ, cos θ and tan θ. From the definition of cos 2A: cos 2A = cos2 A – sin2 A cos A = -√7/4 and sin A = 3/4 from our triangle above. It is a good idea to write down the trigonometric identities

Example 4: Simplify a) b) Use the definitions sin 2θ = 2 sin θ cos θ It is a good idea to write down the trigonometric identities Example 4: Simplify a) b) Use the definitions sin 2θ = 2 sin θ cos θ tan θ = sinθ/cosθ Use the definitions sin 2θ = 2 sin θ cos θ cos 2θ =cos2 θ – sin2 θ Using the Pythagorean identity sin2 θ = 1 – cos2 θ

Example 5: Prove the identity : Using the Pythagorean identity It is a good idea to write down the trigonometric identities Make all the ‘2A’ substitutions Using the Pythagorean identity [cos2 θ + sin2 θ] = 1 Expand brackets. Factorise numerator and denominator Hence,