1. Trigonometric Identities II Double Angles.
By Mr Porter

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

3. 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!.

4. 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!.

5. 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

6. 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

7. 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

8. 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

9. 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,