Unlocking Trigonometric Identities: Double and Compound Angles

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Unlocking Trigonometric Identities: Double and Compound Angles

Week 7 - Engineering Maths

Learning Objective

At the end of the lesson, you will understand double angle and compound angle formulae, and convert products to sums and differences.

What do you already know about trigonometric identities?

Understanding Double Angle Formulae

Double angle formulae:

sin(2A) = 2sin(A)cos(A)

cos(2A) = cos²(A) - sin²(A)

tan(2A) = 2tan(A)/(1-tan²(A))

What is sin(2θ) in terms of sin and cos?

A

sin(θ) + cos(θ)

B

sin²(θ) - cos²(θ)

C

2sin²(θ)

D

2sin(θ)cos(θ)

Compound Angle Formulae

Compound angles: sin(A±B) = sin(A)cos(B) ± cos(A)sin(B), cos(A±B) = cos(A)cos(B) ∓ sin(A)sin(B).

A compound angle is formed when two angles are combined, typically in the context of trigonometry. It usually refers to the sum or difference of two angles. The most common formulas for compound angles are the sine and cosine formulas, which are expressed as follows:

sin(A+B) = sin(A)cos(B) + cos(A)sin(B)

sin(A-B) = sin(A)cos(B) - cos(A)sin(B)

cos(A+B) = cos(A)cos(B) - sin(A)sin(B)

cos(A-B) = cos(A)cos(B) + sin(A)sin(B)

What is cos(A + B) equal to?

A

sin(A + B)

B

cos(A)cos(B) - sin(A)sin(B)

C

cos(A + B)

D

sin(A)cos(B) + cos(A)sin(B)

Using Compound Angle Formulae

Solve equations like sin(75°) using sin(45°+30°) to simplify computations.

Example to try

Let’s find cos⁡(105^∘) using a different pair of angles and confirm that the compound angle formula is correct.

cos(A+B) = cos(A)cos(B) - sin(A)sin(B)

Example to try

Let’s find cos⁡(105^∘) using a different pair of angles.

Using the cosine angle formula:

cos⁡(105^∘) = cos(60^∘ + 45^∘) = cos(60^∘)cos(45^∘) - sin(60^∘)sin(45^∘)

cos⁡(105^∘) = cos(60^∘ + 45^∘) = 0.5 x 0.7071 - 0.8660 x 0.7071

−0.2588 = −0.2588

This confirms that the formula is correct.

How are you finding this topic so far?

A

Good. I am happy with this topic.

B

Fairly good but will need to review it.

C

I am finding it a little difficult

D

Not good. I don't understand it.

Products to Sums and Differences

Formulae:

sin(A)sin(B) = 1/2[cos(A-B) - cos(A+B)]

cos(A)cos(B) = 1/2[cos(A-B) + cos(A+B)]

sin(A)cos(B)=1/2[sin(A+B)+sin(A−B)]

cos(A)sin(B)=1/2​[sin(A+B)−sin(A−B)]

Example

Simplify the expression cos⁡(5x)cos⁡(3x) using the product-to-sum identity.

Example

Simplify the expression cos⁡(5x)cos⁡(3x) using the product-to-sum identity.

Example

Example to try

Simplify the expression sin⁡(4x)cos⁡(2x) using the product-to-sum formula

Write down 3 things you learned in this lesson.

Write down 2 things you want to know more about.