Deriving the Double Angle Formulas

The double angle formulas are derived from the compound angle formulas by setting B = A. They are essential tools for simplifying expressions and solving equations in Year 12 Mathematics.

sin 2A

sin 2A = 2 sinA cosA

Derived from sin(A + A) = sinA cosA + cosA sinA = 2 sinA cosA

cos 2A (Three Forms)

cos 2A = cos^2A - sin^2A

cos 2A = 2cos^2A - 1

(using sin^2A = 1 - cos^2A)

cos 2A = 1 - 2sin^2A

(using cos^2A = 1 - sin^2A)

tan 2A

tan 2A = 2 tanA/1 - tan^2A

Choosing the Right Form of cos 2A

The three forms of cos 2A are all equivalent, but choosing the right one simplifies your work:

cos^2A - sin^2A

Use when both sinA and cosA are known or the expression contains both.

2cos^2A - 1

Use when only cosA is known, or to eliminate sin from an expression.

1 - 2sin^2A

Use when only sinA is known, or to eliminate cos from an expression.

Half-Angle Results

By rearranging the double angle formulas for cos 2A, we obtain the half-angle results, which are useful for integration and simplification:

cos^2A = 1 + cos 2A/2

From cos 2A = 2cos^2A - 1

sin^2A = 1 - cos 2A/2

From cos 2A = 1 - 2sin^2A

Application: These are essential for integrating sin^2(x) and cos^2(x), which cannot be integrated directly but can be integrated once rewritten using these identities.

Key Vocabulary

Double Angle

An expression involving 2A, where the angle is doubled. E.g., sin 2A, cos 2A.

Half-Angle Identity

A rearrangement of a double angle formula that expresses sin^2 or cos^2 in terms of cos 2A.

Power Reduction

Using half-angle identities to rewrite squared trig functions as first-power expressions for integration.

Equivalent Forms

Different algebraic expressions that have the same value for all valid inputs, like the three forms of cos 2A.

Worked Examples

1

If sinA = 3/5 and A is in the first quadrant, find sin 2A and cos 2A.

Step 1: cosA = 4/5 (from 3-4-5 triangle, Q1 so positive)

Step 2: sin 2A = 2 sinA cosA = 2(3/5)(4/5) = 24/25

Step 3: cos 2A = cos^2A - sin^2A = 16/25 - 9/25 = 7/25

Answer: sin 2A = 24/25, cos 2A = 7/25

2

Show that (sinA + cosA)^2 = 1 + sin 2A.

LHS: (sinA + cosA)^2 = sin^2A + 2sinAcosA + cos^2A

= (sin^2A + cos^2A) + 2sinAcosA

= 1 + 2sinAcosA

= 1 + sin 2A

= RHS (proven)

3

Express cos^2(3x) without a squared term (for integration).

Step 1: Use the half-angle identity: cos^2A = (1 + cos 2A)/2

Step 2: Let A = 3x: cos^2(3x) = (1 + cos 6x)/2

Answer: cos^2(3x) = (1 + cos 6x) / 2

Knowledge Check

Select the correct answer for each question. Click "Check Answer" to see if you are right.

Question 1

What is sin 2A equal to?

2 sinA 2 sinA cosA sin^2 A sinA cosA

Question 2

How many different forms does cos 2A have?

1 2 3 4

Question 3

If cosA = 5/13 (A in Q1), what is cos 2A?

10/13 119/169 -1/169 -119/169

Question 4

sin^2(x) can be written as:

(1 - cos 2x) / 2 (1 + cos 2x) / 2 1 - cos x sin(2x) / 2

Question 5

Simplify: 2sin(15°)cos(15°).

sin(15°) sin(30°) = 1/2 cos(30°) = √3/2 2sin(30°) = 1

Double Angle Formulas