Compound Angle Identities Revision Notes for Grade 12 NSC Matric Mathematics

Compound Angle Identities

Understanding compound angles

Compound angles are expressions that involve the sum or difference of two angles, such as $(α + β)$ or $(α - β)$ . These appear frequently in trigonometry problems and require special formulae to solve.

chatImportant

A critical point to understand is that the distributive law does not apply to trigonometric ratios. This means that:

$\cos(α - β) ≠ \cos α - \cos β$

This is a common mistake that many students make.

Let's see why this doesn't work with a simple example:

Consider $\cos(180° - 120°)$ :

Using the incorrect approach: $\cos 180° - \cos 120° = -1 - (-\frac{1}{2}) = -\frac{1}{2}$
The correct answer is actually: $\cos(60°) = \frac{1}{2}$

The answers are completely different! This shows why we need special formulae for compound angles.

Deriving the compound angle formula for cos(α - β)

To understand where the compound angle formulae come from, we can use the unit circle and the distance formula.

infoNote

Understanding the Geometric Approach

The geometric derivation helps us see why the compound angle formulae work. By using the unit circle, we can connect the algebraic formulas to visual geometric relationships.

Consider the unit circle (radius = 1) with two points:

$L(\cos β, \sin β)$
$K(\cos α, \sin α)$

Using the distance formula: $d = \sqrt{(x₁ - x₂)² + (y₁ - y₂)²}$

We can calculate the distance KL in two ways:

Method 1: Using coordinates

$KL² = (\cos α - \cos β)² + (\sin α - \sin β)²$
$= \cos²α - 2\cos α \cos β + \cos²β + \sin²α - 2\sin α \sin β + \sin²β$
$= (\cos²α + \sin²α) + (\cos²β + \sin²β) - 2(\cos α \cos β + \sin α \sin β)$
$= 1 + 1 - 2(\cos α \cos β + \sin α \sin β)$
$= 2 - 2(\cos α \cos β + \sin α \sin β)$

Method 2: Using the cosine rule

$KL² = 1² + 1² - 2(1)(1)\cos(α - β)$
$= 2 - 2\cos(α - β)$

Since both expressions equal $KL²$ , we can equate them:

$2 - 2\cos(α - β) = 2 - 2(\cos α \cos β + \sin α \sin β)$

Simplifying: $\cos(α - β) = \cos α \cos β + \sin α \sin β$

The four compound angle formulae

chatImportant

Essential Compound Angle Identities

Here are the four fundamental formulae that you must memorise:

$\cos(α - β) = \cos α \cos β + \sin α \sin β$
$\cos(α + β) = \cos α \cos β - \sin α \sin β$
$\sin(α - β) = \sin α \cos β - \cos α \sin β$
$\sin(α + β) = \sin α \cos β + \cos α \sin β$

infoNote

Pattern Recognition Tip

Notice the pattern in the signs:

Cosine formulae: The sign matches the angle operation (+ stays +, - stays -)
Sine formulae: The sign is opposite to the angle operation (+ becomes -, - becomes +)

Worked examples

lightbulbExample

Worked Example 1: Deriving cos(α + β)

Question: Derive $\cos(α + β)$ in terms of trigonometric ratios of $α$ and $β$ .

Solution:

Step 1: Use the compound angle formula for $\cos(α - β)$

We start with $\cos(α + β) = \cos(α - (-β))$

Step 2: Apply the formula and use the properties of negative angles

$\cos(α + β) = \cos α \cos(-β) + \sin α \sin(-β)$

Since $\cos(-β) = \cos β$ and $\sin(-β) = -\sin β$ :

$\cos(α + β) = \cos α \cos β + \sin α(-\sin β)$
$\cos(α + β) = \cos α \cos β - \sin α \sin β$

lightbulbExample

Worked Example 2: Proving an identity using special angles

Question: Prove that $\sin 75° = \frac{\sqrt{2}(\sqrt{3}+1)}{4}$ without using a calculator.

Solution:

Step 1: Express $75°$ as a sum of special angles

$75° = 45° + 30°$

Step 2: Apply the compound angle formula for $\sin(α + β)$

$\sin 75° = \sin(45° + 30°)$
$= \sin 45° \cos 30° + \cos 45° \sin 30°$

Step 3: Substitute known values for special angles

$\sin 75° = \left(\frac{1}{\sqrt{2}}\right)\left(\frac{\sqrt{3}}{2}\right) + \left(\frac{1}{\sqrt{2}}\right)\left(\frac{1}{2}\right)$
$= \frac{\sqrt{3}}{2\sqrt{2}} + \frac{1}{2\sqrt{2}}$
$= \frac{\sqrt{3} + 1}{2\sqrt{2}}$

Step 4: Rationalise the denominator

$\sin 75° = \frac{\sqrt{3} + 1}{2\sqrt{2}} × \frac{\sqrt{2}}{\sqrt{2}}$
$= \frac{\sqrt{2}(\sqrt{3} + 1)}{4}$

Therefore, $\sin 75° = \frac{\sqrt{2}(\sqrt{3}+1)}{4}$ ✓

lightbulbExample

Worked Example 3: Evaluating expressions using compound angles

Question: Determine the value of $\cos 65° \cos 35° + \cos 25° \cos 55°$ without using a calculator.

Solution:

Step 1: Use co-functions to simplify

Notice that $\cos 25° = \sin 65°$ and $\cos 55° = \sin 35°$

So our expression becomes: $\cos 65° \cos 35° + \sin 65° \sin 35°$

Step 2: Recognise the compound angle pattern

This matches the formula for $\cos(α - β) = \cos α \cos β + \sin α \sin β$

Therefore: $\cos 65° \cos 35° + \sin 65° \sin 35° = \cos(65° - 35°)$

Step 3: Simplify

$\cos(65° - 35°) = \cos 30° = \frac{\sqrt{3}}{2}$

Therefore, $\cos 65° \cos 35° + \cos 25° \cos 55° = \frac{\sqrt{3}}{2}$

Exam tips and techniques

infoNote

When to use compound angle formulae:

When you see expressions involving sums or differences of angles
When you need to prove trigonometric identities
When evaluating exact values using special angles
When simplifying complex trigonometric expressions

Common exam techniques:

Look for special angle combinations: $75° = 45° + 30°$ , $15° = 45° - 30°$
Use co-functions: $\cos(90° - θ) = \sin θ$
Check your work: When possible, verify answers using a calculator
Pattern recognition: Learn to spot when expressions match compound angle formulae

chatImportant

Common mistakes to avoid:

Don't apply the distributive law to trigonometric functions
Watch your signs carefully - they're different for sine and cosine formulae
Remember that angles must be in the same units (degrees or radians)
Always simplify fully - don't leave answers with unnecessary radicals in denominators

bookmarkSummary

Key Points to Remember:

Compound angle formulae are essential - you cannot solve compound angle problems without them
The distributive law does not work for trigonometric ratios: $\cos(α - β) ≠ \cos α - \cos β$
Learn the sign patterns: cosine formulae keep the same sign, sine formulae flip the sign
Practice with special angles $(30°, 45°, 60°)$ to build confidence with exact values
Always check your work when calculators are allowed to verify your algebraic solutions

Compound Angle Identities (Grade 12 NSC Matric Mathematics): Revision Notes