Derive and use compound angles to solve problems

Derivation of sin(α + β)

Consider two angles α and β on a unit circle. Using geometry, we can show that:

$\sin(\alpha + \beta) = \sin\alpha\cos\beta + \cos\alpha\sin\beta$

This comes from the fact that when we add angles, the vertical component combines the sine of one angle with the cosine of the other.

Derivation of cos(α + β)

Similarly, using the distance formula between points on the unit circle:

$\cos(\alpha + \beta) = \cos\alpha\cos\beta - \sin\alpha\sin\beta$

Formulas for Differences

Using the fact that cosine is an even function (cos(-θ) = cos θ) and sine is an odd function (sin(-θ) = -sin θ), we can derive:

$\sin(\alpha - \beta) = \sin\alpha\cos\beta - \cos\alpha\sin\beta$

$\cos(\alpha - \beta) = \cos\alpha\cos\beta + \sin\alpha\sin\beta$

Example 1: Find sin 75°

Solution:

Write 75° as a sum of special angles: 75° = 45° + 30°

$\sin 75° = \sin(45° + 30°)$ $= \sin 45°\cos 30° + \cos 45°\sin 30°$ $= \frac{\sqrt{2}}{2} \times \frac{\sqrt{3}}{2} + \frac{\sqrt{2}}{2} \times \frac{1}{2}$ $= \frac{\sqrt{6}}{4} + \frac{\sqrt{2}}{4}$ $\sin 75° = \frac{\sqrt{6} + \sqrt{2}}{4}$

Example 2: Find cos 15°

Solution:

Write 15° as a difference of special angles: 15° = 45° - 30°

$\cos 15° = \cos(45° - 30°)$ $= \cos 45°\cos 30° + \sin 45°\sin 30°$ $= \frac{\sqrt{2}}{2} \times \frac{\sqrt{3}}{2} + \frac{\sqrt{2}}{2} \times \frac{1}{2}$ $= \frac{\sqrt{6}}{4} + \frac{\sqrt{2}}{4}$ $\cos 15° = \frac{\sqrt{6} + \sqrt{2}}{4}$

Example 3: Simplify sin 100° cos 50° + cos 100° sin 50°

Solution:

This expression matches the form sin(α + β) = sinα cosβ + cosα sinβ, where α = 100° and β = 50°.

$\sin 100°\cos 50° + \cos 100°\sin 50° = \sin(100° + 50°)$ $= \sin 150° = \frac{1}{2}$

• Always express the given angle as a sum or difference of special angles (30°, 45°, 60°)
• Identify which formula to use based on the operation (addition or subtraction) and the trigonometric function
• Memorize all four formulas as they are frequently used together
• Check your answer using a calculator to verify

In Tanzania, construction engineers use compound angle formulas when calculating diagonal bracing for buildings or bridges. For example, when installing a roof truss on a building in Dar es Salaam, a surveyor might need to find the exact angle where two beams meet. If one beam is angled at 45° from horizontal and another at 30°, the compound angle formula helps calculate the precise measurement for cutting the beams, ensuring the structure is safe and stable.