8.1 Q-4

Exercise 8.1 — Question 4

This question applies the fundamental law of trigonometry (sine and cosine addition/subtraction formulas) to evaluate or simplify trigonometric expressions.

Key Identities Used

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

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

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

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

Strategy

1. Identify the pattern — look at the sign between terms and whether the functions are sine or cosine.
2. Match to the correct identity — e.g., $\sin A\cos B-\cos A\sin B=\sin (A-B)$.
3. Substitute and simplify — combine the angles and evaluate if the result is a standard angle.

Worked Example

Simplify: $\sin 138^{\circ }\cos 46^{\circ }-\cos 138^{\circ }\sin 46^{\circ }$

Step 1: Recognise the pattern matches $\sin (\alpha -\beta )=\sin \alpha \cos \beta -\cos \alpha \sin \beta$.

Step 2: Apply with $\alpha =138^{\circ }$, $\beta =46^{\circ }$:

$\sin (138^{\circ }-46^{\circ })=\sin 92^{\circ }$

Step 3: $\sin 92^{\circ }=\sin (90^{\circ }+2^{\circ })=\cos 2^{\circ }$