Exercise 8.1 — Q1: Fundamental Law of Trigonometry

The Fundamental Law of Trigonometry

The fundamental law of trigonometry (also called the cosine difference formula) states:

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

This is the foundation from which all other sum and difference identities are derived.

---

Sum and Difference Formulas

Sine Formulas

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

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

Cosine Formulas

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

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

Tangent Formulas

$\tan (\alpha +\beta )=\frac{\tan \alpha +\tan \beta }{1-\tan \alpha \tan \beta }$

$\tan (\alpha -\beta )=\frac{\tan \alpha -\tan \beta }{1+\tan \alpha \tan \beta }$

---

Worked Examples

Example 1

Simplify: $\sin 75^{\circ }\cos 15^{\circ }+\cos 75^{\circ }\sin 15^{\circ }$

Solution: This matches the pattern $\sin \alpha \cos \beta +\cos \alpha \sin \beta =\sin (\alpha +\beta )$

$=\sin (75^{\circ }+15^{\circ })=\sin 90^{\circ }=1$

---

Example 2

Simplify: $\cos 80^{\circ }\cos 20^{\circ }+\sin 80^{\circ }\sin 20^{\circ }$

Solution: This matches $\cos \alpha \cos \beta +\sin \alpha \sin \beta =\cos (\alpha -\beta )$

$=\cos (80^{\circ }-20^{\circ })=\cos 60^{\circ }=\frac{1}{2}$

---

Example 3

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

Solution: This matches $\sin \alpha \cos \beta -\cos \alpha \sin \beta =\sin (\alpha -\beta )$

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

---

Key Pattern Recognition