13: Trigonometry

Trigonometry relates angles and lengths of triangles. Figure A.1 shows a right-angled triangle and conventions to label its corners, sides, and angles. In the following, we assume all triangles to have at least one right angle (90 degrees or π/2 ) as all planar triangles can be dissected into two right-angled triangles.

The sum of all angles in any triangle is 180 degrees or 2π, or

\[\alpha +\beta +\gamma =180^{\circ}\]

If the triangle is right-angled, the relationship between edges a, b, and c, where c is the edge opposite of the right angle is

\[a^{2}+b^{2}=c^{2}\]

The relationship between angles and edge lengths are captured by the trigonometric functions:

\[\sin\alpha =\frac{opposite}{hypothenuse}=\frac{a}{c}\]

\[\cos\alpha =\frac{adjacent}{hypothenuse}=\frac{b}{c}\]

\[\tan\alpha =\frac{opposite}{adjacent}=\frac{\sin\alpha }{\cos\alpha }=\frac{a}{b}\]

Here, the hypothenuse is the side of the triangle that is opposite to the right angle. The adjacent and opposite are relative to a specific angle. For example, in Figure 13.1, the adjacent of angle α is side b and the opposite of α is edge a.

Relations between a single angle and the edge lengths are captured by the law of cosines:

\[a^{2}=b^{2}+c^{2}-2bc\cos\alpha \]

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