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13: Trigonometry

Last updated

Aug 10, 2021
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- 12.3: RGB-D Mapping
- 13.1: Inverse trigonometry

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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.

Figure $\PageIndex{1}$ : Left: A right-angled triangle with common notation. Right: Trigonometric relationships on the unit circle and angles corresponding to the four quadrants.

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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