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13: Trigonometry

  • Page ID
    14857
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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.

    clipboard_eb1c91951bc71f8a84c9302596ec82001.png
    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:

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


    This page titled 13: Trigonometry is shared under a CC BY-NC 4.0 license and was authored, remixed, and/or curated by Nikolaus Correll via source content that was edited to the style and standards of the LibreTexts platform; a detailed edit history is available upon request.