Trigonometry
Essential formulas and concepts in trigonometry
Trigonometric Ratios (right triangle definitions)
\(\displaystyle \sin\theta = \frac{\text{opposite}}{\text{hypotenuse}},\quad \cos\theta = \frac{\text{adjacent}}{\text{hypotenuse}},\quad \tan\theta = \frac{\text{opposite}}{\text{adjacent}}\)
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Description/Usage:
Defines sine, cosine, and tangent as ratios of sides in a right-angled triangle.
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Pythagorean Identity
\(\sin^2\theta + \cos^2\theta = 1\)
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Description/Usage:
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Law of Sines (for any triangle)
\(\displaystyle \frac{\sin A}{a} = \frac{\sin B}{b} = \frac{\sin C}{c}\)
Variables:
- \(A, B, C are the angles of a triangle, and a, b, c are the side lengths opposite those angles, respectively.\)
Description/Usage:
Relates the ratios of each side length to the sine of its opposite angle; useful for solving general (non-right) triangles. Usually accompanied by a triangle diagram showing angles and opposite sides.
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Law of Cosines
\(c^2 = a^2 + b^2 - 2ab\cos C\)
Variables:
- \(a, b, c are side lengths of a triangle, and C is the angle opposite side c.\)
Description/Usage:
Generalizes the Pythagorean theorem to any triangle. It finds a third side or an angle, useful when the Law of Sines is ambiguous or not applicable. Visualized by dropping perpendiculars in a non-right triangle to derive the relation.
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Euler’s Formula (complex plane)
\(e^{i\theta} = \cos\theta + i\,\sin\theta\)
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Description/Usage:
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