Formulas & tables/Mathematics/Trigonometry/Trigonometric functions in a right triangle

Trigonometric functions in a right triangle

sin⁡α=ac, cos⁡α=bc, tan⁡α=ab\sin\alpha=\tfrac{a}{c},\ \cos\alpha=\tfrac{b}{c},\ \tan\alpha=\tfrac{a}{b}

Result unit: — (ratio of sides)

What it describes

Definitions of the trigonometric functions of an acute angle in a right triangle.

Symbols

α\alpha
acute angle
aa
leg opposite the angle
bb
leg adjacent to the angle
cc
hypotenuse

Where it comes from

Sine, cosine, and tangent are ratios of sides in a right triangle: sin⁡=oppositehypotenuse\sin=\tfrac{\text{opposite}}{\text{hypotenuse}}, cos⁡=adjacenthypotenuse\cos=\tfrac{\text{adjacent}}{\text{hypotenuse}}, tan⁡=oppositeadjacent\tan=\tfrac{\text{opposite}}{\text{adjacent}}.

When to use it

Right triangles — sides and angles; vectors, slopes.

Example

In a right triangle the side opposite the angle = 3 and the hypotenuse = 5. Find sin⁡α\sin\alpha.
Substitutiona=3a=3, c=5c=5.
Resultsin⁡α=35=0.6\sin\alpha=\dfrac{3}{5}=0.6.

Related formulas

Pythagorean trigonometric identity Tangent as a ratio

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