Trigonometry in right-angled triangles

In a right-angled triangle the trigonometric functions take on an immediate meaning: they are simple ratios between the sides. This is the starting point for solving any triangle.

Property — Relations in right-angled triangles

Let $ABC$ be a right-angled triangle with the right angle at $C$, legs $a = BC$, $b = AC$, hypotenuse $c = AB$, angle $\alpha = \widehat{BAC}$ opposite the leg $a$, angle $\beta = \widehat{ABC}$ opposite the leg $b$. Then:

\sin\alpha &= \frac{\text{cateto opposto ad }\alpha}{\text{ipotenusa}} = \frac{a}{c}, & \cos\alpha &= \frac{\text{cateto adiacente ad }\alpha}{\text{ipotenusa}} = \frac{b}{c}, \\ \tan\alpha &= \frac{\text{cateto opposto}}{\text{cateto adiacente}} = \frac{a}{b}. \end{aligned}$$ With analogous formulae for $\beta$, swapping the roles of $a$ and $b$.

These formulae generalise the definition of $\sin$, $\cos$, $\tan$ given on the trigonometric circle: the right-angled triangle with vertices $O$, $P_\alpha$ and the projection of $P_\alpha$ onto the $x$-axis has hypotenuse $1$ (the radius), and its legs are $\cos\alpha$ (horizontal) and $\sin\alpha$ (vertical).

Links

Topics: Triangle trigonometry
Concepts: Triangle solving · Right-angled triangle
Functions: Cosine · Sine · Tangent
Skills: Synthetic geometry · Using formulae