Take any right triangle with an acute angle of $35^\circ$. Every other right triangle with a $35^\circ$ angle is similar to it, by Proving AA Similarity. Similar triangles have proportional sides, so the ratio of any two sides is the same in all of them. Each ratio depends only on the angle, and that is why we can give the ratios names. You saw exact cases in 45-45-90 and 30-60-90 Triangles.
From an acute angle $A$ of a right triangle, the hypotenuse is across from the right angle, the opposite leg is across from $A$, and the adjacent leg is the leg that helps form $A$.
Key idea
Why it works
Worked example: All three ratios
- From $A$: opposite $= BC = 8$, adjacent $= AC = 15$, hypotenuse $= 17$.
- $\sin A = \frac{8}{17}$, $\cos A = \frac{15}{17}$, $\tan A = \frac{8}{15}$.
Answer: $\sin A = \frac{8}{17}$, $\cos A = \frac{15}{17}$, $\tan A = \frac{8}{15}$
Worked example: From the other angle
- From $B$, the opposite side is $AC = 15$ and the adjacent leg is $BC = 8$.
- $\sin B = \frac{15}{17}$ and $\tan B = \frac{15}{8}$.
Answer: $\sin B = \frac{15}{17}$, $\tan B = \frac{15}{8}$
| Angle | $\sin$ | $\cos$ | $\tan$ |
|---|---|---|---|
| $30^\circ$ | $\frac{1}{2}$ | $\frac{\sqrt{3}}{2}$ | $\frac{\sqrt{3}}{3}$ |
| $45^\circ$ | $\frac{\sqrt{2}}{2}$ | $\frac{\sqrt{2}}{2}$ | $1$ |
| $60^\circ$ | $\frac{\sqrt{3}}{2}$ | $\frac{1}{2}$ | $\sqrt{3}$ |
Common mistake