Solving triangles

Sine, cosine and tangent are more than definitions: they are tools. As soon as you know one angle and one side of a right triangle, you can work out any missing side or angle. The whole trick is to pick the right formula and rearrange it for the quantity you want.

Which function fits?

Look at which sides are involved. If the opposite leg and the hypotenuse appear, use the sine. With the adjacent leg and the hypotenuse, use the cosine. If it is the two legs, tangent is your tool. The same rule always holds: bring the known angle, the known side and the wanted side together in one equation.

Finding a missing side

If you know the hypotenuse $c$ and the acute angle $\alpha$, you find the leg $a$ opposite $\alpha$ by rearranging the sine formula. Multiplying $\sin\alpha = \frac{a}{c}$ by $c$ gives the equation for $a$. Example: $c = 10$ cm and $\alpha = 30°$ give $a = 10 \cdot \sin 30° = 10 \cdot 0.5 = 5$ cm. The other two functions work in exactly the same way: for the leg $b$ adjacent to $\alpha$ you get $b = c \cdot \cos\alpha$, and when you know $b$ and want $a$, $a = b \cdot \tan\alpha$ does the job.

Finding a missing angle

The other way round, you sometimes know two sides and want the angle. Then you need the inverse function, shown on the calculator as $\sin^{-1}$, $\cos^{-1}$ or $\tan^{-1}$. If you know the opposite and adjacent legs, then $\tan\alpha = \frac{a}{b}$ and so $\alpha = \tan^{-1}\!\left(\frac{a}{b}\right)$. For $a = 3$ and $b = 4$ this gives about $36.87°$.