How Do You Use Sine, Cosine, and Tangent to Find Sides and Angles of a Right Triangle?

Use sine, cosine, or tangent by labelling the sides relative to a chosen acute angle, selecting the ratio containing the known and unknown sides, and solving the resulting equation. Use an inverse trigonometric function when the unknown is an angle.

Choosing the Correct Ratio

First identify the hypotenuse, which is opposite the right angle and is always the longest side. The opposite and adjacent sides depend on which acute angle, $\theta$, you are using.

This is often remembered as SOHCAHTOA.

Use this sequence every time: choose the reference angle, label the three sides, identify the pair of sides involved, select the matching ratio, substitute, and rearrange. If finding an angle, isolate the trigonometric ratio first and then apply the corresponding inverse function. Keep full calculator precision until the final answer.

Finding a Side

Suppose $\theta=35^\circ$, the hypotenuse is $12$, and the opposite side is $x$. Since opposite and hypotenuse are involved, use sine:

$$\sin 35^\circ=\frac{x}{12}$$

Rearranging,

$$x=12\sin 35^\circ\approx 6.88.$$

Finding an Angle

If the opposite side is $7$ and the adjacent side is $10$, use tangent:

$$\tan\theta=\frac{7}{10}.$$

Apply inverse tangent:

$$\theta=\tan^{-1}\left(\frac{7}{10}\right)\approx 35.0^\circ.$$

Check that each side is positive and the hypotenuse is longest. An acute angle must satisfy $0^\circ<\theta<90^\circ$. If a result is unreasonable, recheck side labels, algebra, and calculator mode; these checks catch most setup errors.

A common misconception is that “opposite” and “adjacent” are fixed labels. They change when the chosen angle changes; only the hypotenuse remains fixed.

IB Exam Technique

This is Mathematics AA SL, subtopic SL 3.2, and may be tested on Paper 1 or Paper 2. Draw and label the triangle, write the trigonometric equation before calculating, show rearrangement, and check the calculator is in degrees when the question uses degrees; otherwise, IB examinations assume radians.