How Do You Use Sine, Cosine, and Tangent for a Right Triangle?

Use sine, cosine, or tangent by identifying the sides relative to the given angle, choosing the ratio containing the known and unknown sides, and then solving the equation. The mnemonic SOHCAHTOA summarizes the three ratios.

This is IB Mathematics: Applications and Interpretation SL 3.2, applicable to both Paper 1 and Paper 2. A GDC is available on every Math AI paper.

Identify the sides

For an acute angle $\theta$ in a right triangle:

• The hypotenuse is opposite the right angle and is always the longest side.
• The opposite side is directly across from $\theta$.
• The adjacent side touches $\theta$, but is not the hypotenuse.

Worked Example: Finding a Side

Suppose $\theta=35^\circ$, the adjacent side is $8\text{ cm}$, and the opposite side is $x$. Since opposite and adjacent are involved, use tangent:

$$\tan 35^\circ=\frac{x}{8}$$

Therefore,

$$x=8\tan 35^\circ=5.60\text{ cm}$$

Check the result against the triangle: because $35^\circ$ is less than $45^\circ$, the opposite side should be shorter than the adjacent side. The value $5.60<8$ is therefore reasonable. This contextual check helps detect an incorrect ratio or calculator mode.

If the unknown is in the denominator, rearrange carefully before evaluating. For example, $\cos 40^\circ=7/h$ gives $h=7/\cos 40^\circ$, not . Estimating first is useful because the hypotenuse must exceed .

Worked Example: Finding an Angle

If the opposite side is $6\text{ m}$ and the hypotenuse is $10\text{ m}$, use sine:

$$\sin\theta=\frac{6}{10}$$

Apply the inverse sine function:

$$\theta=\sin^{-1}(0.6)=36.9^\circ$$

A common misconception is that the side labels are fixed. They are not: opposite and adjacent depend on which acute angle is chosen.

IB Exam Technique

Draw and label the triangle, write the trigonometric equation before using your GDC, and ensure it is in degree mode at SL. Keep full calculator precision until the final answer, then give the required accuracy and units.