Trigonometric ratios in right-angled trianglesIB MYP Maths Standard: Revision notes

In a right-angled triangle the hypotenuse is the longest side, opposite the right angle. The other two sides are named from the point of view of the angle $\theta$ you are using: the opposite side is across from $\theta$ and the adjacent side is next to $\theta$ (not the hypotenuse). If you switch to the other acute angle, opposite and adjacent swap. The hypotenuse never changes.

The three trigonometric ratios link an angle with two sides: $\sin\theta=\frac{\text{opposite}}{\text{hypotenuse}},\quad\cos\theta=\frac{\text{adjacent}}{\text{hypotenuse}},\quad\tan\theta=\frac{\text{opposite}}{\text{adjacent}}.$ The memory aid is SOH CAH TOA. To choose a ratio, mark the two sides involved (the one you know and the one you want, or the two you know) and pick the ratio that uses exactly those two sides.

Write the ratio, substitute the numbers, then solve. Example: the hypotenuse is $10$ cm and the angle is $30^\circ$. The side opposite is wanted, so use sine: $\sin30^\circ=\frac{x}{10}$ and $x=10\sin30^\circ=5$ cm. If the unknown side is on the bottom of the fraction, divide instead: $\tan25^\circ=\frac{60}{d}$ gives $d=\frac{60}{\tan25^\circ}=128.7$ m. Make sure your calculator is in degree mode.

When you know two sides and want the angle, use the inverse function: $\sin^{-1}$, $\cos^{-1}$ or $\tan^{-1}$ on your calculator. Example: opposite $7$ and adjacent $9$, so $\tan\theta=\frac{7}{9}$ and $\theta=\tan^{-1}\left(\frac79\right)=37.9^\circ$. For the other acute angle in the same triangle use $90^\circ-37.9^\circ=52.1^\circ$. Round angles to $1$ decimal place unless told otherwise.

The angle of elevation is measured upwards from the horizontal to the line of sight. The angle of depression is measured downwards from the horizontal. The two angles are equal for the same line of sight, because they are alternate angles between parallel horizontal lines. So an observer on a cliff who sees a boat at a depression of $25^\circ$ means that the boat sees the observer at an elevation of $25^\circ$.

For each problem: draw a clear right-angled triangle, label the sides, choose the ratio, write the working, and check the answer makes sense (the hypotenuse is the longest side). Give lengths to $3$ significant figures and angles to $1$ decimal place unless told otherwise. If a measurement is rounded, a calculated answer is only approximate, so allow for that when you comment on accuracy.