IB Maths AI 3.2 Notes

Right-angled triangle trigonometry

Trigonometry studies the relationship between the sides and angles of triangles. It is especially useful for finding unknown lengths and angles in geometry, modelling, navigation and three-dimensional problems. This section begins with right-angled triangles, then extends to any triangle using the sine rule, cosine rule and area formula.

In a right-angled triangle, one angle is $90^\circ$. Relative to a chosen acute angle:

• The hypotenuse is the longest side, opposite the right angle
• The opposite side is across from the chosen angle
• The adjacent side is next to the chosen angle

Image description: right-angled triangle labelled with hypotenuse, opposite and adjacent relative to an angle $\theta$.

The three basic trigonometric ratios are:

$\sin\theta=\frac{\text{opposite}}{\text{hypotenuse}}$

$\cos\theta=\frac{\text{adjacent}}{\text{hypotenuse}}$

$\tan\theta=\frac{\text{opposite}}{\text{adjacent}}$

A common memory aid is SOH CAH TOA.

A common problem will ask you to find a missing side. To do this, use the ratio that connects the angle to the side you know and the side you want.

In a right-angled triangle, an angle is $35^\circ$ and the hypotenuse is $10$. Find the side opposite the angle.

Use sine because opposite and hypotenuse are involved: $\sin35^\circ=\frac{x}{10}$.

$x=10\sin35^\circ$.

$x\approx5.74$.

So the opposite side is about $5.74$.

In a right-angled triangle, an angle is $42^\circ$ and the adjacent side is $8$. Find the hypotenuse.

Use cosine because adjacent and hypotenuse are involved: $\cos42^\circ=\frac{8}{h}$.

$h=\frac{8}{\cos42^\circ}$.

$h\approx10.77$.

So the hypotenuse is about $10.77$.

Another common problem will ask you to find a missing angle. To find an angle, use an inverse trigonometric function.

In a right-angled triangle, the opposite side is $7$ and the hypotenuse is $12$. Find the angle $\theta$.

Set up the ratio: $\sin\theta=\frac{7}{12}$.

Use inverse sine: $\theta=\sin^{-1}\left(\frac{7}{12}\right)$.

$\theta\approx35.7^\circ$.

So the angle is about $35.7^\circ$.

To know which function to use, just remember SOH CAH TOA:

• Use sine when you have opposite and hypotenuse.
• Use cosine when you have adjacent and hypotenuse.
• Use tangent when you have opposite and adjacent.

Additionally, sketching and labelling the triangle clearly helps avoid mistakes.