Right-Angle Trigonometry (AA SL)
Before you ever reach for the sine or cosine rule, most triangle problems start here: a right angle, one known angle or ratio, and a missing side to find. This page covers the three trig ratios (SOH CAH TOA), the elevation/depression setups examiners like to dress them up in, and the slip-ups that cost easy marks. It's part of the broader Triangles (Sine/Cosine Rules) topic.
11 questions on this sub-topic.
The trig ratios
Covered under IB syllabus reference SL3.2: use of sine, cosine and tangent ratios to find the sides and angles of right-angled triangles. These three ratios aren't in the formula booklet - they're assumed prior knowledge, so you do need to know them cold.
The three ratios
\(\sin\theta=\dfrac{\text{opp}}{\text{hyp}}\quad\cos\theta=\dfrac{\text{adj}}{\text{hyp}}\quad\tan\theta=\dfrac{\text{opp}}{\text{adj}}\)
Label the sides relative to the angle you're working with, decide which two you know or want, and pick the matching ratio.
Elevation and depression
\(\tan\theta=\dfrac{\text{height}}{\text{horizontal distance}}\)
Elevation and depression problems are almost always tangent problems in disguise, once you sketch the right-angled triangle formed by the horizontal and the line of sight.
Ready for triangles without a right angle? See the full topic page for sine rule, cosine rule and GDC guidance.
Worked examples
A right-angled triangle has the angle \(30^\circ\) and hypotenuse 12 cm.
Find the length of the side opposite the \(30^\circ\) angle.
Worked solution
Opposite and hypotenuse are involved, so use sine: \(\sin 30^\circ=\dfrac{\text{opp}}{12}.\) M1
Rearrange and evaluate: opp \(=12\sin 30^\circ=12\times\tfrac12=6\) cm. A1
From a point 50 m from the base of a tower, the angle of elevation of the top is \(32^\circ\).
Find the height of the tower.
Worked solution
Height is opposite, base distance is adjacent, so use tangent: \(\tan 32^\circ=\dfrac{h}{50}.\) M1
Rearrange and evaluate: \(h=50\tan 32^\circ\approx 31.2\) m. A1
Common mistakes
- Labelling the sides before checking the reference angle. Opposite and adjacent only make sense relative to a specific angle - if the question asks about the other acute angle, the labels swap and so does the ratio you need.
- Wrong angle mode on the GDC. \(\sin(30)\) means something completely different in radian mode. Always confirm degree mode before touching a trig button in a right-angle problem.
- Mixing up elevation and depression. Both give the same tangent equation once you've drawn the triangle, but sketching the horizontal line first stops you writing the height and distance the wrong way round.
- Reaching for Pythagoras when an angle is involved. A right-angled triangle question that gives one side and one angle needs a trig ratio (SOH CAH TOA) - Pythagoras' theorem only applies once two sides are already known.
Ready to practise properly?
11 right-angle trigonometry questions, marked instantly like the real exam.
Quick answers
How do I know which trig ratio to use?
Label the two sides you know or want relative to the given angle as opposite, adjacent or hypotenuse, then pick sine (opp/hyp), cosine (adj/hyp) or tangent (opp/adj) accordingly.
What is the difference between angle of elevation and angle of depression?
Angle of elevation is measured upward from the horizontal to an object above; angle of depression is measured downward from the horizontal to an object below. By alternate angles the two are equal for the same line of sight.