Right-Angled Trigonometry (AI SL)
Right-angled trigonometry turns a triangle with one \(90^\circ\) corner into a labelled diagram you can solve with SOHCAHTOA or Pythagoras' theorem. It covers angles of elevation and depression, and 3D problems where the right angle sits hidden inside a solid. This page is part of the broader Triangle Trigonometry topic.
33 questions on this sub-topic.
The two formulas
Covered under IB syllabus reference SL3.3: applications of right-angled trigonometry, including Pythagoras' theorem, angles of elevation and depression, and constructing labelled diagrams from written statements. Neither tool below is new to the Diploma, but knowing when to reach for which one is the real skill.
Trig ratios (SOHCAHTOA)
\(\sin\theta=\dfrac{\text{opp}}{\text{hyp}},\quad \cos\theta=\dfrac{\text{adj}}{\text{hyp}},\quad \tan\theta=\dfrac{\text{opp}}{\text{adj}}\)
Pick the ratio that connects the side you already know to the side or angle you're finding.
Not in booklet - prior knowledgePythagoras' theorem
\(a^2+b^2=c^2\)
Relates the two shorter sides to the hypotenuse in any right-angled triangle - useful before or after a trig step.
Not in booklet - prior knowledgeNeed the wider syllabus context and GDC settings for trig? See Triangle Trigonometry's GDC guidance.
Worked examples
In a right-angled triangle the angle is \(50^\circ\) and the hypotenuse is 20 cm.
Find the side opposite the angle (3 significant figures).
Worked solution
The wanted side is opposite the \(50^\circ\) angle; the hypotenuse is 20 cm. Opposite + hypotenuse → sine. M1
\(\sin 50^\circ = \dfrac{\text{opp}}{20}\), so A1
\(\text{opp} = 20\sin 50^\circ = 15.32\ldots \approx 15.3 \text{ cm (3 significant figures).}\) A1
From the top of a cliff 80 m high, the angle of depression to a boat at sea is \(22^\circ\). The boat then moves directly towards the cliff. After moving, the angle of depression is \(47^\circ\).
Find the distance the boat travelled.
Worked solution
Initial distance from base of cliff.
\(d_1=\dfrac{80}{\tan22^\circ}\) M1
\(d_1\approx198.1\) m. A1
Final distance from base.
\(d_2=\dfrac{80}{\tan47^\circ}\approx74.6\) m. Distance moved \(=198.1-74.6\) M1
\(\approx123\) m. A1
Common mistakes
- Mixing up opposite and adjacent. Relabel the two known sides relative to the angle you're using each time - a side that's "adjacent" for one angle in the triangle is "opposite" for the other.
- Measuring depression from the wrong line. Both elevation and depression are measured from the horizontal at eye level, not from the vertical or the ground - draw the horizontal dashed line first.
- Calculator left in radian mode. IB right-angled trigonometry questions are set in degrees unless stated otherwise - check the angle mode before evaluating any sin, cos or tan.
Ready to practise properly?
38 right-angled trigonometry questions, marked instantly like the real exam.
Quick answers
How do I know which trig ratio to use in a right-angled triangle?
Label the two sides you know or want relative to the angle - opposite, adjacent or hypotenuse - and pick sin (opp/hyp), cos (adj/hyp) or tan (opp/adj) accordingly, remembered as SOHCAHTOA.
What's the difference between angle of elevation and angle of depression?
Elevation is measured upward from the horizontal to an object above; depression is measured downward from the horizontal to an object below. Both are measured from the horizontal, and alternate angle facts often link them between two positions.