IB Maths AA SLTopic 3 — Geometry & TrigPaper 1 & 2~10 min read
Pythagoras & Right-Angled Trigonometry
If a triangle has a right angle, you’ve got two huge tools at your disposal: Pythagoras’ theorem for sides, and SOH CAH TOA for angles. Master both and you’ll handle any right-angled triangle problem in seconds.
📘 What you need to know
Pythagoras’ theorem: for any right-angled triangle, a2 + b2 = c2, where c is the hypotenuse (longest side).
SOH CAH TOA: sin = opphyp, cos = adjhyp, tan = oppadj.
Use trig to find a side when you have an angle and one side. Use inverse trig (sin⁻¹, cos⁻¹, tan⁻¹) to find an angle when you have two sides.
Angle of elevation is measured upward from horizontal; angle of depression is measured downward from horizontal.
Always check your calculator mode — degrees or radians — before computing trig values.
Pythagoras’ Theorem
The most famous result in geometry: in a right-angled triangle, the square of the hypotenuse equals the sum of the squares of the other two sides. This works for any right-angled triangle — full stop.
Pythagoras’ Theorema2 + b2 = c2
A right-angled triangle
The hypotenuse is always the longest side — it’s the one opposite the right angle. Identify it first; everything else depends on knowing which side is c.
Finding a missing side
If you know two sides of a right-angled triangle, you can always find the third using Pythagoras. Just rearrange the formula based on what you’re solving for:
Solving for any side
Hypotenuse: c = √(a2 + b2)
Shorter side: a = √(c2 − b2)
If you’re finding the hypotenuse, you’re adding the squares. If you’re finding a shorter side, you’re subtracting. The longest side is always the biggest, so when you subtract you take the smaller from the bigger.
Right-Angled Trigonometry — SOH CAH TOA
Pythagoras only relates sides. To bring angles into the picture, use trigonometry. There are three trig ratios you need: sine, cosine, and tangent — each of which connects an angle to two specific sides.
Step 1: Label the sides relative to your angle
Pick the angle you care about (call it θ). Now label the three sides:
H
Hypotenuse
Longest side, opposite the right angle
O
Opposite
The side opposite the angle θ
A
Adjacent
The side next to θ (not the hypotenuse)
Labelling sides relative to angle θ
The hypotenuse never changes regardless of which angle you pick — it’s always opposite the right angle. But “opposite” and “adjacent” depend on which angle you’ve chosen. If you switch to the other (non-right) angle, opp and adj swap.
Step 2: Use SOH CAH TOA
SOH
sin θ = OH
Sine = Opposite over Hypotenuse
CAH
cos θ = AH
Cosine = Adjacent over Hypotenuse
TOA
tan θ = OA
Tangent = Opposite over Adjacent
🧠 Memory hooks for SOH CAH TOA
“Some Old Hippie / Caught Another Hippie / Tripping On Acid” — silly but it sticks. Or pick whichever phrase makes you remember S-O-H, C-A-H, T-O-A.
Side or angle? — pick the right tool
Two routes through every right-angled trig question. Look at what you’re given and what you’re finding:
🔍 Finding a SIDE?
You’ll have an angle + one side. Pick the trig ratio that matches the two sides you have/want.
Rearrange and solve directly:
side = sin θ × H or Osin θ
🔍 Finding an ANGLE?
You’ll have two sides. Pick the trig ratio that matches them, then use inverse trig.
Apply sin⁻¹, cos⁻¹, or tan⁻¹:
θ = sin⁻¹(OH) etc.
The 3-step method: (1) label the sides relative to your angle, (2) tick which two sides you have or want — that tells you S, C, or T, (3) substitute and solve.
Angles of elevation and depression
Both are measured from the horizontal. Looking up at something gives you an angle of elevation. Looking down gives you an angle of depression. They appear in word problems about towers, planes, ships — anywhere you’ve got a “height + distance” setup.
Elevation vs depression
🤔 The angles are equal across the line of sight
If you’re standing on the ground and your friend is on a roof, the angle of elevation from you up to them equals the angle of depression from them down to you. They’re alternate angles between two parallel horizontal lines. Useful when a question gives you one and asks for the other.
Worked examples
WE 1
Pythagoras — find the hypotenuse
A right-angled triangle has shorter sides of 5 cm and 12 cm. Find the length of the hypotenuse.
A right-angled triangle has hypotenuse 17 cm and one shorter side of 8 cm. Find the other shorter side.
Step 1: Rearrange Pythagorasb2 = c2 − a2Step 2: Substitute c = 17, a = 8b2 = 172 − 82 = 289 − 64 = 225Step 3: Square rootb = √225 = 15 cm8-15-17 is another classic Pythagorean triple
WE 3
Trig — find a side using cosine
A right-angled triangle has hypotenuse 10 cm and one of its angles is 35°. Find the length of the side adjacent to the 35° angle.
Step 1: Label the sides
Hypotenuse = 10, want the adjacent side.
Step 2: Pick the right ratio
Adjacent + Hypotenuse → use cosine (CAH).
Step 3: Set up and solvecos 35° = A10A = 10 × cos 35° = 10 × 0.8192…A ≈ 8.19 cm (3 s.f.)make sure your calculator is in DEG mode!
WE 4
Inverse trig — find an angle
A right-angled triangle has its opposite side = 7 cm and hypotenuse = 12 cm. Find the angle θ.
Step 1: Identify the sides
Have Opposite (7) and Hypotenuse (12), finding angle θ.
Step 2: Use SOH (since O and H)sin θ = 712Step 3: Apply inverse sineθ = sin⁻¹(712) = sin⁻¹(0.5833…)θ ≈ 35.7° (3 s.f.)finding angles → always use inverse trig (sin⁻¹, cos⁻¹, tan⁻¹)
WE 5
Angle of elevation (real-world)
From a point on horizontal ground, the angle of elevation to the top of a tower is 28°. The point is 50 m from the base of the tower. Find the height of the tower to 3 s.f.
Step 1: Sketch + label28° angle, 50 m adjacent (horizontal), height = oppositeStep 2: Pick the ratio
Have A = 50, want O → use TAN (TOA).
Step 3: Set up and solvetan 28° = O50O = 50 × tan 28° = 50 × 0.5317…Height ≈ 26.6 m (3 s.f.)always sketch the situation first — even a rough triangle helps
💡 Top tips
Always sketch the triangle first. Mark the right angle, the known angle, and the known sides — this stops you using the wrong ratio.
Label H, O, A on your sketch relative to the angle of interest. Once you’ve labelled, picking S/C/T is automatic.
Memorise the Pythagorean triples: 3-4-5, 5-12-13, 8-15-17, 7-24-25. They appear constantly in IB problems.
Calculator mode matters. Most IB SL trig is in degrees for right-angled work, but radians elsewhere — check before evaluating.
Use the 3-step method: label the sides → pick the right ratio (S, C, or T) → solve.
For inverse trig, the buttons are usually shift + sin/cos/tan on a calculator.
For elevation/depression problems, draw both the observer and the target, plus the horizontal — the angles always sit at the horizontal line.
⚠ Common mistakes
Calculator in the wrong mode. Computing sin 30° in radian mode gives a totally different answer. Always check DEG vs RAD.
Using Pythagoras when there’s no right angle. Pythagoras only applies to right-angled triangles — for others you need the sine or cosine rule.
Confusing opposite and adjacent. The opposite is “across from” the angle; the adjacent is “next to” the angle (and not the hypotenuse).
Treating the hypotenuse as a “shorter side”. The hypotenuse is always the longest side and always opposite the right angle.
Forgetting to take the square root. After computing a2 + b2, the answer is c2, not c.
Using sin instead of inverse sin (or vice versa). sin gives you a ratio from an angle; sin⁻¹ gives you the angle from a ratio.
Mixing up elevation and depression. Elevation is looking up; depression is looking down — both from the horizontal.
Right-angled trig is the foundation for everything trigonometric you’ll meet in IB. Get fluent with SOH CAH TOA and the rest of trig — sine rule, cosine rule, identities — comes much more easily.
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