IB Maths AA SLTopic 3 — Geometry & TrigPaper 1 & 2~9 min read
Solving Equations Using Trigonometric Graphs
Your calculator’s sin⁻¹, cos⁻¹, and tan⁻¹ buttons only give you one answer to a trig equation — but most trig equations have many. The graph shows you exactly where the rest are. Sketch, draw a horizontal line, find the others.
📘 What you need to know
Inverse trig (sin⁻¹, cos⁻¹, tan⁻¹) gives only the principal value. Other solutions come from the graph’s symmetry and periodicity.
For sin x = k or cos x = k (with −1 < k < 1, k ≠ 0): expect 2 solutions per 360°.
For tan x = k: expect 1 solution per 180°.
Symmetry shortcuts: sin is symmetric about x = 90°; cos is symmetric about x = 0°; tan repeats every 180°.
Always sketch the graph and the horizontal liney = k first — the intersections ARE the solutions.
Why one calculator answer isn’t enough
Type sin⁻¹(0.5) into your calculator and you’ll get 30°. But sin x = 0.5 has infinitely many solutions: 30°, 150°, 390°, 510°, … and so on going forever in both directions. Your calculator picks just one — the principal value — and ignores the rest. The graph shows you all of them at once.
y = sin x cuts the line y = 0.5 at every solution
The dashed red line y = 0.5 cuts the sine curve in two places over [0°, 360°]: at 30° and 150°.
How many solutions to expect
Before you start hunting for answers, work out roughly how many solutions you should find. This stops you missing any.
Equation
Solutions per cycle
Cycle length
sin x = k (−1 < k < 1, k ≠ 0)
2
every 360°
cos x = k (−1 < k < 1, k ≠ 0)
2
every 360°
tan x = k
1
every 180°
sin/cos x = ±1
1
every 360° (the peak/trough)
sin/cos x = 0
2
every 360° (the zeros)
Quick check: divide the width of your interval by 360° (or 180° for tan). If it’s a whole number, double it for sin/cos to get the exact count. If not, the closest whole numbers give you the minimum and maximum counts.
For example, in [0°, 720°] you’d expect 720 ÷ 360 = 2 full cycles of sin, so 4 solutions for sin x = 0.5. In [0°, 540°] (a 1.5-cycle interval), you’d expect 2 or 3 — the exact number depends on where the interval starts and ends.
The 4-step method
1
SKETCH
Draw the trig graph over the given interval. Check degrees vs radians.
2
DRAW LINE
Add the horizontal line y = k. Solutions are the intersections.
3
PRINCIPAL
Use inverse trig on your calculator to get the first solution.
4
SYMMETRY
Use the graph’s symmetry/periodicity to find all the others.
The symmetry shortcuts
Once you’ve got the principal value, each trig function has its own quick rule for finding the second solution (and beyond):
y = sin x
Symmetric about x = 90°. Repeats every 360°.
If x = α, then 180° − α
also a solution. Add ±360° for more.
y = cos x
Symmetric about x = 0°. Repeats every 360°.
If x = α, then −α
also a solution. Add ±360° for more.
y = tan x
Repeats every 180°.
If x = α, then α + 180°
also a solution. Add ±180° for more.
🤔 Why these specific symmetries?
The sine wave reflects across its peak — so if 30° gives sin = 0.5, then 90° + (90° − 30°) = 150° also gives sin = 0.5. The cosine wave reflects across the y-axis — so if 60° works, so does −60°. Tan doesn’t reflect at all; it just translates by 180° because that’s its period.
🧠 Memory trick — “subtract from 180” or “negate”
For sin: second solution = 180° − principal. For cos: second solution = −principal (or equivalently 360° − principal). For tan: second solution = principal + 180°.
Cosine — the symmetry across the y-axis
y = cos x with y = 0.5 — symmetric solutions
cos x = 0.5 in [−360°, 360°] has solutions at −300°, −60°, 60°, 300°.
Worked examples
WE 1
Solve a sine equation in [0°, 360°]
Solve sin x = 0.7 for 0° ≤ x ≤ 360°. Give answers to 1 d.p.
Step 1: Find the principal valuex = sin⁻¹(0.7) = 44.4°Step 2: Apply the sin symmetry rule
Second solution = 180° − 44.4° = 135.6°Step 3: Check both fit the interval
Both 44.4° and 135.6° lie in [0°, 360°] ✓
x = 44.4° or 135.6°expected 2 solutions, found 2 — done!
WE 2
Solve a cosine equation in [0°, 360°]
Solve cos x = 0.3 for 0° ≤ x ≤ 360°. Give answers to 1 d.p.
Step 1: Principal valuex = cos⁻¹(0.3) = 72.5°Step 2: Cos symmetry — second solution is −72.5°but that’s outside [0°, 360°]Step 3: Add 360° to bring it into range−72.5° + 360° = 287.5°x = 72.5° or 287.5°when the negative version is out of range, add 360°
WE 3
Solve a tangent equation
Solve tan x = 2 for 0° ≤ x ≤ 360°. Give answers to 1 d.p.
Step 1: Principal valuex = tan⁻¹(2) = 63.4°Step 2: Apply tan periodicity (every 180°)Next solution = 63.4° + 180° = 243.4°Step 3: Check no others fit
63.4° + 360° = 423.4° (out of range) ✗
x = 63.4° or 243.4°tan only has 1 solution per 180° — so 2 in 360°
WE 4
Cosine equation over a wider interval (SME-style)
One solution to cos x = 0.5 is 60°. Find all the other solutions in the range −360° ≤ x ≤ 360°.
Step 1: Apply cos symmetry
Cos is symmetric about x = 0° → if 60° works, so does −60°.
Step 2: Apply periodicity (±360°)60° + 360° = 420° (out of range)60° − 360° = −300° ✓−60° + 360° = 300° ✓−60° − 360° = −420° (out of range)Step 3: List all solutions in [−360°, 360°]x = −300°, −60°, 60°, 300°interval width is 720° = 2 full cycles → expect 4 solutions ✓
WE 5
Solve in radians
Solve sin x = √32 for 0 ≤ x ≤ 2π. Give exact values.
Step 1: Recognise the exact valuesin π3 = √32 → principal value = π3Step 2: Apply sin symmetry: π − principalSecond solution = π − π3 = 2π3Step 3: Both inside [0, 2π] ✓x = π3 or 2π3in radians, 180° becomes π — same logic, different units
💡 Top tips
Always sketch the graph first with the horizontal line. The intersections show every solution at a glance — no missed answers.
Predict the count. Divide the interval by 360° (or 180° for tan) before solving, so you know how many to expect.
Memorise the 3 symmetry rules: sin → 180° − α, cos → −α, tan → α + 180°.
Add or subtract multiples of 360° (or 180° for tan) to find solutions outside the principal range.
Calculator mode matters. Always check DEG vs RAD before evaluating inverse trig.
For radian problems, the rules become: sin → π − α, cos → −α, tan → α + π. Same pattern, π replaces 180°.
Mark each solution on your sketch as you find it — visually confirms you haven’t missed any.
⚠ Common mistakes
Only writing down the calculator answer. sin⁻¹/cos⁻¹/tan⁻¹ give just the principal value — there are usually more solutions.
Wrong symmetry rule. Sin uses 180° − α, cos uses −α. Mixing these up is very common.
Forgetting tan repeats every 180°, not 360°. So tan = k has 2 solutions in [0°, 360°], not 1.
Solutions outside the interval. Always check that each candidate is actually within the given range.
Missing solutions when the interval is wide (e.g. [−360°, 720°] = 3 cycles → up to 6 solutions for sin/cos).
Calculator in the wrong mode. If your principal value looks weird, you’re probably in the wrong DEG/RAD mode.
Special-case errors. If k = ±1 there’s only 1 solution per 360° (the peak/trough). Don’t double it by mistake.
Sketching the graph first makes solving trig equations almost automatic. Don’t trust the calculator alone — let the picture confirm how many answers you should have, then use the symmetry rules to nail them down.
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