IB Maths AI HL Trig Identities & Equations Paper 1 & 2 Multi-solution equations ~9 min read

Solving Equations Using Trigonometric Graphs

A calculator’s inverse-trig button gives just one answer — the principal value. The graph supplies all the others. Draw the curve, draw the horizontal line y = k, and find every intersection in the given interval. For sin and cos that’s two intersections per 360°; for tan, one per 180°.

📘 What you need to know

Principal value: the single answer your calculator returns for sin⁻¹, cos⁻¹, tan⁻¹. It’s just one of (usually) many solutions.
Second solution — sin x = k: if x = α is a solution, so is x = 180° − α (or π − α). Add ±360° (or ±2π) for more.
Second solution — cos x = k: if x = α is a solution, so is x = −α (or equivalently 360° − α). Add ±360° for more.
Tan x = k: solutions are α, α + 180°, α + 360°, … (period 180°/π).
How many solutions: sin/cos give roughly 2 per 360°; tan gives 1 per 180°. Edge cases: k = ±1 (tangent to peak/trough, one solution per 360° for sin/cos); k = 0 (zero crossings).
Always sketch first — the graph confirms the count and lets you spot whether endpoints are included.

The graph method, step by step

The principal value from your calculator is one solution. To find the rest in a given interval, sketch the curve, draw the horizontal line y = k, and look for every intersection. For sin and cos, the two solutions in a single revolution are symmetric — about x = 90° for sin (the peak), and about x = 0° for cos (the maximum). For tan, solutions repeat every 180°, so once you have one, just add 180° to get the next.

Example: sin x = 0.6 in [0°, 360°] has two solutions, α ≈ 36.9° (principal) and 180° − α ≈ 143.1°, symmetric about the peak at x = 90°.

Second-solution shortcuts sin x = k: α, 180°−α · cos x = k: α, −α · tan x = k: α, α+180° add ±360° (sin/cos) or ±180° (tan) until you cover the interval

Counting the solutions in any interval

Before solving, work out how many solutions to expect — that way you’ll spot if you’ve missed one. Divide the interval width by the period: 360° for sin and cos, 180° for tan. For sin x = k or cos x = k with strict −1 < k < 1 (excluding 0), expect about 2 solutions per 360°. For tan x = k, expect about 1 per 180°. Edge cases (k = ±1 or 0) need a quick sketch check to count exactly.

Quick count rule: for sin/cos with −1 < k < 1 and k ≠ 0, multiply the interval length (in degrees) by 2/360 to estimate the number of solutions. For tan, multiply by 1/180. Round to the closest whole number(s) to get the possible counts.

🧭 Recipe — solve a trig equation

Rearrange to sin/cos/tan = k: isolate the trig function.
Principal value: use sin⁻¹, cos⁻¹, or tan⁻¹ on your calculator (match degree/radian mode to the question).
Second solution (in one revolution): 180°−α for sin, −α (= 360°−α) for cos, or α+180° for tan.
Fill the interval: add ±360° (or ±180° for tan) to each known solution until you’ve covered the whole range.
Check the count: sketch quickly and confirm the number of solutions matches expectations.

Worked examples

WE 1

Basic sin equation in one revolution

Solve sin x = 0.6 for x in the interval 0° ≤ x ≤ 360°. Give answers to 1 d.p.

principal value x = sin⁻¹(0.6) = 36.87° second solution: 180° − α x = 180° − 36.87° = 143.13° both in [0°, 360°] — no need to add 360° x = 36.9°, 143.1° two solutions per 360° as expected (since 0 < 0.6 < 1).

WE 2

cos equation in an asymmetric interval

Solve cos x = 0.3 for x in the interval −180° ≤ x ≤ 360°. Give answers to 1 d.p.

principal value α = cos⁻¹(0.3) = 72.54° second solution: −α x = −72.54° fill the interval by adding ±360° −72.54° + 360° = 287.46° ✓ 72.54° + 360° = 432.54° (out) −72.54° − 360° = −432.54° (out) x = −72.5°, 72.5°, 287.5° 540° interval → expect ~3 solutions for cos; matches ✓

WE 3

tan equation in one revolution

Solve tan x = −1.5 for x in the interval 0° ≤ x ≤ 360°. Give answers to 1 d.p.

principal value α = tan⁻¹(−1.5) = −56.31° (in [−90°, 90°]) shift into [0°, 360°] by adding 180° −56.31° + 180° = 123.69° next solution: + 180° 123.69° + 180° = 303.69° x = 123.7°, 303.7° two solutions per 360° for tan (one per 180°).

WE 4

sin equation in radians, negative k

Solve sin x = −0.8 for x in the interval 0 ≤ x ≤ 2π. Give answers to 3 s.f.

principal value (calc in radians) α = sin⁻¹(−0.8) = −0.9273 rad (in [−π/2, 0] — outside our target interval) sin is negative in Q3 and Q4 Q3: π − (−0.9273) = π + 0.9273 = 4.069 Q4: 2π + (−0.9273) = 5.356 x ≈ 4.07, 5.36 rad when the principal value is outside the target interval, shift to the standard 180° − α or 360° + α form to bring it in.

WE 5

cos equation over two full revolutions

Solve cos x = −0.4 for x in the interval −360° ≤ x ≤ 360°. Give answers to 1 d.p.

principal value α = cos⁻¹(−0.4) = 113.58° cos symmetry: −α also a solution −113.58° add ±360° to each 113.58° − 360° = −246.42° −113.58° + 360° = 246.42° (other ±360° shifts are outside) x = −246.4°, −113.6°, 113.6°, 246.4° 720° interval → 4 solutions for cos. ✓

WE 6

Linear-rearrange to a sin equation — exact answers

Solve 2 sin x + 1 = 0 for x in the interval 0° ≤ x ≤ 360°. Give exact answers.

rearrange 2 sin x = −1 → sin x = −1/2 principal value (exact, from memory) α = sin⁻¹(−1/2) = −30° sin negative in Q3 and Q4 Q3: 180° − (−30°) = 210° Q4: 360° + (−30°) = 330° x = 210°, 330° since sin(30°) = 1/2 is a standard exact value, no calculator needed.

💡 Top tips

Sketch first, then solve — a quick graph confirms the expected number of solutions and prevents missed ones.
Match calculator mode to the question: degrees if the interval is in degrees; radians if it’s in π.
Standard exact values: sin(30°) = ½, sin(45°) = √2/2, sin(60°) = √3/2 (and cos in the opposite order). Recognising these saves calculator steps.
For tan equations, period is 180° — add 180° (not 360°) to step to the next solution.
For sin x = k with k negative, the principal value lies in [−90°, 0°]; for cos x = k with k negative, in [90°, 180°]; for tan, in [−90°, 0°].

⚠ Common mistakes

Reporting only the principal value — the calculator gives one; the graph supplies the rest.
Using 360° − α for sin — that’s the cos rule. For sin use 180° − α.
Forgetting to add ±360° when the interval extends past one revolution.
Including out-of-range candidates — always filter to the stated interval at the end.
Mode mismatch: solving “in radians” with calculator in degree mode (or vice versa) corrupts every value.

Chapter complete — you now have all four Trigonometric Identities & Equations sub-topics: the Unit Circle, Simple Identities, Graphs of Trig Functions, and Solving Equations Using Trig Graphs. Together they cover every trig question that asks “find all values of x such that …”.

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