IB Maths AA SLTopic 3 — Trig Equations & IdentitiesPaper 1 & 2~12 min read
Linear Trigonometric Equations
When you see something like sin x = 0.5 or 2cos(2x − 30°) = −1, you’re solving a linear trig equation. The trap is that there are many answers in any given range — your calculator only gives you one. This note shows you exactly how to find them all.
📘 What you need to know
A trig equation like sin x = k has infinitely many solutions — you list the ones in the given range.
Step 1 is always to isolate the trig term (e.g. divide both sides by 2 if needed) and then take inverse to get the principal value.
For sin: second value = 180° − principal | For cos: second value = −principal (or 360° − principal) | For tan: just add 180° each time.
Then add or subtract 360° (180° for tan) to find every solution in the given range.
For sin(ax + b) = k, substitute y = ax + b, transform the range too, solve, then convert back.
Check whether the question uses degrees or radians and set your calculator to match.
Why are there so many answers?
The sin, cos, and tan functions are periodic — they repeat themselves over and over. So if sin x = 0.5 is true at x = 30°, it’s also true at x = 150°, at x = 390°, at x = 510°, and so on forever in both directions.
Your calculator only gives you one of these — the principal value. Your job is to find all the others that fit inside the range the question asks for.
y = sin x meets y = 0.5 in many places
Every solution to sin x = 0.5 is a place where the wavy sin curve meets the flat line y = 0.5. There are infinitely many — you just take the ones inside the range you’ve been asked about.
Method for sin x = k and cos x = k
The 3-step method works the same way for both. The only thing that changes is how you find the second value.
The 3-step method
Find the principal value. Use sin−1 or cos−1 on your calculator. (Make sure you’re in the right mode — degrees or radians.)
Find the second value using the rule for that ratio (see cards below).
Add or subtract 360° (or 2π in radians) to those two values until you’ve collected every answer that fits the given range.
The second-value rule for each ratio
sin x = k
Step 2: second value
180° − principal
e.g. sin x = ½ → 30° & 150°
Step 3: then ± 360°
cos x = k
Step 2: second value
− principal or 360° − principal
e.g. cos x = ½ → 60° & −60°
Step 3: then ± 360°
tan x = k
Just one rule:
add ± 180° each time
e.g. tan x = 1 → 45°, 225°…
tan repeats every 180°
🤔 Why is tan only one rule, while sin and cos have two?
Because the tan curve repeats itself every 180° (its period is shorter), every “second value” you’d want is just 180° added to the first. Sin and cos have a period of 360°, so within one period they each hit the same value twice — once on the way up, once on the way down — hence two values to start from.
📍
Watch out: cos can give a negative principal value
If k is negative, your calculator may give a principal value bigger than 90° — for example, cos−1(−½) = 120°. The “second value” −principal would be −120°. Both work; pick whichever falls in your range.
Transformed equations: sin(ax + b) = k
What if the equation has 2x, or 3x + 60°, instead of just x? You can’t just take inverse-sin and stop — you have to deal with the inside of the bracket carefully.
The fix is a substitution: let the whole inside of the bracket equal a new variable y. Solve for y first, then convert back to x at the end.
The substitution method — 4 steps
Original
sin(2x + 60°) = √32
→
Let y = 2x + 60°
sin y = √32
Substitutey = (whatever’s inside the bracket).
Transform the range too — apply the same operations to the original x-range. E.g. if 0° ≤ x ≤ 360° and y = 2x + 60°, multiply by 2 then add 60° → 60° ≤ y ≤ 780°.
Find every y in that new range using the 3-step method from above.
Convert back to x using y = 2x + 60° → x = (y − 60°) ÷ 2.
🧠
Memory trick: “Do the same to the range”
Whatever you do to x to get y, do exactly the same to the x-range to get the y-range. Multiply, then add — in that order. Many students mess this up by adding first.
A coefficient like 2x means twice as many solutions in the same range. So sin(2x) = 0.5 in 0° to 360° will give you 4 answers, not 2 — because y sweeps through twice the range.
Worked examples
WE 1
Solve sin x = ½ for 0° ≤ x ≤ 360°
Find all values of x in the given range.
sin x = ½Sin equation, range is one full period (0° to 360°).Principal value:x = sin−1(½) = 30°Second value (180° − 30°):x = 150°Add ± 360°: 30° + 360° = 390° (out of range), 150° + 360° = 510° (out of range)Both originals are in range, no extras needed.x = 30°, 150°always check whether each “extra” is inside the range — only keep ones that are
WE 2
Solve 2cos x = −1 for −π ≤ x ≤ 3π
Find all values of x in radians.
2 cos x = −1Range is wider than 2π — expect more than 2 answers. Calculator in radians!Isolate cos:cos x = −½Principal value:x = cos−1(−½) = 2π3Second value (− principal):x = − 2π3Add ± 2π to find all in [−π, 3π]:From 2π/3:2π3 + 2π = 8π3 ✓From −2π/3:−2π3 + 2π = 4π3 ✓x = − 2π3, 2π3, 4π3, 8π3range is 4π wide → got 4 answers. Makes sense (2 per period × 2 periods).
WE 3
Solve tan x = √3 for −180° ≤ x ≤ 360°
Find all values of x in the given range.
tan x = √3Tan equation — only one rule: just add 180° each time.Principal value:x = tan−1(√3) = 60°Add 180°:60° + 180° = 240° ✓Subtract 180°:60° − 180° = −120° ✓Try more:240° + 180° = 420° (out), −120° − 180° = −300° (out)x = −120°, 60°, 240°tan repeats every 180°, so answers are spaced exactly 180° apart
WE 4
Solve sin(2x) = ½ for 0° ≤ x ≤ 360°
Find all values of x in the given range.
sin(2x) = ½Coefficient 2 inside — substitute y = 2x and transform the range.Let y = 2x.Range: 0° ≤ x ≤ 360° → 0° ≤ y ≤ 720°Equation becomes:sin y = ½Principal:y = sin−1(½) = 30°Second value (180° − 30°):y = 150°Add 360° to each:y = 390°, 510°All four are in [0°, 720°]:y = 30°, 150°, 390°, 510°Convert back: x = y ÷ 2x = 15°, 75°, 195°, 255°a “2x” inside doubles the number of solutions in the same x-range
WE 5
Solve 2cos(2x − 30°) = −1 for −360° ≤ x ≤ 360°
Find all values of x in the given range.
2 cos(2x − 30°) = −1Coefficient 2 and −30° inside — substitute and transform the range carefully.Let y = 2x − 30°.Transform range (× 2 then − 30°):−360° ≤ x ≤ 360° → −720° ≤ 2x ≤ 720° → −750° ≤ y ≤ 690°Isolate cos:cos y = −½Principal:y = cos−1(−½) = 120°Second value (− principal):y = −120° (or +240°)Add ± 360° to find all in [−750°, 690°]:From 120°: 120°, 120° − 360° = −240°, 120° − 720° = −600°, 120° + 360° = 480°From 240°: 240°, 240° − 360° = −120°, 240° − 720° = −480°, 240° + 360° = 600°8 solutions for y:−600°, −480°, −240°, −120°, 120°, 240°, 480°, 600°Convert back: x = (y + 30°) ÷ 2x = −285°, −225°, −105°, −45°, 75°, 135°, 255°, 315°range × 2 = 4π wide for y, so expect ~8 cos solutions. The count is your reality check!
💡 Top tips
Set your calculator mode first. Degrees or radians? If your principal value looks weird, this is usually why.
Always isolate the trig term first. 5 sin x = 3 isn’t ready to solve — divide by 5 first to get sin x = 3/5.
Use the count as a sanity check. A range of 360° gives 2 sin/cos solutions and 2 tan solutions. A range of 720° gives 4 of each. Coefficient 2 inside doubles those again.
For sin equations, the second value is 180° − principal (or π − principal in radians).
For cos equations, the easiest second value is just − principal — the same number with a flipped sign.
For tan equations, you only need one principal value, then keep adding ± 180° until you leave the range.
For transformed equations, do the operations to the range in the same order as the substitution — multiply by the coefficient first, then add the constant.
If using a graph, sketch it lightly so you can see how many intersections exist in your range — that’s your answer count.
⚠ Common mistakes
Stopping at the principal value. Your calculator gives one answer. The exam wants every answer in the range. Always go through the full method.
Wrong calculator mode. Solving in degrees when the question is in radians (or vice versa) will give you an unreadable number. Check the mode badge in the corner of your GDC.
Forgetting to transform the range. If y = 2x + 60° and x ∈ [0°, 360°], then y ∈ [60°, 780°] — not [0°, 360°]. This single mistake costs the most marks.
Doing the range transformation in the wrong order. If y = 2x + 60°, multiply by 2 first, then add 60°. The reverse (subtracting 60° then dividing by 2) is what you do for the final answer, not the range.
Wrong second-value rule. For sin it’s 180° − principal. For cos it’s −principal. Mixing these up is super common under pressure — write the rule above your working before you start.
Adding 180° instead of 360° (or vice versa). Sin and cos repeat every 360°. Tan repeats every 180°. Don’t mix them up.
Counting answers wrong. A 720° range with cos(2x) gives 8 solutions. If you only have 2, you’ve missed something — go back and add more multiples.
Including answers outside the range. If the question says 0° ≤ x ≤ 360°, then 360° is allowed but 360.001° is not. Always re-check inclusivity.
Linear trig equations are pure pattern-matching once you’ve drilled them. Practice 5 problems of each type and you’ll never be slow at this section again.
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