IB Maths AA SL
Topic 3 β Trig Equations & Identities
Paper 1 & 2
~12 min read
Quadratic Trigonometric Equations
If you spot a sin2, cos2, or tan2 in an equation, it’s secretly a quadratic in disguise. Once you see the disguise, you can crack open these problems with the same factorising tools you’ve used a hundred times before.
π What you need to know
- A quadratic trig equation is just a normal quadratic with sin, cos, or tan in place of x.
- Single ratio: substitute y = sin x (or cos / tan), solve like a normal quadratic, then put the trig back in.
- Mixed ratios with sinΒ² or cosΒ²: use sin2x + cos2x = 1 to get everything in one ratio.
- Mixed with tan: use tan x = sin x / cos x to convert tan into sin and cos.
- Always reject values like sin x = 2 or cos x = β3 β those are impossible (sin and cos must be between β1 and 1).
- tan x = anything is fine β tan can take any value.
What’s a “quadratic” trig equation?
Look at this normal quadratic: 2y2 + 3y β 2 = 0. You’d factorise it to (2y β 1)(y + 2) = 0 and get y = Β½ or y = β2. Nothing scary.
Now imagine someone replaces every y with cos x:
It looks scarier now β but it’s the exact same equation. The trick is to mentally “undo” the disguise: let y = cos x, factorise, then translate back at the end.
The substitution flow
Start
2 cos2 x + 3 cos x β 2 = 0
β
Let y = cos x
2y2 + 3y β 2 = 0
β
Solve for x
cos x = Β½ β x = β¦
Method for single-ratio quadratics
If your equation has only one trig function (e.g. just sin, or just cos, or just tan) and there’s a squared term + a linear term, this is the fastest path:
The 4-step method
- Substitute a new variable y equal to the trig ratio.
- Solve the resulting quadratic β factorise, complete the square, use the formula, or use your GDC.
- Reject any values that aren’t possible for that trig ratio (see card below).
- Solve each remaining trig equation for x in the given range.
Which values are possible?
Sin and cos are stuck between β1 and 1. Tan can be anything. So when you finish solving the quadratic, immediately throw out any answer that’s outside the allowed range:
sin x
β1 β€ sin x β€ 1
Reject any sin = Β±1.5, Β±2, etc.
cos x
β1 β€ cos x β€ 1
Reject any cos = Β±1.5, Β±2, etc.
tan x
any real number
Never reject tan values β they can be anything.
If your quadratic gives you sin x = 2, don’t panic β just write “no solution” next to it and move on. Examiners want to see you noticed it was impossible.
Method for mixed-ratio equations
What if your equation has two different trig ratios? Like sin2 and cos, or sin and tan? You need to convert one of them so the equation becomes single-ratio first. Two situations come up most:
Situation 1 β equation has sin2 or cos2 mixed with linear
Use the Pythagorean identity sin2x + cos2x = 1 to swap one out.
π§ Memory trick: “Match the linear, swap the squared”
Look at the linear term. If it’s sin x, swap any cos2x for (1 β sin2x). If the linear is cos x, swap any sin2x for (1 β cos2x). The squared term changes β the linear stays.
Situation 2 β equation has tan mixed with sin or cos
Use tan x = sin xcos x to rewrite the tan in terms of sin and cos. Then clear the fraction by multiplying through by cos x, factorise, and solve.
Decoder β what to do for which equation type
If the equation looks likeβ¦
β
β¦do this
2cos2x + 3 cos x β 2 = 0
β
Single ratio (cos): let y = cos x, factorise, solve.
3cos2x + 5 sin x = 1
β
Mixed (cos2 + sin): swap cos2x = 1 β sin2x to make it all-sin.
2 sin x = tan x
β
Mixed (sin + tan): rewrite tan = sin/cos, clear fractions, factorise.
tan2x = 3
β
Single ratio (tan): square root β tan x = Β±β3, solve each.
πAfter converting, treat it like Linear Trig Equations
Once you’ve factorised and got things like sin x = Β½ or cos x = β1, the rest is exactly the technique from the Linear Trig Equations note β find principal value, second value, and add Β± 360Β° (or 180Β° for tan).
Worked examples
WE 1Solve a quadratic in cos x
Solve 2cos2 x + 3 cos x β 2 = 0 for 0Β° β€ x β€ 360Β°.
2cos2 x + 3 cos x β 2 = 0
Single ratio (only cos) β substitute and factorise.
Let y = cos x: 2y2 + 3y β 2 = 0
Factorise: (2y β 1)(y + 2) = 0
Solve: y = Β½ or y = β2
Replace y with cos x:
cos x = Β½ β
cos x = β2 REJECT (out of range)
Solve cos x = Β½ in [0Β°, 360Β°]:
Principal: x = cosβ1(Β½) = 60Β°
Second (β principal + 360Β°): x = 300Β°
x = 60Β°, 300Β°
always check whether each value of y is even possible before solving!
WE 2Solve a quadratic in sin x
Solve 2sin2 x β sin x β 1 = 0 for 0Β° β€ x β€ 360Β°.
2sin2 x β sin x β 1 = 0
Single ratio (only sin) β substitute, factorise, solve.
Let y = sin x: 2y2 β y β 1 = 0
Factorise: (2y + 1)(y β 1) = 0
Solve: y = β Β½ or y = 1
Both are in [β1, 1] β keep both.
sin x = β Β½: principal = β30Β°, second = 180Β° β (β30Β°) = 210Β°
Add 360Β° to β30Β°: x = 330Β°
In range [0Β°, 360Β°]: x = 210Β°, 330Β°
sin x = 1: x = 90Β°
x = 90Β°, 210Β°, 330Β°
sin x = 1 only has ONE solution per period (the peak of the curve)
WE 3Solve a mixed equation with cos2 and sin
Solve 11 sin x β 7 = 5cos2 x for 0 β€ x β€ 2Ο. Give answers to 3 s.f.
11 sin x β 7 = 5cos2 x
Mixed ratios (sin and cosΒ²). Linear is sin β swap out the cosΒ² to make it all-sin.
Use the identity cos2 x = 1 β sin2 x
Substitute: 11 sin x β 7 = 5(1 β sin2 x)
Expand: 11 sin x β 7 = 5 β 5sin2 x
Move all to one side: 5sin2 x + 11 sin x β 12 = 0
Factorise: (5 sin x β 4)(sin x + 3) = 0
Solve: sin x = 45 or sin x = β3
sin x = β3 REJECT (out of range)
Solve sin x = 4/5 in [0, 2Ο]:
Principal: x = sinβ1(0.8) β 0.927
Second (Ο β principal): x β Ο β 0.927 β 2.21
x β 0.927, 2.21 (3 s.f.)
in radians β make sure your calculator is set to RAD mode!
WE 4Solve a mixed equation with sin and tan
Solve 2 sin x = tan x for 0Β° β€ x β€ 360Β°.
2 sin x = tan x
Mixed (sin and tan) β replace tan using tan = sin/cos.
Use the identity tan x = sin xcos x
Substitute: 2 sin x = sin xcos x
Γ cos x: 2 sin x cos x = sin x
Move to one side: 2 sin x cos x β sin x = 0
Factor sin x: sin x (2 cos x β 1) = 0
Set each factor = 0:
sin x = 0: x = 0Β°, 180Β°, 360Β°
cos x = Β½: x = 60Β°, 300Β°
x = 0Β°, 60Β°, 180Β°, 300Β°, 360Β°
never divide by sin x! you’d lose the sin x = 0 solutions
WE 5Solve an equation with sin2 on its own
Solve 4sin2 x β 3 = 0 for 0Β° β€ x β€ 360Β°.
4sin2 x β 3 = 0
No linear term β just rearrange and square root. Watch for the Β± !
Rearrange: sin2 x = 34
Square root: sin x = Β± β32
Both Β± values are in [β1, 1] β solve each.
sin x = β32: principal = 60Β°, second = 120Β°
sin x = β β32: principal = β60Β° (out), +360Β° β 300Β°; second = 180Β° β (β60Β°) = 240Β°
x = 60Β°, 120Β°, 240Β°, 300Β°
when squaring or square-rooting, the Β± nearly always doubles your number of answers
π‘ Top tips
- Spot the disguise. Whenever you see sin2, cos2, or tan2 in an equation, your first thought should be: “this is a quadratic β let me substitute”.
- Always reject impossible values. Sin and cos are stuck between β1 and 1. Tan can be anything. Cross out impossible solutions clearly so the marker sees you noticed.
- Match the linear, swap the squared. If the equation has sin2 and a linear cos, swap the sin2 using 1 β cos2. Same the other way.
- For tan + sin/cos equations, always rewrite tan as sin/cos first, then clear the fraction, then factorise.
- Don’t divide both sides by a trig term. Factorise instead β dividing throws away solutions where that term equals zero.
- If you can’t spot a factorisation, the quadratic formula works just as well after substitution. Don’t be a hero β use it.
- Once you reach things like sin x = Β½, treat the rest exactly like the linear trig equations note: principal value, second value, then Β± 360Β° or 180Β°.
- For each surviving root, find every angle in the range β it’s easy to forget the second one when you’re rushing.
β Common mistakes
- Trying to factorise without substituting. Substitution makes the algebra obvious. Don’t skip it just because you can β the chance of a sign error goes way up.
- Forgetting to reject impossible values. If you write sin x = 2 and then try to solve it, you’ll waste minutes and lose marks. Spot it and write “no solutions” instantly.
- Using the wrong identity to convert. Mixed sin2 and cos? Swap the squared, not the linear. Many students “simplify” sin x using sin = β(1 β cos2) β don’t.
- Dividing both sides by sin x or cos x. Equations like 2 sin x cos x = sin x should always be factorised, never divided.
- Forgetting the Β± when square-rooting. sin2x = ΒΎ gives sin x = Β± β3/2, not just the positive one. The Β± often doubles your number of solutions.
- Stopping after one solution per branch. Each branch (sin x = Β½, cos x = βΒ½, etc.) usually has 2 solutions per 360Β° period. Find them all.
- Wrong calculator mode. Radians or degrees? Check before you start, and check again before pressing inverse-sin.
- Not multiplying through fractions. When tan = sin/cos creates a fraction, multiply through by cos x immediately β leaving the fraction makes factorising impossible.
That’s the full Trig Equations & Identities chapter done. The big idea across all five notes is the same β get an equation into a single language (one ratio, one angle) and the rest is mechanical. Practice this and Paper 1 trig becomes a guaranteed scoring zone.
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