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Compound and double angle equations

Solve compound and double angle equations by expansion, factorisation and quadratic substitution. Keep zero roots, check tangent domains and find every answer in the interval.

Before you startCompound and double angle formulae, quadratic equations and complete trig root sets

01 / Keep the equation and its domain together

A correct rearrangement must preserve all allowed roots.

Compound-angle equations combine algebra with periodic functions. First simplify the angle expressions, then solve the algebra, then recover every angle in the stated interval. Throughout, keep track of any division or substitution that might exclude an original input.

Original domain → simplify → factor or substitute → interval roots → original check

Use the model to compare equations and intervals. Some roots disappear if you divide by a trig factor. A tangent substitution can also miss angles where the original equation still makes sense.

Keep every allowed rootExplore

EquationInterval

Solutions: 0°, 60°, 180°, 300°, 360°.

No undefined inputs for this equation.

Dividing by sin x would lose: 0°, 180°, 360°.

The lists include only angles inside the selected interval. Gold dots mark exact solutions, not estimates read from a graph.

02 / Expand both compound expressions

Collect sine and cosine before choosing the next step.

Solve sin(x + 30°) = cos(x − 30°) for 0° ≤ x < 360°Worked example

sin x cos 30° + cos x sin 30° = cos x cos 30° + sin x sin 30°

Use the correct signs on both sides.

(√3 − 1)(sin x − cos x) = 0

Multiply by 2 and collect terms.

sin x = cos x

The constant √3 − 1 is nonzero.

cos x = 0 gives no solution to this equation

At those angles sine is ±1, not zero.

tan x = 1, so x = 45°, 225°

Division by cos x is safe after that check.

03 / Factor instead of cancelling a trig function

A factor equal to zero gives an entire branch of solutions.

Solve sin 2x = sin x for 0° ≤ x ≤ 360°Worked example

2sin x cos x − sin x = 0

Bring everything to one side.

sin x(2cos x − 1) = 0

Factor the common sine.

sin x = 0 gives x = 0°, 180°, 360°

Both endpoints are included.

cos x = 1/2 gives x = 60°, 300°

Keep both cosine roots.

All solutions: 0°, 60°, 180°, 300°, 360°

Dividing by sin x at the start would lose three valid answers.

Watch the zero-factor roots disappear and return

Pause, replay or seek freely. The notes explain the same idea and stay in view.

04 / Choose a cosine form to create one unknown

Substitute a single ratio after removing the doubled angle.

Solve cos 2x = sin x for 0° ≤ x < 360°Worked example

1 − 2sin² x = sin x

Choose the sine-only cosine form.

Let u = sin x: 2u² + u − 1 = 0

The quadratic is in u, not in x.

(2u − 1)(u + 1) = 0

Thus u = 1/2 or u = −1.

sin x = 1/2 gives 30°, 150°; sin x = −1 gives 270°

Both quadratic values lie in [−1, 1].

x = 30°, 150°, 270°

Use each branch exactly once.

05 / Reject an impossible trig value before finding angles

Real sine and cosine lie between −1 and 1.

Solve 2cos 2x − 3cos x − 1 = 0 for 0 ≤ x < 2πWorked example

Use cos 2x = 2cos² x − 1

This creates a quadratic in cosine.

4cos² x − 3cos x − 3 = 0

Let u = cos x.

u = (3 ± √57)/8

The positive choice is greater than 1, so reject it.

Let α = arccos((3 − √57)/8)

This principal angle lies in (π/2, π).

x = α or 2π − α

Both angles satisfy the original equation.

Keep the exact inverse-trig expression until a decimal answer is requested. Rejecting an impossible value is different from losing a valid zero factor through division.

06 / Transform the interval with the input angle

Solve in the new angle, then convert every answer back.

Solve sin(2x − π/3) = 1/2 for 0 ≤ x ≤ πWorked example

Let u = 2x − π/3

The interval becomes −π/3 ≤ u ≤ 5π/3.

sin u = 1/2 gives u = π/6, 5π/6 in this interval

No other periodic copies fit.

2x = π/2 or 7π/6

Add π/3 to each root.

x = π/4 or 7π/12

Both lie in the original interval.

07 / Solve a tangent equation on its original domain

Cross-multiply only after stating excluded angles.

Solve tan 2x = tan x for 0° ≤ x < 360°Worked example

Exclude x = 45° + 90°k and x = 90° + 180°k

The original equation requires both tan 2x and tan x to exist.

Let t = tan x, so 2t/(1 − t²) = t

Here t exists and t² ≠ 1 by the original domain.

2t = t − t³, so t(t² + 1) = 0

There are no real roots of t² + 1 = 0.

t = 0 gives x = 0°, 180°

Both are allowed and satisfy the original equation.

08 / Check inputs omitted by a substitution

The formula involving tan x may have extra holes.

Solve tan 2x = 0 for 0° ≤ x ≤ 360°Worked example

Directly, 2x = 180°k

Tangent is zero at integer half-turns.

x = 0°, 90°, 180°, 270°, 360°

All five make the original tan 2x equal zero.

Using t = tan x alone would only find 0°, 180°, 360°

The substitution is undefined at 90° and 270°.

Check those omitted inputs in the original equation

They are valid roots and must be restored.

That restoration is correct for tan 2x = 0. It would be wrong for tan 2x = tan x, because the right side of that original equation is undefined at the same inputs.

09 / Check transformed candidates in the original equation

Squaring can introduce roots that satisfy only a changed equation.

Solve sin x = cos x for 0° ≤ x < 360° by squaring, then checkingWorked example

Squaring gives sin² x = cos² x

Equivalently cos 2x = 0.

Candidates: 45°, 135°, 225°, 315°

Squaring has removed sign information.

At 135° and 315°, sine and cosine have opposite signs

These fail the original unsquared equation.

Can x = 90° be restored after a tangent substitution in both tan 2x = 0 and tan 2x = tan x?

Hint

Evaluate each original equation at 90°.

Worked solution

Only in tan 2x = 0: tan 180° = 0. In tan 2x = tan x, the right side tan 90° is undefined, so 90° is not in the original domain.

11 / Recap

Preserve roots while simplifying.

Expand compound angles and choose a double-angle form that uses one ratio.
Factor before dividing by a trig expression.
Reject quadratic values outside the range of sine or cosine.
Transform the interval along with the angle expression.
Track original tangent domains separately from substitution restrictions.
Check squared candidates and omitted substitution inputs in the original equation.

Section 1 of 11 · Keep the equation and its domain together