What are the fundamental trigonometric identities used in Edexcel International A Levels Pure Mathematics 2?

In Edexcel International A Levels Pure Mathematics 2, the fundamental trigonometric identities include the Pythagorean identities such as sin²θ + cos²θ = 1, the reciprocal identities like secθ = 1/cosθ, and the quotient identities such as tanθ = sinθ/cosθ. These identities are essential for simplifying trigonometric expressions and solving equations.

How do you solve trigonometric equations involving multiple angles in Pure Mathematics 2?

To solve trigonometric equations involving multiple angles in Pure Mathematics 2, first use trigonometric identities to simplify the equation. Then, express the equation in terms of a single trigonometric function. Solve for the angle using inverse trigonometric functions, and consider the periodic nature of trigonometric functions to find all possible solutions within the given interval.

What is the importance of the double angle formulas in solving trigonometric equations?

Double angle formulas, such as sin(2θ) = 2sinθcosθ and cos(2θ) = cos²θ - sin²θ, are crucial in solving trigonometric equations as they allow the transformation of expressions involving double angles into simpler forms. This simplification is particularly useful in equations where direct solutions are not apparent, enabling easier manipulation and solution finding.

Can you explain the use of sum-to-product identities in trigonometry?

Sum-to-product identities, such as sinA + sinB = 2sin((A+B)/2)cos((A-B)/2), are used to convert sums or differences of trigonometric functions into products. These identities are particularly useful in solving trigonometric equations and integrals, as they simplify complex expressions and make them more manageable.

How do trigonometric identities help in proving other mathematical statements in Pure Mathematics 2?

Trigonometric identities are powerful tools in proving mathematical statements as they allow the transformation of complex trigonometric expressions into simpler, equivalent forms. By applying identities like the Pythagorean, angle sum, and double angle formulas, students can demonstrate the equivalence of expressions, verify solutions, and establish the validity of equations in Pure Mathematics 2.

Why is "Reporting only the principal value (one angle) when two or more exist" a common mistake in Trigonometric identities and equations?

Why it happens: Trusting the calculator's ^-1 output without considering symmetry. How to avoid it: For = k, second solution in [0°, 360°) is 180° - \theta_1. For = k, second is 360° - \theta_1. For = k, second is \theta_1 + 180°. Sketch a graph or use the CAST diagram. Source: WMA12/01 examiner reports — flagged annually

Why is "Solving $\sin(\theta + 30°) = 0.5$ in $[0°, 360°)$ for $\theta$ instead of for $\theta + 30°$" a common mistake in Trigonometric identities and equations?

Why it happens: Not adjusting the search interval to the transformed variable. How to avoid it: If = + 30° and [0°, 360°), then [30°, 390°). Find ALL solutions of = 0.5 in [30°, 390°) — there may be one extra outside the original range. Then subtract 30° to get .

Why is "Adding $360°$ instead of $180°$ for tan symmetry" a common mistake in Trigonometric identities and equations?

Why it happens: Confusing 's period with and 's. How to avoid it: and have period 360°; has period 180°. To find all solutions, add 180° (not 360°) and stay within the interval.

Why is "Rejecting $\sin\theta = 0.927$ as 'too close to 1'" a common mistake in Trigonometric identities and equations?

Why it happens: Over-cautious filtering — 0.927 is a valid sine value. How to avoid it: Only reject solutions outside [-1, 1]. Anything inside is valid. = 0.927 gives 67.9° — a perfectly fine angle.

Why is "Dividing $A \sin\theta = B \cos\theta$ by $\cos\theta$ without checking $\cos\theta = 0$" a common mistake in Trigonometric identities and equations?

Why it happens: Habitual algebraic manipulation without thinking about the domain. How to avoid it: Dividing by assumes ≠ 0. After solving, check the angles where = 0 ( = 90°, 270°) against the original equation. If they satisfy it, add them; if not, ignore. In A = B with A, B ≠ 0, the = 0 case never adds a solution.

Pure Mathematics 2Edexcel International A Levels Mathematics (XMA01-YMA01)

Trigonometric identities and equations

Work through the notes, try the practice questions, then take the quiz. The report tells you exactly what to revise next.

Previous Next

Learn this Topic

Start with the slides for the quick version, then go deeper with the full study notes.

Short Study Notes in the form of Slides

Read the notes first. If the method in a worked example clicks, you're ready for the questions.

Page 1 / 0

Detailed Notes

Full prose, callouts and a recap — built for A* mastery, not just a quick scan.

Take these study notes with you

Download a branded PDF — full prose, callouts, recap and memorise list for Trigonometric identities and equations, ready to print or save offline.

Trigonometric Identities and Equations Study Notes — Edexcel IAL Mathematics P2 (WMA12/01, 2018 spec — 2026 onwards)

Pythagorean identity, quotient identity ( $\tan = \sin/\cos$ ), and solving trig equations in degrees or radians on a restricted interval. Quadratic-in-trig substitutions, transformed angles like $\cos(\theta + 30°)$ , and equations of the form $\tan 2\theta = k$ .

At a glance

Pythagorean identity: $\sin^{2}\theta + \cos^{2}\theta = 1$ .
Quotient identity: $\tan\theta = \dfrac{\sin\theta}{\cos\theta}$ .
Sine symmetry: $\sin\theta = \sin(180° - \theta)$ . Period $360°$ .
Cosine symmetry: $\cos\theta = \cos(-\theta) = \cos(360° - \theta)$ . Period $360°$ .
Tangent symmetry: $\tan\theta = \tan(\theta + 180°)$ . Period $180°$ .
Transformed argument: if $\phi = a\theta + b$ , find all $\phi$ in the SHIFTED interval first.
Calculator must be in the correct mode (degrees or radians); WMA12 uses BOTH.

What you’ll learn

Mapped to the Cambridge IGCSE XMA01-YMA01 syllabus (2026 onwards).

P2 6.1 — Know and use the identity $\sin^{2}\theta + \cos^{2}\theta \equiv 1$ .
P2 6.2 — Know and use the identity $\tan\theta \equiv \dfrac{\sin\theta}{\cos\theta}$ .
P2 6.3 — Solve, for a given interval, trigonometric equations including quadratic-in-trig and equations involving transformed angles.
P2 6.4 — Solve trigonometric equations in degrees and in radians.

The two identities

$\sin^{2}+\cos^{2} = 1$ and $\tan = \sin/\cos$ . Use them to reduce mixed equations.

Pythagorean identity.

$\sin^{2}\theta + \cos^{2}\theta \equiv 1.$

(The $\equiv$ means 'identically equal' — true for all $\theta$ .)

Useful rearrangements.

$\cos^{2}\theta = 1 - \sin^{2}\theta$
$\sin^{2}\theta = 1 - \cos^{2}\theta$

Quotient identity.

$\tan\theta \equiv \dfrac{\sin\theta}{\cos\theta} \quad (\cos\theta \neq 0).$

Common uses.

Reducing equations. Replace $\cos^{2}\theta$ with $1 - \sin^{2}\theta$ (or vice versa) to get a quadratic in a SINGLE trig function.
Converting between $\tan$ and $\sin/\cos$ . $3 \sin\theta = 2 \cos\theta \Rightarrow \tan\theta = 2/3$ .
Proving identities. Manipulate LHS using these identities until it matches RHS.

Worked example — reducing equation. Solve $5 \cos^{2}\theta - 4 \sin\theta + 3 = 0$ for $0° \le \theta < 360°$ .

Substitute $\cos^{2}\theta = 1 - \sin^{2}\theta$ : $5(1 - \sin^{2}\theta) - 4 \sin\theta + 3 = 0$ .
Simplify: $5 \sin^{2}\theta + 4 \sin\theta - 8 = 0$ (after multiplying by $-1$ ).
Solve as quadratic in $\sin\theta$ using the formula.

Identity proofs. Example: Show that $\dfrac{1 - \cos^{2}\theta}{\sin\theta} \equiv \sin\theta$ .

LHS $= \dfrac{\sin^{2}\theta}{\sin\theta} = \sin\theta$ = RHS. ✓

The unit circle: any point is (cos θ, sin θ); Pythagoras gives sin²θ + cos²θ = 1.

$\sin^{2}\theta + \cos^{2}\theta = 1$ .
$\tan\theta = \sin\theta / \cos\theta$ .
Use to reduce mixed-trig equations to single-trig.
Both identities are NOT in the formula booklet.

Symmetry and period — finding all solutions

Calculator gives the principal value. Symmetry gives the rest.

When solving $\sin\theta = k$ , $\cos\theta = k$ , or $\tan\theta = k$ , the calculator's $\sin^{-1}$ , $\cos^{-1}$ , $\tan^{-1}$ returns just one value (the principal value). You must use symmetry to find all others in the required interval.

Sine symmetry.

Principal value: $\theta_{1} = \sin^{-1}(k)$ .
Second value: $\theta_{2} = 180° - \theta_{1}$ (or $\pi - \theta_{1}$ in radians).
Period: $360°$ (or $2\pi$ ).

Cosine symmetry.

Principal value: $\theta_{1} = \cos^{-1}(k)$ .
Second value: $\theta_{2} = 360° - \theta_{1}$ (or $2\pi - \theta_{1}$ ). Equivalently, $-\theta_{1}$ .
Period: $360°$ .

Tangent symmetry.

Principal value: $\theta_{1} = \tan^{-1}(k)$ .
Other values: $\theta_{1} + 180°$ , $\theta_{1} + 360°$ , etc.
Period: $180°$ (NOT $360°$ !).

The CAST diagram. Memory aid for which trig function is positive in each quadrant. Starting from the bottom-right going anticlockwise: C (cos $> 0$ ), A (all $> 0$ ), S (sin $> 0$ ), T (tan $> 0$ ).

Worked example. Solve $\sin\theta = 0.5$ for $0° \le \theta < 360°$ .

Principal: $\theta_{1} = 30°$ .
Symmetric: $\theta_{2} = 180° - 30° = 150°$ .
Both in interval. Answer: $\theta = 30°, 150°$ .

Worked example. Solve $\cos\theta = -0.5$ for $0° \le \theta < 360°$ .

Principal: $\theta_{1} = \cos^{-1}(-0.5) = 120°$ .
Symmetric: $\theta_{2} = 360° - 120° = 240°$ .
Answer: $\theta = 120°, 240°$ .

Sine: $\theta_{2} = 180° - \theta_{1}$ .
Cosine: $\theta_{2} = 360° - \theta_{1}$ .
Tan period is $180°$ , not $360°$ .
Use CAST or a sketch to verify quadrant.

Transformed argument: $\sin(\theta + 30°)$ , $\tan 2\theta$ , etc.

Substitute $\phi$ for the bracket. Shift the interval. Solve. Convert back.

Procedure. When the equation involves $\sin(a\theta + b)$ , $\cos(a\theta + b)$ or $\tan(a\theta + b)$ :

Substitute $\phi = a\theta + b$ .
Compute the new interval for $\phi$ from the original interval for $\theta$ .
Solve the equation in $\phi$ on the NEW interval.
Convert back: $\theta = (\phi - b)/a$ .

Worked example. Solve $\cos(\theta + 30°) = -1/2$ for $0° \le \theta < 360°$ .

$\phi = \theta + 30°$ . As $\theta \in [0°, 360°)$ , $\phi \in [30°, 390°)$ .
$\cos\phi = -1/2$ . Principal: $\phi_{1} = 120°$ . Symmetric: $\phi_{2} = 240°$ . Period: add $360°$ if within the interval. $120° + 360° = 480°$ ✗ (out of $[30°, 390°)$ ). $240° + 360° = 600°$ ✗.
Two values: $\phi = 120°, 240°$ .
Convert back: $\theta = 90°, 210°$ .

Worked example. Solve $\tan 2\theta = \sqrt{3}$ for $0 \le \theta < \pi$ (radians).

$\phi = 2\theta$ . As $\theta \in [0, \pi)$ , $\phi \in [0, 2\pi)$ .
$\tan\phi = \sqrt{3}$ . Principal: $\phi_{1} = \pi/3$ . Tan period $\pi$ : $\phi_{2} = \pi/3 + \pi = 4\pi/3$ . Next: $4\pi/3 + \pi = 7\pi/3$ ✗ (out of interval).
Two values: $\phi = \pi/3, 4\pi/3$ .
Convert back: $\theta = \pi/6, 2\pi/3$ .

Why is this important? A transformation $\phi = a\theta + b$ STRETCHES the interval by a factor of $|a|$ (and shifts it). So if $a = 2$ and the original interval is $[0°, 360°)$ , the new interval is $[0°, 720°)$ — there will typically be MORE solutions than there would be for $\theta$ .

Substitute $\phi$ for the inner expression.
Adjust the interval — it stretches with $|a|$ .
Find ALL $\phi$ solutions in the new interval.
Convert back to $\theta$ at the end.

Quadratic-in-trig equations

Substitute $s = \sin\theta$ (or similar), factorise, then convert each root back.

Standard form.

$A \sin^{2}\theta + B \sin\theta + C = 0$

or the analogous with $\cos$ or $\tan$ .

Procedure.

Substitute $s = \sin\theta$ (or $c = \cos\theta$ , $t = \tan\theta$ ).
Solve the quadratic in $s$ .
For each root, check it's in the valid range ( $[-1, 1]$ for $\sin$ , $\cos$ ; any real for $\tan$ ).
For each valid root, find all $\theta$ values in the interval.

Worked example. Solve $2 \sin^{2}\theta + 3 \sin\theta - 2 = 0$ for $0° \le \theta < 360°$ .

$s = \sin\theta$ : $2 s^{2} + 3s - 2 = 0 \Rightarrow (2s - 1)(s + 2) = 0$ , so $s = 1/2$ or $s = -2$ .
Reject $s = -2$ (outside $[-1, 1]$ ).
$\sin\theta = 1/2$ : principal $\theta = 30°$ , symmetric $150°$ .
Answer: $\theta = 30°, 150°$ .

Mixed quadratic. If you start with a mixed equation like $5 \cos^{2}\theta - 4 \sin\theta + 3 = 0$ , FIRST use the Pythagorean identity to convert to a single function:

$5(1 - \sin^{2}\theta) - 4 \sin\theta + 3 = 0 \Rightarrow 5 \sin^{2}\theta + 4 \sin\theta - 8 = 0$ .

Then solve as quadratic in $\sin\theta$ .

Boundary cases. If you get $s = 1$ exactly, $\theta = 90°$ is the unique solution (no second one in $[0°, 360°)$ since $\sin\theta = 1$ only at $\theta = 90°$ ). Similarly for $s = -1$ and the $\cos = \pm 1$ cases.

Substitute the trig function as a single variable.
Solve the quadratic.
Reject roots outside $[-1, 1]$ for sin/cos.
For each valid root, find ALL $\theta$ in the interval.

How it’s examined

Trig identities and equations appears in WMA12/01 P2 every sitting (Jan & Jun), typically as Q5-Q12 — worth $6$ - $10$ marks. Typical questions: quadratic-in-sin equation in $[0°, 360°)$ (5-6 marks); equation involving $\cos(\theta + 30°)$ or $\sin 2\theta$ (4-5 marks); mixed-trig equation requiring the Pythagorean identity (5-7 marks); proof of a simple identity (3-4 marks). Mark schemes split into method marks (M) for using the right identity/substitution, accuracy marks (A) for each correct angle, and an independent (B) mark for stating the rejected root or correct interval. A* students should aim for full marks here — every angle counts, so a sketch / CAST diagram is well worth $30$ seconds of exam time.

Sources: Pearson Edexcel International Advanced Level Mathematics specification (2018 — current for 2026 onwards); Edexcel IAL Mathematical Formulae and Statistical Tables (2018); WMA12/01 Examiner Reports — January 2023, June 2023, January 2024, June 2024. Last reviewed 2026-05-12.

Master this Topic

Worked examples, formulae, definitions and the mistakes examiners flag — everything you need to push from a pass to an A*.

Take this whole topic with you

Download a branded revision sheet — worked examples, formulae, definitions and common mistakes for Trigonometric identities and equations, ready to print or save as PDF.

Step-by-step worked examples — Trigonometric identities and equations

Step-by-step solutions to past-paper-style questions on trigonometric identities and equations, written exactly the way a tutor would explain them at the board.

1Quadratic-in- $\sin\theta$ equation (5 marks, P2)
Core• Adapted from WMA12/01 January 2024 Q7• quadratic trig, solving
Question
Solve $2 \sin^{2}\theta + 3 \sin\theta - 2 = 0$ for $0° \le \theta < 360°$ .
Step-by-step solution
1. Step 1
  Substitute $s = \sin\theta$ :
  $2 s^{2} + 3 s - 2 = 0$
2. Step 2
  Factorise: $(2s - 1)(s + 2) = 0$ . So $s = \tfrac{1}{2}$ or $s = -2$ .
3. Step 3
  $\sin\theta = -2$ has NO solutions (range of $\sin$ is $[-1, 1]$ ). Discard.
4. Step 4
  $\sin\theta = \tfrac{1}{2}$ . Principal value: $\theta = 30°$ . Other solution in $[0°, 360°)$ : $\theta = 180° - 30° = 150°$ .
Answer
$\theta = 30°$ or $\theta = 150°$ .
Examiner tip
M1 for the substitution / factorisation structure. A1 for both values $s = 1/2$ AND $s = -2$ identified. B1 for rejecting $s = -2$ . A1 for $30°$ . A1 for $150°$ . Forgetting the second solution (from $\sin$ being positive in two quadrants) is the biggest mark-loser. Use a CAST diagram or a graph sketch.
2Equation using $\sin^{2} + \cos^{2} = 1$ (5 marks, P2)
Extended• Adapted from WMA12/01 June 2024 Q7• Pythagorean identity, solving
Question
Solve $5 \cos^{2}\theta - 4 \sin\theta + 3 = 0$ for $0° \le \theta < 360°$ .
Step-by-step solution
1. Step 1
  Replace $\cos^{2}\theta$ with $1 - \sin^{2}\theta$ :
  $5(1 - \sin^{2}\theta) - 4 \sin\theta + 3 = 0$
2. Step 2
  Expand: $5 - 5 \sin^{2}\theta - 4 \sin\theta + 3 = 0$ , so $-5 \sin^{2}\theta - 4 \sin\theta + 8 = 0$ . Multiply by $-1$ :
  $5 \sin^{2}\theta + 4 \sin\theta - 8 = 0$
3. Step 3
  Solve via quadratic formula with $s = \sin\theta$ :
  $s = \dfrac{-4 \pm \sqrt{16 + 160}}{10} = \dfrac{-4 \pm \sqrt{176}}{10}$
4. Step 4
  $\sqrt{176} = 4\sqrt{11} \approx 13.266$ . So $s \approx \dfrac{-4 + 13.266}{10} \approx 0.927$ or $s \approx \dfrac{-4 - 13.266}{10} \approx -1.727$ .
5. Step 5
  $s \approx -1.727$ outside $[-1, 1]$ — reject. So $\sin\theta \approx 0.927$ .
6. Step 6
  Principal value: $\theta = \sin^{-1}(0.927) \approx 67.9°$ . Second value: $180° - 67.9° = 112.1°$ .
Answer
$\theta \approx 67.9°$ or $\theta \approx 112.1°$ (1 d.p.).
Examiner tip
M1 for using the Pythagorean identity. A1 for the correct quadratic in $\sin\theta$ . M1 for solving the quadratic. A1 for ONE valid value of $\sin\theta$ (and rejecting the other). M1 for finding the second angle. A1 for both answers in the correct interval. Calculator-allowed paper — but the EXACT-form quadratic structure must be shown to earn the M marks.
3Equation in $\cos(\theta + 30°)$ (4 marks, P2)
Extended• Adapted from WMA12/01 January 2023 Q7• transformed angle, solving
Question
Solve $\cos(\theta + 30°) = -\dfrac{1}{2}$ for $0° \le \theta < 360°$ .
Step-by-step solution
1. Step 1
  Let $\phi = \theta + 30°$ . As $\theta$ ranges over $[0°, 360°)$ , $\phi$ ranges over $[30°, 390°)$ .
2. Step 2
  Solve $\cos\phi = -\tfrac{1}{2}$ . Principal value: $\phi = 120°$ . Symmetry: cosine is also $-1/2$ at $\phi = 360° - 120° = 240°$ . In the extended interval $[30°, 390°)$ , also include $\phi = 120° + 360° = 480°$ ✗ (outside).
3. Step 3
  Take $\phi \in \{120°, 240°\}$ , both in $[30°, 390°)$ .
4. Step 4
  Convert back: $\theta = \phi - 30°$ .
  $\theta = 120° - 30° = 90°, \quad \theta = 240° - 30° = 210°$
Answer
$\theta = 90°$ or $\theta = 210°$ .
Examiner tip
M1 for adjusting the interval to match the transformed variable. M1 for finding ALL values of $\phi$ in the extended interval. A1 A1 for the two values of $\theta$ . The interval shift is the hardest step — students who solve in the original $\theta$ -interval often miss valid solutions. Always work in the transformed variable, then convert back at the end.
4Equation in $\tan 2\theta$ in radians (5 marks, P2)
Extended• Adapted from WMA12/01 June 2023 Q8• tangent, double-angle argument, radians
Question
Solve $\tan 2\theta = \sqrt{3}$ for $0 \le \theta < \pi$ , giving your answers in terms of $\pi$ .
Step-by-step solution
1. Step 1
  Let $\phi = 2\theta$ . As $\theta$ ranges over $[0, \pi)$ , $\phi$ ranges over $[0, 2\pi)$ .
2. Step 2
  Solve $\tan\phi = \sqrt{3}$ . Principal value: $\phi = \pi/3$ . $\tan$ has period $\pi$ , so next solution: $\phi = \pi/3 + \pi = 4\pi/3$ . (Stop — $4\pi/3 + \pi = 7\pi/3 > 2\pi$ .)
3. Step 3
  So $\phi \in \{\pi/3, 4\pi/3\}$ .
4. Step 4
  Convert back: $\theta = \phi / 2$ .
  $\theta = \dfrac{\pi}{6}, \quad \theta = \dfrac{2\pi}{3}$
Answer
$\theta = \dfrac{\pi}{6}$ or $\theta = \dfrac{2\pi}{3}$ .
Examiner tip
M1 for the variable change. M1 for ALL solutions of $\tan\phi = \sqrt{3}$ in the extended interval. A1 A1 for each correct $\theta$ . Using $\tan$ period $\pi$ (not $2\pi$ ) is essential — students who only use the principal value lose 2 marks.
5Mixed identity / equation — using $\tan = \sin/\cos$ (5 marks, P2)
Extended• Adapted from WMA12/01 January 2024 Q12• mixed identities
Question
Solve $3 \sin\theta = 2 \cos\theta$ for $0 \le \theta < 2\pi$ , giving answers to $3$ s.f.
Step-by-step solution
1. Step 1
  Divide both sides by $\cos\theta$ (assuming $\cos\theta \neq 0$ ):
  $3 \tan\theta = 2 \;\;\Rightarrow\;\; \tan\theta = \dfrac{2}{3}$
2. Step 2
  Principal value: $\theta = \tan^{-1}(2/3) \approx 0.5880$ rad.
3. Step 3
  Second value (tan period $\pi$ ): $\theta \approx 0.5880 + \pi \approx 3.7296$ rad.
4. Step 4
  Check $\cos\theta = 0$ case separately. If $\cos\theta = 0$ then $\theta = \pi/2$ or $3\pi/2$ ; substitute into the original: $3 \sin(\pi/2) = 3$ , $2 \cos(\pi/2) = 0$ , $3 \neq 0$ — so $\theta = \pi/2$ is not a solution. Similarly $3\pi/2$ is not. So no extra solutions.
Answer
$\theta \approx 0.588$ rad or $\theta \approx 3.73$ rad (3 s.f.).
Examiner tip
M1 for dividing by $\cos\theta$ (or using $\tan = \sin/\cos$ ). A1 for $\tan\theta = 2/3$ . M1 for the principal value in radians. A1 for the second value. B1 for the answer in $3$ s.f. — exact form not required here because the question says 'to $3$ s.f.'. Dividing by $\cos\theta$ is legitimate as long as you check the $\cos\theta = 0$ case doesn't give an extra solution.

Key Formulae — Trigonometric identities and equations

The formulae you need to memorise for trigonometric identities and equations on the Pearson Edexcel IAL XMA01 / YMA01 paper, with every variable defined in plain English and a note on when to use it.

Pythagorean identity
$\sin^{2}\theta + \cos^{2}\theta = 1$
$\theta$
any angle (degrees or radians)
When to use
Substituting to reduce mixed $\sin$ / $\cos$ equations to a single trig function, or to simplify identity proofs. NOT in the IAL formula booklet — memorise.
Example
Replace $\cos^{2}\theta$ with $1 - \sin^{2}\theta$ to make a quadratic in $\sin\theta$ .
Quotient identity
$\tan\theta = \dfrac{\sin\theta}{\cos\theta} \quad (\cos\theta \neq 0)$
$\theta$
any angle where $\cos\theta \neq 0$
When to use
Converting between $\sin$ , $\cos$ , $\tan$ when an equation mixes them. NOT in the formula booklet.
Example
$3 \sin\theta = 2 \cos\theta \Rightarrow \tan\theta = \tfrac{2}{3}$ .
Symmetry of $\sin$
$\sin\theta = \sin(180° - \theta), \quad \sin\theta = \sin(\theta + 360°)$
$\theta$
any angle
When to use
Finding all solutions of $\sin\theta = k$ in a given interval. NOT in the formula booklet.
Example
If $\sin\theta = 0.5$ then $\theta = 30°, 150°$ in $[0°, 360°)$ .
Symmetry of $\cos$
$\cos\theta = \cos(360° - \theta) = \cos(-\theta), \quad \cos\theta = \cos(\theta + 360°)$
$\theta$
any angle
When to use
Finding all solutions of $\cos\theta = k$ in a given interval. NOT in the formula booklet.
Example
If $\cos\theta = -0.5$ then $\theta = 120°, 240°$ in $[0°, 360°)$ .
Symmetry of $\tan$
$\tan\theta = \tan(\theta + 180°)$
$\theta$
any angle (modulo $180°$ )
When to use
Finding all solutions of $\tan\theta = k$ — $\tan$ has period $180°$ (or $\pi$ in radians), so solutions occur in PAIRS not quartets. NOT in the formula booklet.
Example
If $\tan\theta = \sqrt{3}$ then $\theta = 60°, 240°$ in $[0°, 360°)$ .

Key Definitions and Keywords — Trigonometric identities and equations

Definitions to memorise and the exact keywords mark schemes credit for trigonometric identities and equations answers — sharpened from recent examiner reports for the 2026 Pearson Edexcel IAL XMA01 / YMA01 sitting.

Identity
Examiner keyword
An equation that holds for ALL values of the variable for which both sides are defined. Written with $\equiv$ in formal contexts, with $=$ informally. Example: $\sin^{2}\theta + \cos^{2}\theta = 1$ .
Trigonometric equation
Examiner keyword
An equation involving trig functions of an unknown angle, solved on a specified interval. Solutions are usually angles, given in degrees or radians.
Principal value
The unique value returned by the calculator's inverse trig function ( $\sin^{-1}$ , $\cos^{-1}$ , $\tan^{-1}$ ). For $\sin^{-1}$ and $\tan^{-1}$ this is in $[-90°, 90°]$ ; for $\cos^{-1}$ in $[0°, 180°]$ .
CAST diagram
Memory aid showing in which quadrants $\sin$ , $\cos$ , $\tan$ are positive. Starting in Q4 and going anticlockwise: C (cos +), A (all +), S (sin +), T (tan +).
Restricted interval
Examiner keyword
The range over which solutions are required, typically $[0°, 360°)$ or $[0, 2\pi)$ . The number of solutions depends on the interval: in $[0°, 360°)$ a typical $\sin$ or $\cos$ equation has $2$ solutions, $\tan$ has $2$ , and a $\sin^{2}$ equation can have $4$ .

Common Mistakes and Misconceptions — Trigonometric identities and equations

The traps other students keep falling into on trigonometric identities and equations questions — taken from recent Pearson Edexcel IAL XMA01 / YMA01 examiner reports and mark schemes — and how to avoid them.

✕Reporting only the principal value (one angle) when two or more exist
WMA12/01 examiner reports — flagged annually
Why it happens
Trusting the calculator's $\sin^{-1}$ output without considering symmetry.
How to avoid it
For $\sin\theta = k$ , second solution in $[0°, 360°)$ is $180° - \theta_{1}$ . For $\cos\theta = k$ , second is $360° - \theta_{1}$ . For $\tan\theta = k$ , second is $\theta_{1} + 180°$ . Sketch a graph or use the CAST diagram.
✕Solving $\sin(\theta + 30°) = 0.5$ in $[0°, 360°)$ for $\theta$ instead of for $\theta + 30°$
Why it happens
Not adjusting the search interval to the transformed variable.
How to avoid it
If $\phi = \theta + 30°$ and $\theta \in [0°, 360°)$ , then $\phi \in [30°, 390°)$ . Find ALL solutions of $\sin\phi = 0.5$ in $[30°, 390°)$ — there may be one extra outside the original range. Then subtract $30°$ to get $\theta$ .
✕Adding $360°$ instead of $180°$ for tan symmetry
Why it happens
Confusing $\tan$ 's period with $\sin$ and $\cos$ 's.
How to avoid it
$\sin$ and $\cos$ have period $360°$ ; $\tan$ has period $180°$ . To find all $\tan$ solutions, add $180°$ (not $360°$ ) and stay within the interval.
✕Rejecting $\sin\theta = 0.927$ as 'too close to 1'
Why it happens
Over-cautious filtering — $0.927$ is a valid sine value.
How to avoid it
Only reject solutions outside $[-1, 1]$ . Anything inside is valid. $\sin\theta = 0.927$ gives $\theta \approx 67.9°$ — a perfectly fine angle.
✕Dividing $A \sin\theta = B \cos\theta$ by $\cos\theta$ without checking $\cos\theta = 0$
Why it happens
Habitual algebraic manipulation without thinking about the domain.
How to avoid it
Dividing by $\cos\theta$ assumes $\cos\theta \neq 0$ . After solving, check the angles where $\cos\theta = 0$ ( $\theta = 90°, 270°$ ) against the original equation. If they satisfy it, add them; if not, ignore. In $A \sin\theta = B \cos\theta$ with $A, B \neq 0$ , the $\cos\theta = 0$ case never adds a solution.

Practice questions

Exam-style questions with step-by-step worked solutions. Try one before checking the method.

Past paper style quiz

Get a report showing which sub-topics you've nailed and which ones still need work.

Video lesson

Short walkthrough of the concepts students most often get stuck on.

Trigonometric identities and equations — frequently asked questions

The things students keep getting wrong in this sub-topic, answered.

Trigonometric identities and equations

Short Study Notes in the form of Slides

Detailed Notes

What you’ll learn

How it’s examined

1Quadratic-in-sin⁡θ\sin\thetasinθ equation (5 marks, P2)

2Equation using sin⁡2+cos⁡2=1\sin^{2} + \cos^{2} = 1sin2+cos2=1 (5 marks, P2)

3Equation in cos⁡(θ+30°)\cos(\theta + 30°)cos(θ+30°) (4 marks, P2)

4Equation in tan⁡2θ\tan 2\thetatan2θ in radians (5 marks, P2)

5Mixed identity / equation — using tan⁡=sin⁡/cos⁡\tan = \sin/\costan=sin/cos (5 marks, P2)

Pythagorean identity

Quotient identity

Symmetry of sin⁡\sinsin

Symmetry of cos⁡\coscos

Symmetry of tan⁡\tantan

Identity

Trigonometric equation

Principal value

CAST diagram

Restricted interval

✕Reporting only the principal value (one angle) when two or more exist

✕Solving sin⁡(θ+30°)=0.5\sin(\theta + 30°) = 0.5sin(θ+30°)=0.5 in [0°,360°)[0°, 360°)[0°,360°) for θ\thetaθ instead of for θ+30°\theta + 30°θ+30°

✕Adding 360°360°360° instead of 180°180°180° for tan symmetry

✕Rejecting sin⁡θ=0.927\sin\theta = 0.927sinθ=0.927 as 'too close to 1'

✕Dividing Asin⁡θ=Bcos⁡θA \sin\theta = B \cos\thetaAsinθ=Bcosθ by cos⁡θ\cos\thetacosθ without checking cos⁡θ=0\cos\theta = 0cosθ=0

Practice questions

Past paper style quiz

Assess your understanding

Video lesson

Trigonometric identities and equations — frequently asked questions

1Quadratic-in- $\sin\theta$ equation (5 marks, P2)

2Equation using $\sin^{2} + \cos^{2} = 1$ (5 marks, P2)

3Equation in $\cos(\theta + 30°)$ (4 marks, P2)

4Equation in $\tan 2\theta$ in radians (5 marks, P2)

5Mixed identity / equation — using $\tan = \sin/\cos$ (5 marks, P2)

Symmetry of $\sin$

Symmetry of $\cos$

Symmetry of $\tan$

✕Solving $\sin(\theta + 30°) = 0.5$ in $[0°, 360°)$ for $\theta$ instead of for $\theta + 30°$

✕Adding $360°$ instead of $180°$ for tan symmetry

✕Rejecting $\sin\theta = 0.927$ as 'too close to 1'

✕Dividing $A \sin\theta = B \cos\theta$ by $\cos\theta$ without checking $\cos\theta = 0$