Trigonometry

Edexcel International A-Level Mathematics· 2018 Specification (Issue 3), P2 §6.1-6.2· 40 min read

1. Core P2 Trigonometric Identities★★☆☆☆⏱ 10 min

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📘 Definition

P2 Required Trigonometric Identities

Two identities that you must memorize (they are not provided in the exam formula booklet) for all P2 trigonometry problems:

1. \\ \text{tan}\theta = \frac{\text{sin}\theta}{\text{cos}\theta} \quad (\text{valid when } \text{cos}\theta \neq 0)

2. \\ \text{sin}^2\theta + \text{cos}^2\theta = 1

You can rearrange the second identity to substitute for squared trigonometric ratios: $sin^{2} θ = 1 - cos^{2} θ$ or $cos^{2} θ = 1 - sin^{2} θ$ , which is critical for solving quadratic trig equations later.

📐 Worked Example

Given $sin θ = \frac{5}{13}$ and $θ$ is an acute angle, find the exact values of $cos θ$ and $tan θ$ .

1
Step 1: Use the $sin^{2} θ + cos^{2} θ = 1$ identity, substitute $sin θ = \frac{5}{13}$ :
$(\frac{5}{13})^2 + \text{cos}^2\theta = 1$
2
Rearrange to solve for $cos^{2} θ$ :
$\text{cos}^2\theta = 1 - \frac{25}{169} = \frac{144}{169}$
3
Since $θ$ is acute, $cos θ$ is positive, so $cos θ = \frac{12}{13}$
4
Step 2: Use the $tan θ = \frac{sin θ}{cos θ}$ identity:
$\text{tan}\theta = \frac{\frac{5}{13}}{\frac{12}{13}} = \frac{5}{12}$

2. Solving Trig Equations with Transformed Arguments★★★☆☆⏱ 15 min

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P2 trig equations include a single linear transformation of the angle (argument), e.g. $x + 3 0^{o}$ , $2 x$ , $2 (x + \frac{π}{6})$ . Follow this 4-step method to solve them correctly:

Substitute $u$ for the transformed argument, and adjust the given interval by applying the same transformation to the upper and lower bounds
Find all solutions for $u$ in the adjusted interval
Rearrange to solve for $x$ from each valid $u$ value
Verify all $x$ solutions fall within the original given interval

📐 Worked Example

Solve $sin (x + \frac{π}{2}) = \frac{3}{4}$ for $0 < x < 2 π$ . Give your answers to 2 decimal places.

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Step 1: Let $u = x + \frac{π}{2}$ . Adjust the interval: $0 + \frac{π}{2} < u < 2 π + \frac{π}{2} \to \frac{π}{2} < u < \frac{5 π}{2}$
2
Step 2: Solve $sin u = 0.75$ . Principal solution: $u = arcsin (0.75) \approx 0.848$ rad, which is below the lower interval bound. Next valid solutions:
$u = \text{π} - 0.848 ≈ 2.294 \text{ rad}, \quad u = 2\text{π} + 0.848 ≈ 7.131 \text{ rad}$
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Step 3: Solve for $x$ : $x = u - \frac{π}{2}$
$x ≈ 2.294 - 1.571 = 0.72, \quad x ≈ 7.131 - 1.571 = 5.56$
4
Step 4: Verify both 0.72 and 5.56 fall within $0 < x < 2 π$ (≈6.28), so they are valid solutions.

3. Solving Quadratic Trigonometric Equations★★★★☆⏱ 15 min

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Quadratic trig equations include a mix of a squared trig ratio and a linear trig ratio, e.g. $6 cos^{2} x + sin x - 5 = 0$ . Use the $sin^{2} θ + cos^{2} θ = 1$ identity to rewrite the equation as a quadratic in a single trig ratio, then solve as normal.

📐 Worked Example

Solve $6 cos^{2} x + sin x - 5 = 0$ for $0 \leq x < 36 0^{o}$ . Give non-exact answers to 1 decimal place.

1
Step 1: Replace $cos^{2} x$ with $1 - sin^{2} x$ to get a quadratic in $sin x$ :
$6(1 - \text{sin}^2x) + \text{sin}x -5 = 0 → 6\text{sin}^2x - \text{sin}x - 1 = 0$
2
Step 2: Let $y = sin x$ , solve the quadratic equation:
$(3y + 1)(2y - 1) = 0 → y = -\frac{1}{3} \text{ or } y = \frac{1}{2}$
3
Step 3: Solve $sin x = \frac{1}{2}$ : solutions in interval are $x = 3 0^{o}, 15 0^{o}$
4
Step 4: Solve $sin x = - \frac{1}{3}$ : principal solution ≈ -19.47°, so valid solutions in interval are $x = 180 + 19.47 = 199. 5^{o}, 360 - 19.47 = 340. 5^{o}$
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Final solutions: $3 0^{o}, 15 0^{o}, 199. 5^{o}, 340. 5^{o}$

4. Common Pitfalls

Wrong move:

Forgetting to adjust the interval when solving equations with transformed arguments

Why:

You will miss valid solutions or include solutions outside the original interval, losing up to half the marks for the question

Correct move:

First substitute $u$ for the transformed argument, apply the same transformation to the original interval bounds to get the valid range for $u$ before solving

Wrong move:

Using P3 trig identities (sec, cosec, double angle, compound angle) to solve P2 equations

Why:

These are not required for P2, and you risk making unnecessary errors or wasting time in the exam

Correct move:

Only use the two core P2 identities ( $tan θ = \frac{sin θ}{cos θ}$ and $sin^{2} θ + cos^{2} θ = 1$ ) for all P2 trig problems

Wrong move:

Rejecting negative trig ratio solutions without checking the interval

Why:

Angles in quadrants 2, 3 and 4 have negative sine or cosine values, so you will eliminate up to half of your valid solutions

Correct move:

Find all solutions for the trig ratio in the adjusted interval, including those where the ratio is negative, before solving for $x$

Wrong move:

Dividing both sides of a trig equation by $sin x$ or $cos x$ to simplify

Why:

This removes all solutions where the divided ratio equals zero, leading to lost marks

Correct move:

Rearrange all terms to one side of the equation and factor out the common trig ratio instead of dividing

5. Quick Reference Cheatsheet

Concept	Formula/Method	Key Exam Note
Core Identity 1	$sin^{2} θ + cos^{2} θ = 1$	Must memorize, not in formula booklet
Core Identity 2	$tan θ = \frac{sin θ}{cos θ}$	Valid only when $cos θ \neq = 0$
Transformed Argument Equations	Adjust interval for $u$ 2. Solve for $u$ 3. Solve for $x$	Adjust interval first to avoid missing solutions
Quadratic Trig Equations	Replace squared ratio with identity → solve quadratic → solve trig equations	Never divide by a trig ratio, factor instead

Going deeper

formula_bookEdexcel IAL Maths Formula BookletNo P2 trig identities are included in this booklet, you must memorize them

What's Next

Now that you have mastered P2 trigonometry, you are ready to progress to more advanced trigonometric content in Pure Mathematics 3 (P3), including reciprocal trigonometric ratios, compound and double angle identities, and R-form expressions. These skills build directly on the core identities and equation-solving methods you learned here, so make sure you are confident solving all P2-style trig equations before moving on. You can also practice applying these skills to P2 past paper questions to familiarize yourself with Edexcel's exam phrasing and mark scheme expectations, and ensure you are avoiding the common pitfalls outlined in this guide.