Skip to main content
MathidiaMathematics Academy
A Level synthesis
Lesson 71 of 75
Introduction

Pure Mathematics synthesis

Objective. Connect functions, trigonometry, vectors and calculus.

  1. 1. Learn
  2. 2. Worked example
  3. 3. Practice
  4. 4. Feedback
  5. 5. Continue

Learn: the key idea

Pure Mathematics synthesis brings together functions, trigonometry, vectors and calculus from across the A Level course to solve multi-step problems that do not announce which technique to use, requiring recognition of structure, such as spotting a composite function that needs the chain rule, or a trigonometric identity that simplifies before differentiating. Strong synthesis means selecting an efficient route, for example converting a trigonometric equation into a quadratic in sin x before solving, or using vector methods to find an angle via the dot product rather than coordinate geometry. For example, finding the gradient of a curve defined trigonometrically at a given point may require both an identity and the chain rule together.

Worked example

Find the gradient of the curve y = sin(2x) + x² at the point where x = π/4, combining trigonometric and calculus techniques from this pure synthesis topic.

2 … 4
Coordinate grid with the line 2 … 4
  1. 1Differentiate: dy/dx = 2cos(2x) + 2x
  2. 2At x = π/4: 2x = π/2, so cos(π/2) = 0
  3. 3First term: 2 × 0 = 0
  4. 4dy/dx = 0 + 2(π/4) = π/2

Answer: Gradient = π/2

Concept mastery

Based on your past attempts at this lesson — the question types to practise again come first.

Complete a practice set to see your concept breakdown here.

Practice

Practice happens on its own screen, one question at a time. Answers stay hidden until you submit yours, and you can stop and pick up at the same question later.

Start practising

Common mistake. Students attempt a single technique throughout a multi-part synthesis question even when the structure changes, for example trying only algebraic substitution when a trigonometric identity is needed first. Read the full problem before choosing a method, since synthesis questions often require switching technique mid-solution.

Your progress

Tick each step as you finish it. Your exact reading position is saved automatically, so Continue drops you back on the same line.

0 of 5 steps complete

Your place saves automatically as you read.

My dashboard