This module covers how to solve equations such as sin x = 0.5 or cos 2x = −0.3 on a stated interval, and how to read the graphs of sine, cosine and tangent when they are stretched, shifted or moved up and down. The two halves belong together: the graph tells you how many answers to expect, and the equation tells you where they are.
Check the current Cambridge Additional Mathematics 0606 syllabus for the exact content, notation and calculator rules in your exam year. Our Additional Mathematics learning guide shows where this module sits among the others.
What should you already know?
You need the graphs of sin x, cos x and tan x between 0° and 360°, the exact values for 30°, 45° and 60°, and the idea of a function being transformed. If exact values are shaky, revise them first.
The trigonometric identities module is a useful neighbour, because many equations need an identity before they can be solved. If your course uses radians, radians, arcs and sectors covers the units.
An orienting example
Solve 2 sin x = −1 for 0° ≤ x ≤ 360°.
Step 1, isolate: sin x = −1/2.
Step 2, basic angle: ignore the sign and find sin⁻¹(1/2) = 30°.
Step 3, quadrants: sine is negative in the third and fourth quadrants, so x = 180° + 30° = 210° and x = 360° − 30° = 330°.
Check: sin 210° = −0.5 and sin 330° = −0.5. A sketch of y = sin x shows the line y = −0.5 meeting the curve twice between 0° and 360°, which matches.
That one question used the core habit of the whole module: find every angle in the interval, then confirm the count with a picture.
In which order should you study it?
- Solve a trigonometric equation on a stated interval: the basic angle and quadrant method that all later lessons rely on.
- Track a phase shift and period: reads a graph’s stretch and slide straight from its equation.
- Find all roots after a variable substitution: handles cases like sin(2x − 30°) and quadratics in sin x.
- Use a sketch to detect missing solutions: a quick count that catches the answers you forgot.
- Interpret an amplitude change separately from translation: separates the height of a wave from the line it oscillates around.
Then work through the mixed practice set. One lesson a day and the practice set at the weekend is a steady pace.
Which traps catch most students here?
- Stopping at the calculator value, which gives only one of the solutions.
- Dividing the interval wrongly after a substitution such as 2x, so solutions are lost or invented.
- Reading a shift from the wrong bracket, for example treating 2x − 60° as a shift of 60° when it is 30°.
- Confusing amplitude with the range, especially when the graph has been moved up or down.
- Forgetting that tan has a different period from sin and cos.
Each lesson shows one of these slips in full and then corrects it.
How should you use the practice set?
Attempt each question on paper before you open the answer, and sketch the graph whenever a question says “all solutions”. The non-calculator working trainer helps you keep exact values sharp. The mistake log and retest queue is a good place to record which step went wrong.
When you get something wrong, read the routing notes at the end of the practice set and go back to the lesson it names. Fix the lesson, then retry a fresh question a few days later.
If you keep missing solutions in the interval, a teacher in online one-to-one Additional Mathematics tuition can work through your sketches and answers with you.