Ratios and Graphs of Trigonometric Functions · Form 5
Ratios and Graphs of Trigonometric Functions: Paper 2 Answering Guide
How Ratios and Graphs of Trigonometric Functions appears in Mathematics Paper 2 (Subjective), and how to lay out your working so you earn every method mark.
How it is examined
Paper 2 often asks you to sketch a sine or cosine graph over a stated range, then use it to find angles that satisfy an equation. The graph must be neat and correctly scaled; the sign of a ratio in the second, third or fourth quadrant is where marks quietly go.
Showing your working
Mathematics Paper 2 (Subjective) is worth 100 marks and gives marks for the steps, not only the answer. Write each line clearly: state the formula or rule, substitute the numbers, then simplify.
If the question carries units, carry them through to the final line.
The question
Question: (a) Find the value of sin 315°, giving your answer in surd form. (b) Solve the equation 2 cos x + 1 = 0 for 0° ≤ x ≤ 360°.
Mark-earning layout, line by line
- (a) State the reference angle: 315° lies in the 4th quadrant, so the reference angle = 360° − 315° = 45°.
- (a) Apply the quadrant sign: sine is negative in the 4th quadrant, so sin 315° = −sin 45°.
- (a) Write the surd value: sin 45° = 1/√2, therefore sin 315° = −1/√2 = −√2/2.
- (b) Rearrange to isolate the ratio: 2 cos x + 1 = 0, so cos x = −1/2.
- (b) Reference angle: cos⁻¹(1/2) = 60°.
- (b) Identify the quadrants: cosine is negative in the 2nd and 3rd quadrants, so x = 180° − 60° and x = 180° + 60°.
- (b) Answer: x = 120° and x = 240°, both within 0° ≤ x ≤ 360°.
Source:SPM: Format Pentaksiran mulai 2021, Matematik (1449)
Frequently asked questions
How should I lay out a 'solve for 0° ≤ x ≤ 360°' answer to secure the method marks?
Set it out in stages: rearrange to cos x = −1/2, state the reference angle 60°, name the quadrants where cosine is negative, then write both solutions 120° and 240°. Each stage carries a mark, so even a slip in the final arithmetic still leaves the method marks above it intact.
Do I lose marks for giving the answer as −0.707 instead of −1/√2?
If the question says 'surd form' or 'exact value', then yes, −0.707 is not accepted and loses the accuracy mark. Leave the answer as −1/√2 or the rationalised −√2/2.
A decimal is only fine when the question asks for an answer correct to a stated number of decimal places.
In part (a) must I show the reference angle, or can I just write the value down?
Show it. Writing the value straight down risks everything on one line, if it is wrong you score zero.
Stating the reference angle 45° and the quadrant sign first earns method marks even if the final surd slips, and it makes your reasoning easy for the marker to follow.