Trigonometric ratios and their graphs

Q: Do I need to memorise exact 30/45/60° trig values for the exam?

Yes, these values are not provided on the 4PM1 formula sheet, so you must recall them to answer exact value questions.

Q: What is the difference between the period of sin/cos and tan?

Sin and cos have a period of 360° (2π radians), while tan has a shorter period of 180° (π radians).

Edexcel International GCSE Further Pure Mathematics· 10B· 45 min read

1. Trigonometric ratios for angles of any magnitude★★☆☆☆⏱ 10 min

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📘 Definition

Trigonometric ratios for any angle

For an angle θ measured counterclockwise from the positive x-axis, with a point $(x, y)$ on the terminal arm at distance $r$ from the origin: $sin θ = \frac{y}{r}$ , $cos θ = \frac{x}{r}$ , $tan θ = \frac{y}{x}$

This definition works for angles greater than 90°, negative angles, and angles measured in both degrees and radians, removing the restriction to right-angled triangles. The sign of each ratio depends on which quadrant the terminal arm of the angle falls into.

📐 Worked Example

State the sign of $sin 21 0^{\circ}$ , $cos 33 0^{\circ}$ , and $tan 15 0^{\circ}$ .

1
Identify the quadrant for each angle: 210° falls in Q3 (180°–270°), 330° falls in Q4 (270°–360°), 150° falls in Q2 (90°–180°)
2
Apply the CAST rule: Q3 only has positive tan, so $sin 21 0^{\circ}$ is negative. Q4 only has positive cos, so $cos 33 0^{\circ}$ is positive. Q2 only has positive sin, so $tan 15 0^{\circ}$ is negative.

✓ Quick check

What is the sign of $cos 22 5^{\circ}$ ?
- Positive
- Negative
- Zero
Reveal answer
Negative —
225° falls in Q3, where only tan ratios are positive, so cos 225° is negative.

2. Exact trigonometric values for 30°, 45°, 60°★★☆☆☆⏱ 10 min

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You must recall exact surd values for these common angles, as they are not provided on the 4PM1 formula sheet. You can derive them using 30-60-90 and 45-45-90 right triangles if needed.

Angle (°)	Angle (rad)	$sin θ$	$cos θ$	$tan θ$
30	$\frac{π}{6}$	$\frac{1}{2}$	$\frac{3}{2}$	$\frac{1}{3}$
45	$\frac{π}{4}$	$\frac{2}{2}$	$\frac{2}{2}$	$1$
60	$\frac{π}{3}$	$\frac{3}{2}$	$\frac{1}{2}$	$3$

📐 Worked Example

Find the exact value of $cos 3 0^{\circ} + sin 4 5^{\circ}$ , leaving your answer in surd form.

1
Recall exact values: $cos 3 0^{\circ} = \frac{3}{2}$ , $sin 4 5^{\circ} = \frac{2}{2}$
2
Add the fractions: $\frac{3}{2} + \frac{2}{2} = \frac{3 + 2}{2}$
3
No further simplification is possible, so the final answer is $\frac{3 + 2}{2}$

3. Using symmetry and CAST to find ratios of related angles★★★☆☆⏱ 12 min

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You can use graph symmetry or the CAST diagram to find trig values for angles outside the 0–90° range by relating them to their acute reference angle (the smallest angle between the terminal arm and the x-axis).

Q2 (90° < θ < 180°): Reference angle = $18 0^{\circ} - θ$ , sin positive, cos and tan negative
Q3 (180° < θ < 270°): Reference angle = $θ - 18 0^{\circ}$ , tan positive, sin and cos negative
Q4 (270° < θ < 360°): Reference angle = $36 0^{\circ} - θ$ , cos positive, sin and tan negative

📐 Worked Example

Find the exact value of $sin 12 0^{\circ}$ and $tan 30 0^{\circ}$ .

1
For $sin 12 0^{\circ}$ : 120° is in Q2, reference angle = $180 - 120 = 6 0^{\circ}$ . Sin is positive in Q2, so $sin 12 0^{\circ} = sin 6 0^{\circ} = \frac{3}{2}$
2
For $tan 30 0^{\circ}$ : 300° is in Q4, reference angle = $360 - 300 = 6 0^{\circ}$ . Tan is negative in Q4, so $tan 30 0^{\circ} = - tan 6 0^{\circ} = - 3$

✓ Quick check

What is the exact value of $cos 21 0^{\circ}$ ?
- $\frac{3}{2}$
- $- \frac{3}{2}$
- $\frac{1}{2}$
- $- \frac{1}{2}$
Reveal answer
$-\frac{\sqrt{3}}{2}$ —
210° is in Q3, reference angle = 30°, cos is negative in Q3, so $cos 21 0^{\circ} = - cos 3 0^{\circ} = - \frac{3}{2}$

4. Graphs of sin x, cos x, tan x★★★☆☆⏱ 15 min

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You must be able to sketch the three trigonometric graphs with correct period, amplitude, intercepts, and clearly label axes as degrees or radians as required.

Function	Period	Amplitude	Range	Key intercepts (0 ≤ x ≤ 2π)
$y = sin x$	$2 π$ (360°)	1	$- 1 \leq y \leq 1$	$(0, 0), (π, 0), (2 π, 0)$
$y = cos x$	$2 π$ (360°)	1	$- 1 \leq y \leq 1$	$(\frac{π}{2}, 0), (\frac{3 π}{2}, 0)$
$y = tan x$	$π$ (180°)	Undefined	All real numbers	$(0, 0), (π, 0), (2 π, 0)$

📐 Worked Example

Sketch the graph of $y = sin x$ for $0^{\circ} \leq x \leq 36 0^{\circ}$ , labelling all intercepts, maximum and minimum points.

1
Draw x-axis labelled x (degrees) from 0 to 360, y-axis from -1 to 1
2
Mark intercepts at $(0, 0)$ , $(180, 0)$ , $(360, 0)$
3
Mark maximum point at $(90, 1)$ and minimum point at $(270, - 1)$
4
Draw a smooth periodic wave connecting these points, matching the standard sine curve shape

5. Common Pitfalls

Wrong move:

Using a calculator to find decimal values when exact surd answers are required

Why:

Exact value questions test recall of core knowledge, decimal approximations lose accuracy marks

Correct move:

Use memorised exact 30/45/60° values and CAST symmetry for exact value questions, unless told otherwise

Wrong move:

Stating the period of tan x is 360° / 2π

Why:

Tan x repeats every 180° / π, unlike sin and cos which repeat every 360° / 2π

Correct move:

Memorise: period of sin/cos = 360° (2π), period of tan = 180° (π)

Wrong move:

Assigning the wrong sign to trig ratios in non-acute quadrants

Why:

Forgetting CAST rules leads to sign errors that lose all marks for a question

Correct move:

First identify the angle's quadrant, use CAST to confirm the sign before calculating the ratio from the reference angle

Wrong move:

Sketching tan x as a continuous curve across asymptotes

Why:

Tan x is undefined at odd multiples of 90° / π/2, so it has breaks at these points

Correct move:

Draw separate disconnected branches of tan x on either side of each vertical asymptote

Wrong move:

Forgetting to label graph axes as degrees or radians

Why:

The same x value has different meaning in degrees vs radians, examiners require clear labelling for full marks

Correct move:

Always add an explicit label to the x-axis (e.g. x (degrees) or x (radians) when sketching trig graphs

6. Quick Reference Cheatsheet

Concept	Key Details
Exact values (30/45/60°)	sin30=1/2, sin45=√2/2, sin60=√3/2; cos30=√3/2, cos45=√2/2, cos60=1/2; tan30=1/√3, tan45=1, tan60=√3
CAST Diagram	Q1: All +ve; Q2: Sin +ve; Q3: Tan +ve; Q4: Cos +ve
sin x / cos x properties	Period = 360° (2π), Amplitude = 1, Range = [-1, 1]
tan x properties	Period = 180° (π), No amplitude, Range = all real numbers, Asymptotes at 90°+180°n

7. Frequently Asked

Do I need to memorise exact 30/45/60° trig values for the exam?

Yes, these values are not provided on the 4PM1 formula sheet, so you must recall them to answer exact value questions.

What is the difference between the period of sin/cos and tan?

Sin and cos have a period of 360° (2π radians), while tan has a shorter period of 180° (π radians).

Going deeper

formula sheetEdexcel 4PM1 Formula SheetOfficial formula sheet for the 4PM1 exam

What's Next

Now that you have mastered trigonometric ratios and their graphs, you are ready to move on to core trigonometric identities, which build on your knowledge of sin, cos and tan relationships to simplify expressions and solve exam-style problems. This topic is a foundational prerequisite for all subsequent trigonometry content in the Edexcel IGCSE Further Pure Math syllabus, including solving trigonometric equations and applying addition formulae. Make sure you practise sketching graphs and using the CAST diagram regularly to retain these core skills for your exam.