# Trigonometric ratios and their graphs

> Edexcel International GCSE Further Pure Mathematics · 4PM1 2016 Spec (Higher)
> Source: https://www.owlsprep.com/study/edexcel-igcse-further-pure-math-s10-trigonometric-ratios-and-their-graphs/

This guide covers core trigonometric ratio properties, sin/cos/tan graph shapes, exact 30/45/60° values, and CAST diagram use for angles of any magnitude, aligned to the Edexcel 4PM1 specification.

**Prerequisites:** [Basic right-angled trigonometry](https://www.owlsprep.com/study/edexcel-igcse-further-pure-math-s10-right-angled-trigonometry/); [Degree and radian angle measurement](https://www.owlsprep.com/study/edexcel-igcse-further-pure-math-s10-angle-measurement-degrees-radians/)

## Learning objectives

- Define sin, cos, tan for angles of any magnitude (degrees or radians) and sketch their graphs with correct period, amplitude and intercepts
- Recall exact trigonometric values for 30°, 45°, 60° and their radian equivalents
- Apply the CAST diagram and graph symmetry to find trig ratios for related angles (e.g. 120°, 300°)

## Trigonometric ratios for angles of any magnitude

**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: $\sin\theta = \frac{y}{r}$, $\cos\theta = \frac{x}{r}$, $\tan\theta = \frac{y}{x}$

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.

> **mnemonic**
>
> CAST: Starting from the 4th quadrant and moving counterclockwise: **C**osine positive, **A**ll positive, **S**ine positive, **T**angent positive. This tells you which ratios are positive in each quadrant.

**Worked example:** State the sign of $\sin 210^\circ$, $\cos 330^\circ$, and $\tan 150^\circ$.

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 $\sin 210^\circ$ is negative. Q4 only has positive cos, so $\cos 330^\circ$ is positive. Q2 only has positive sin, so $\tan 150^\circ$ is negative.

**Check your understanding**

1. What is the sign of $\cos 225^\circ$?

   - Positive
   - Negative
   - Zero

   *Why:* 225° falls in Q3, where only tan ratios are positive, so cos 225° is negative.

*Calculator:* allowed

## Exact trigonometric values for 30°, 45°, 60°

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) | $\sin\theta$ | $\cos\theta$ | $\tan\theta$ |
| --- | --- | --- | --- | --- |
| 30 | $\frac{\pi}{6}$ | $\frac{1}{2}$ | $\frac{\sqrt{3}}{2}$ | $\frac{1}{\sqrt{3}}$ |
| 45 | $\frac{\pi}{4}$ | $\frac{\sqrt{2}}{2}$ | $\frac{\sqrt{2}}{2}$ | $1$ |
| 60 | $\frac{\pi}{3}$ | $\frac{\sqrt{3}}{2}$ | $\frac{1}{2}$ | $\sqrt{3}$ |

**Worked example:** Find the exact value of $\cos 30^\circ + \sin 45^\circ$, leaving your answer in surd form.

1. Recall exact values: $\cos 30^\circ = \frac{\sqrt{3}}{2}$, $\sin 45^\circ = \frac{\sqrt{2}}{2}$
2. Add the fractions: $\frac{\sqrt{3}}{2} + \frac{\sqrt{2}}{2} = \frac{\sqrt{3} + \sqrt{2}}{2}$
3. No further simplification is possible, so the final answer is $\frac{\sqrt{3} + \sqrt{2}}{2}$

> **tip**
>
> Always leave answers in exact surd form unless the question explicitly asks for a decimal approximation, to avoid losing accuracy marks.

*Calculator:* allowed

## Using symmetry and CAST to find ratios of related angles

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 = $180^\circ - \theta$, sin positive, cos and tan negative
- Q3 (180° < θ < 270°): Reference angle = $\theta - 180^\circ$, tan positive, sin and cos negative
- Q4 (270° < θ < 360°): Reference angle = $360^\circ - \theta$, cos positive, sin and tan negative

**Worked example:** Find the exact value of $\sin 120^\circ$ and $\tan 300^\circ$.

1. For $\sin 120^\circ$: 120° is in Q2, reference angle = $180 - 120 = 60^\circ$. Sin is positive in Q2, so $\sin 120^\circ = \sin 60^\circ = \frac{\sqrt{3}}{2}$
2. For $\tan 300^\circ$: 300° is in Q4, reference angle = $360 - 300 = 60^\circ$. Tan is negative in Q4, so $\tan 300^\circ = -\tan 60^\circ = -\sqrt{3}$

**Check your understanding**

1. What is the exact value of $\cos 210^\circ$?

   - $\frac{\sqrt{3}}{2}$
   - $-\frac{\sqrt{3}}{2}$
   - $\frac{1}{2}$
   - $-\frac{1}{2}$

   *Why:* 210° is in Q3, reference angle = 30°, cos is negative in Q3, so $\cos 210^\circ = -\cos 30^\circ = -\frac{\sqrt{3}}{2}$

*Calculator:* allowed

## Graphs of sin x, cos x, tan x

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) |
| --- | --- | --- | --- | --- |
| $y = \sin x$ | $2\pi$ (360°) | 1 | $-1 ≤ y ≤ 1$ | $(0,0), (\pi,0), (2\pi,0)$ |
| $y = \cos x$ | $2\pi$ (360°) | 1 | $-1 ≤ y ≤ 1$ | $(\frac{\pi}{2},0), (\frac{3\pi}{2},0)$ |
| $y = \tan x$ | $\pi$ (180°) | Undefined | All real numbers | $(0,0), (\pi,0), (2\pi,0)$ |

**Worked example:** Sketch the graph of $y = \sin x$ for $0^\circ ≤ x ≤ 360^\circ$, 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

> **warning**
>
> The tan x graph has vertical asymptotes at $90^\circ, 270^\circ, \frac{\pi}{2}, \frac{3\pi}{2}$ etc. Do not connect the two branches of the tan graph across these asymptotes, as this will cost you marks.

*Calculator:* allowed

## Common pitfalls

- **Wrong:** Using a calculator to find decimal values when exact surd answers are required
  - Why it fails: Exact value questions test recall of core knowledge, decimal approximations lose accuracy marks
  - Correct: Use memorised exact 30/45/60° values and CAST symmetry for exact value questions, unless told otherwise
- **Wrong:** Stating the period of tan x is 360° / 2π
  - Why it fails: Tan x repeats every 180° / π, unlike sin and cos which repeat every 360° / 2π
  - Correct: Memorise: period of sin/cos = 360° (2π), period of tan = 180° (π)
- **Wrong:** Assigning the wrong sign to trig ratios in non-acute quadrants
  - Why it fails: Forgetting CAST rules leads to sign errors that lose all marks for a question
  - Correct: First identify the angle's quadrant, use CAST to confirm the sign before calculating the ratio from the reference angle
- **Wrong:** Sketching tan x as a continuous curve across asymptotes
  - Why it fails: Tan x is undefined at odd multiples of 90° / π/2, so it has breaks at these points
  - Correct: Draw separate disconnected branches of tan x on either side of each vertical asymptote
- **Wrong:** Forgetting to label graph axes as degrees or radians
  - Why it fails: The same x value has different meaning in degrees vs radians, examiners require clear labelling for full marks
  - Correct: Always add an explicit label to the x-axis (e.g. `x (degrees)` or `x (radians)` when sketching trig graphs

## 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 |

## 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.

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