# Trigonometry (Edexcel IAL Maths P1)

> Edexcel International A-Level Mathematics · IAL P1 2018
> Source: https://www.owlsprep.com/study/edexcel-ial-math-p1-trigonometry/

This guide covers all Edexcel IAL P1 trigonometry content, including sine/cosine rules, triangle area calculation, radian measure for arc and sector calculations, and properties of $\text{sin }x$, $\text{cos }x$, $\text{tan }x$ graphs, aligned to the 2018 specification.

**Prerequisites:** [Basic triangle properties and Pythagoras' theorem](https://www.owlsprep.com/study/edexcel-ial-math-p1-basic-geometry/); [Unit conversion skills](https://www.owlsprep.com/study/edexcel-ial-math-p1-algebra-basics/)

## Learning objectives

- Apply sine rule, cosine rule, and $\frac{1}{2}ab\text{sin }C$ formula to solve triangle problems, including the ambiguous sine rule case
- Convert between degrees and radians, calculate arc length and sector area using radian measure
- Recognise and describe the graphs of $\text{sin }x$, $\text{cos }x$, $\text{tan }x$, their symmetry and periodicity
- Accurately recall formulae not provided in the P1 formula booklet for exam use
- Avoid common exam pitfalls when applying trigonometric concepts to structured problems

## 1. Sine Rule, Cosine Rule, and Triangle Area Formula

This section covers core triangle-solving formulae for P1 trigonometry. You are expected to memorise all formulae except the cosine rule, which is provided in the formula booklet.

**Sine Rule** — Relates side lengths of any triangle to sines of opposite angles, used when you have two angles and one side, or two sides and a non-included angle

*Notation:* $\frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C}$ where $a, b, c$ are sides opposite angles $A, B, C$ respectively

*Example:* If $a=5$cm, $A=30^\circ$, $B=45^\circ$, $b = \frac{5 \sin 45^\circ}{\sin 30^\circ} = 5\sqrt{2}$ cm

> **Ambiguous Case of Sine Rule**
>
> When using the sine rule to find an angle given two sides and a non-included angle, there may be two valid angles (one acute, one obtuse) if the side opposite the given angle is shorter than the other given side.

**Cosine Rule** — Used to find a side when two sides and the included angle are known, or to find an angle when all three sides are known

*Notation:* $a^2 = b^2 + c^2 - 2bc \cos A$

**Area of a Triangle** — Calculates area of any triangle when you know two sides and the included angle between them

*Notation:* $Area = \frac{1}{2}ab \sin C$

**Worked example:** In triangle ABC, AB=7 cm, BC=5 cm, angle BAC=35°. Find the two possible sizes of angle ACB, correct to 1 decimal place.

1. Step 1: Label the triangle: side $a = BC =5$ cm opposite angle $A=35^\circ$, side $c = AB=7$ cm opposite angle $C$ (to find)

   $$\frac{a}{\sin A} = \frac{c}{\sin C}$$
2. Step 2: Rearrange to solve for $\sin C$

   $$\sin C = \frac{7 \sin 35^\circ}{5} \approx 0.8030$$
3. Step 3: Find the first acute solution

   $$C = \sin^{-1}(0.8030) \approx 53.4^\circ$$
4. Step 4: Find the second obtuse solution using $\sin C = \sin(180^\circ - C)$

   $$C = 180^\circ - 53.4^\circ = 126.6^\circ$$
5. Step 5: Verify both angles are valid: $35^\circ + 53.4^\circ = 88.4^\circ < 180^\circ$, $35^\circ + 126.6^\circ = 161.6^\circ < 180^\circ$, so both are correct.

> **Exam tip:** Always check for the ambiguous case of the sine rule: if the side opposite the given angle is shorter than the other given side, you must give both possible angle values unless the question specifies the triangle is acute/obtuse.

*Calculator:* allowed

## 2. Radian Measure, Arc Length and Sector Area

Radian measure is an alternative unit for angles that simplifies circular calculations. You must memorise conversion factors and arc/sector formulae, as these are not provided in the P1 formula booklet.

**Radian** — One radian is the angle subtended at the centre of a circle by an arc equal in length to the radius of the circle. $\pi$ radians = 180°

*Example:* Convert 60° to radians: $60 \times \frac{\pi}{180} = \frac{\pi}{3}$ radians. Convert $\frac{\pi}{2}$ radians to degrees: $\frac{\pi}{2} \times \frac{180}{\pi} = 90^\circ$

**Arc Length** — Calculates arc length of a circle, where $r$ is radius and $\theta$ is central angle in radians

*Notation:* $s = r\theta$

**Sector Area** — Calculates area of a sector, where $r$ is radius and $\theta$ is central angle in radians

*Notation:* $A = \frac{1}{2}r^2\theta$

**Worked example:** A sector of a circle has radius 8 cm and perimeter 30 cm. Find the area of the sector, giving your answer to 3 significant figures.

1. Step 1: Perimeter of a sector = $2r + s$, where $s$ is arc length

   $$Perimeter = 2(8) + s = 30$$
2. Step 2: Solve for arc length $s$

   $$s = 30 - 16 = 14 \text{ cm}$$
3. Step 3: Use $s = r\theta$ to find central angle $\theta$ in radians

   $$\theta = \frac{14}{8} = 1.75 \text{ radians}$$
4. Step 4: Calculate sector area

   $$A = 0.5(8^2)(1.75) = 56.0 \text{ cm}^2$$

> **Exam tip:** Always convert angles from degrees to radians before using arc length and sector area formulae. Multiply degrees by $\frac{\pi}{180}$ to get radians.

*Calculator:* allowed

## 3. Graphs of $\text{sin }x$, $\text{cos }x$, $\text{tan }x$: Symmetry and Periodicity

You need to recognise the three core trigonometric graphs, describe their key properties (range, period, symmetry) and identify key points including intercepts, maxima, minima, and asymptotes for $\text{tan }x$.

**Graph of $y = \text{sin }x$** — Sinusoidal wave with amplitude 1, period 360° (2π radians). Odd function, symmetric about origin, range $[-1, 1]$. Maxima at $90^\circ + 360^\circ k$, minima at $270^\circ + 360^\circ k$ for integer $k$.

**Graph of $y = \text{cos }x$** — Sinusoidal wave with amplitude 1, period 360° (2π radians). Even function, symmetric about y-axis, range $[-1,1]$. Maxima at $0^\circ + 360^\circ k$, minima at $180^\circ + 360^\circ k$ for integer $k$.

**Graph of $y = \text{tan }x$** — Periodic function with period 180° (π radians). Vertical asymptotes at $x = 90^\circ + 180^\circ k$ for integer $k$. Range $(-\infty, +\infty)$.

**Worked example:** State the period of the function $y = \text{tan }3x$, and give the x-coordinates of the first two positive asymptotes in degrees.

1. Step 1: Recall period of $y = \text{tan }(kx) = \frac{180^\circ}{k}$

   $$Period = \frac{180^\circ}{3} = 60^\circ$$
2. Step 2: Asymptotes of $\text{tan }3x$ occur where $3x = 90^\circ + 180^\circ k$ for integer $k$
3. Step 3: Calculate first two positive asymptotes: for $k=0$, $x=30^\circ$; for $k=1$, $x=90^\circ$

> **Exam tip:** Only $\text{tan }x$ has a period of 180°; $\text{sin }x$ and $\text{cos }x$ both have 360° periods. Questions often ask you to match graphs to functions, so memorise these key differences.

## 4. Exam Strategy for P1 Trigonometry Questions

**Exam command terms**

Edexcel P1 trigonometry questions use standard command terms with clear response expectations:

- **Find** — Calculate the required value, showing all working. Marks are awarded for method as well as final answer. *(Find the length of side AB correct to 2 decimal places.)*

- **Show that** — Prove the given statement is true using valid steps. You must use provided information to arrive exactly at the stated result. *(Show that the area of the sector is $12\pi$ cm².)*

- **State** — Give the answer without working, though you may use working to check. These are usually 1 mark questions. *(State the period of $y = \text{cos }x$.)*

> **Formula Recall Checklist**
>
> Before your exam, confirm you can write all required formulae from memory: sine rule, $\frac{1}{2}ab\text{sin }C$, arc length $s=r\theta$, sector area $A=\frac{1}{2}r^2\theta$. Only the cosine rule is provided for P1.

**Worked example:** A student calculates the area of a triangle with sides 9 cm, 12 cm, included angle 40° as $0.5*9*12*\text{sin }40$, but their calculator is in radian mode. State the error and find the correct area to 3 significant figures.

1. Step 1: Error: The calculator is in radian mode, so it is calculating $\text{sin }40$ radians instead of $\text{sin }40$ degrees.
2. Step 2: Correct calculation in degree mode

   $$Area = 0.5*9*12*\sin 40^\circ \approx 34.7 \text{ cm}^2$$

> **Exam tip:** Always check your calculator is in the correct mode (degrees or radians) before starting trigonometry questions. Most triangle questions use degrees, while sector/arc questions use radians.

## Common pitfalls

- **Wrong:** Forgetting to check for the ambiguous case of the sine rule, only giving one angle solution
  - Why it fails: When two sides and a non-included angle are given, and the side opposite the given angle is shorter than the other given side, two valid triangles exist, so you lose marks for only giving one answer
  - Correct: Calculate both the acute and obtuse solution for the angle, then check if both sum with the given angle to less than 180° to confirm validity
- **Wrong:** Using degree measure in the arc length or sector area formulae
  - Why it fails: The formulae $s=r\theta$ and $A=\frac{1}{2}r^2\theta$ only work when $\theta$ is in radians, so using degrees gives a massively incorrect answer
  - Correct: Always convert any degree angle to radians first by multiplying by $\frac{\pi}{180}$ before using these formulae
- **Wrong:** Assuming the period of $\text{tan }x$ is 360° like $\text{sin }x$ and $\text{cos }x$
  - Why it fails: $\text{tan }x$ repeats every 180°, so the period of $\text{tan}(kx)$ is $\frac{180^\circ}{k}$, not $\frac{360^\circ}{k}$, leading to wrong answers for period and asymptote questions
  - Correct: Memorise that $\text{tan }x$ has a unique period of 180° (π radians), separate from $\text{sin }x$ and $\text{cos }x$
- **Wrong:** Trying to use trigonometric identities like $\text{sin}^2x + \text{cos}^2x =1$ in P1 questions
  - Why it fails: These identities are part of the P2 syllabus, so P1 questions are designed to be solved without them, wasting time or leading to errors
  - Correct: Use only the sine rule, cosine rule, area formula, radian formulae, and graph properties specified for P1
- **Wrong:** Forgetting to show working steps for "show that" questions
  - Why it fails: Edexcel awards marks for each valid step of the proof, so even if you get the final answer correct, you lose marks if you skip key steps
  - Correct: Write every step of your working clearly, starting from the given information, to arrive exactly at the stated result

## Cheatsheet

| Concept | Formula/Property | Memorise Required? |
| --- | --- | --- |
| Sine Rule | $\frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C}$ | Yes |
| Cosine Rule | $a^2 = b^2 + c^2 - 2bc \cos A$ | No (provided in booklet) |
| Triangle Area | $Area = \frac{1}{2}ab \sin C$ | Yes |
| Degrees to Radians | $\text{Radians} = \text{Degrees} \times \frac{\pi}{180}$ | Yes |
| Arc Length | $s = r\theta$ (θ in radians) | Yes |
| Sector Area | $A = \frac{1}{2}r^2\theta$ (θ in radians) | Yes |
| $y = \text{sin }x$ period | 360° / 2π radians | Yes |
| $y = \text{cos }x$ period | 360° / 2π radians | Yes |
| $y = \text{tan }x$ period | 180° / π radians | Yes |
| $\text{tan }x$ asymptotes | At odd multiples of 90° | Yes |

## What's next

Now that you have mastered core P1 trigonometry, you are ready to practice applying these concepts to structured exam-style questions. Trigonometry is a core, regularly-assessed part of the P1 exam, so consistent practice will help you secure these marks easily. You should also review how trigonometry is combined with other P1 topics like coordinate geometry and differentiation in longer Section B questions, which are common on past papers. Once you are confident with P1 content, you can move on to P2 trigonometry, which covers identities and solving trigonometric equations.

---

From [OwlsPrep](https://www.owlsprep.com) — free study guides for A-Level, IB, AP and IGCSE, written against the official syllabus. Canonical page: https://www.owlsprep.com/study/edexcel-ial-math-p1-trigonometry/