Trigonometry

Mathematics (YMA01) · Pearson Edexcel International A Level · 9 min read · Updated: Sep 7, 2026

Welcome to Trigonometry: Your Guide to Angles and Shapes!

Hello future mathematician! This chapter, Trigonometry, is fundamental to Pure Mathematics 1 (P1). Don't worry if the term sounds complicated—it just means "measurement of triangles."

In this unit, we will move beyond simple right-angled triangles and learn how to solve any triangle problem. We'll also explore the powerful periodic nature of sine, cosine, and tangent, which underpins everything from sound waves to electrical current.

Let's dive in and make these concepts crystal clear!

1. Review: Right-Angled Triangles and SOH CAH TOA

Before tackling P1 trigonometry, we quickly need to review the basics. This knowledge is your foundation!

Understanding the Sides

In a right-angled triangle, relative to a specific angle $\theta$:

Hypotenuse (H): The longest side, opposite the right angle.
Opposite (O): The side directly across from angle $\theta$.
Adjacent (A): The side next to angle $\theta$ (that is not the hypotenuse).

The Key Ratios

These ratios define the relationships between the sides and the angle:

Memory Aid: SOH CAH TOA

SOH: Sine $\theta = \frac{\text{Opposite}}{\text{Hypotenuse}}$
CAH: Cosine $\theta = \frac{\text{Adjacent}}{\text{Hypotenuse}}$
TOA: Tangent $\theta = \frac{\text{Opposite}}{\text{Adjacent}}$

Key Takeaway: SOH CAH TOA only works for triangles containing a 90° angle! P1 requires methods for triangles without one.

2. Solving Non-Right-Angled Triangles (The Rules)

When you encounter a triangle that isn't right-angled, you need two special tools: the Sine Rule and the Cosine Rule.

Standard Triangle Notation

To use the rules correctly, we use standard notation:

The angles are labeled with capital letters (A, B, C).
The sides opposite those angles are labeled with the corresponding lowercase letters (a, b, c).

The Sine Rule

The Sine Rule is used when you have a matching pair of side and opposite angle, and you need to find one other missing element.

The Formula

To find a side:
$$\frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C}$$

To find an angle (just flip the formula!):
$$\frac{\sin A}{a} = \frac{\sin B}{b} = \frac{\sin C}{c}$$

When to Use It (The Cases)

Finding a side: You need two angles and one side (AAS).
Finding an angle: You need two sides and one non-included angle (SSA).

Example: If you know angle A and side a, and you know angle B, you can easily find side b.

The Cosine Rule

The Cosine Rule is more powerful but also slightly more complicated. Think of the Cosine Rule as your tool when you don't have a matching pair.

Case 1: Finding a Side (SAS - Side-Angle-Side)

You need two sides and the included angle (the angle between them).
The formula to find side a is:
$$a^2 = b^2 + c^2 - 2bc \cos A$$

Note how the single side squared ($a^2$) is opposite the angle used in the cosine term ($\cos A$).

Case 2: Finding an Angle (SSS - Side-Side-Side)

If you know all three sides, you can rearrange the formula above to find any angle. This is often the preferred form for finding angles:
$$\cos A = \frac{b^2 + c^2 - a^2}{2bc}$$

Common Mistake to Avoid: When using the Cosine Rule to find an angle, be careful with the signs. The side opposite the angle you are trying to find ($a^2$) must be subtracted at the end of the numerator.

Key Takeaway: Use the Sine Rule when you have a side/angle pair. Use the Cosine Rule when you have SSS or SAS.

3. Area of a Triangle

We all know the standard area formula: $Area = \frac{1}{2} \times \text{base} \times \text{height}$. But in P1, we often don't know the perpendicular height!

The Trigonometric Area Formula

This formula allows you to find the area using two sides and the angle included between them (SAS).

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

Analogy: Imagine two arms (a and b) holding an angle (C). The formula measures the space they enclose.

Step-by-Step Area Calculation:

Identify the two sides you know (e.g., $a$ and $b$).
Identify the angle C that is exactly between those two sides.
Substitute the values into the formula.

Key Takeaway: Always ensure the angle used in the area formula is the one trapped between the two sides you use.

4. Trigonometric Identities

Identities are equations that are true for all possible values of $\theta$. In P1, you must master two core identities. These are crucial for proving relationships and simplifying complex equations.

Identity 1: The Quotient Identity

This links sine, cosine, and tangent:

$$\tan \theta \equiv \frac{\sin \theta}{\cos \theta}$$

Did you know? This identity explains why the tangent graph has asymptotes. Whenever $\cos\theta = 0$, the fraction becomes "division by zero," creating a vertical line the graph can never touch!

Identity 2: The Pythagorean Identity

This identity comes directly from applying the Pythagorean theorem ($x^2 + y^2 = r^2$) to the unit circle (where $r=1$, $x=\cos\theta$, and $y=\sin\theta$).

$$\sin^2 \theta + \cos^2 \theta \equiv 1$$

Note on Notation: $\sin^2 \theta$ simply means $(\sin \theta)^2$.

Using the Identities

These identities allow you to express one trig function in terms of another.

Example: If you need to solve an equation involving both $\sin \theta$ and $\cos \theta$, you might rearrange Identity 2 to substitute $\sin^2 \theta = 1 - \cos^2 \theta$, allowing you to solve the equation entirely in terms of cosine.

Key Takeaway: Memorize $\tan \theta = \frac{\sin \theta}{\cos \theta}$ and $\sin^2 \theta + \cos^2 \theta = 1$. They are your algebraic superpowers in trigonometry.

5. Graphs of Sine, Cosine, and Tangent (0° to 360°)

Understanding the graphs is essential because they show how angles relate to multiple solutions, which is critical when solving equations. We focus on the range $0^\circ \le \theta \le 360^\circ$.

The Sine Function: $y = \sin \theta$

Shape: Starts at 0, goes up, comes down, and returns to 0 (a smooth wave).
Period: 360° (It repeats every 360°).
Maximum Value: 1 (at 90°).
Minimum Value: -1 (at 270°).
Key Points: (0°, 0), (90°, 1), (180°, 0), (270°, -1), (360°, 0).

The Cosine Function: $y = \cos \theta$

Shape: Starts at its peak (1), goes down, and returns to the peak (1).
Period: 360°.
Maximum Value: 1 (at 0° and 360°).
Minimum Value: -1 (at 180°).
Key Points: (0°, 1), (90°, 0), (180°, -1), (270°, 0), (360°, 1).

Did you know? The cosine graph is simply the sine graph shifted 90° to the left! $\cos \theta \equiv \sin (\theta + 90^\circ)$.

The Tangent Function: $y = \tan \theta$

The tangent graph behaves very differently because it uses the ratio $\frac{\sin \theta}{\cos \theta}$.

Period: 180° (It repeats every 180°).
Range: $y$ can be any real number (it goes from $-\infty$ to $+\infty$).
Asymptotes (Vertical Lines): At 90° and 270°. These are the points where $\cos \theta = 0$.
Key Points: (0°, 0), (180°, 0), (360°, 0).

Key Takeaway: Sine and Cosine have a period of 360° and oscillate between -1 and 1. Tangent has a period of 180° and approaches infinity.

6. Solving Trigonometric Equations in the Range 0° to 360°

The biggest hurdle in P1 trigonometry is realizing that for a given ratio (like $\sin \theta = 0.5$), there is usually more than one angle between 0° and 360° that satisfies it.

We use the CAST diagram to find all possible solutions.

Introduction to the CAST Diagram

The 360° rotation is divided into four quadrants. The CAST diagram tells you which trigonometric function is positive in each quadrant.

Visualizing the Diagram (Start in the bottom right and move counter-clockwise):

C (Quadrant IV: 270° to 360°): Cosine is Positive.
A (Quadrant I: 0° to 90°): All (Sine, Cosine, Tangent) are Positive.
S (Quadrant II: 90° to 180°): Sine is Positive.
T (Quadrant III: 180° to 270°): Tangent is Positive.

Step-by-Step Guide to Solving Equations

Follow this precise method every time to ensure you find all solutions for $0^\circ \le \theta \le 360^\circ$.

Example: Solve $\sin \theta = 0.5$

Step 1: Find the Principal Angle ($\alpha$)

Ignore the sign (if it's negative, treat it as positive for this step). This gives you the acute angle in Quadrant A (0° to 90°), which we call the base angle ($\alpha$).
$\alpha = \sin^{-1} (0.5) = 30^\circ$.

Step 2: Identify the Quadrants

Look at the original equation. Since $\sin \theta = 0.5$ is positive, sine is positive in Quadrants A and S.

Step 3: Calculate the Final Solutions

Use the base angle $\alpha$ and the quadrant rules:

Quadrant A (0° to 90°): $\theta_1 = \alpha$
$\theta_1 = 30^\circ$
Quadrant S (90° to 180°): $\theta_2 = 180^\circ - \alpha$
$\theta_2 = 180^\circ - 30^\circ = 150^\circ$

The solutions are $\theta = 30^\circ$ and $\theta = 150^\circ$.

Rules for Each Quadrant

If the base angle is $\alpha$:

Quadrant I (A): Solution = $\alpha$
Quadrant II (S): Solution = $180^\circ - \alpha$
Quadrant III (T): Solution = $180^\circ + \alpha$
Quadrant IV (C): Solution = $360^\circ - \alpha$

Handling Negative Values

If the ratio is negative (e.g., $\cos \theta = -0.8$), you still find the base angle $\alpha$ using the positive value ($\cos^{-1} (0.8)$). Then, you select the quadrants where that function is negative.

Example: If $\cos \theta$ is negative, you must choose the quadrants where C is NOT positive: S and T (180 - $\alpha$ and 180 + $\alpha$).

Solving Equations with Functions of $k\theta$

If the equation involves $\sin(2\theta)$ or $\cos(3\theta)$, you must adjust your range first.

If $0^\circ \le \theta \le 360^\circ$, then $0^\circ \le 2\theta \le 720^\circ$.

This means you must cycle around the CAST diagram twice to find all the solutions for $2\theta$. Once you have found all solutions for $2\theta$, divide by 2 to find $\theta$.

Key Takeaway: The CAST diagram is mandatory for finding multiple solutions. Always find the base angle $\alpha$ first, then use the quadrant rules (180 ± $\alpha$, 360 - $\alpha$).

Quick P1 Trigonometry Review

Checklist of Core P1 Skills:

Solve SSS and SAS triangles using the Cosine Rule.
Solve AAS and SSA triangles using the Sine Rule.
Calculate area using $Area = \frac{1}{2}ab \sin C$.
Manipulate the identities $\tan \theta = \frac{\sin \theta}{\cos \theta}$ and $\sin^2 \theta + \cos^2 \theta = 1$.
Sketch and recognize the shapes of $y=\sin \theta$, $y=\cos \theta$, and $y=\tan \theta$ between 0° and 360°.
Use the CAST diagram method to find all solutions to simple trig equations in the range $0^\circ \le \theta \le 360^\circ$.

Quick check

Can you answer these now?

Open each question to check the key ideas from this chapter.

In what specific type of triangle is SOH CAH TOA applicable?

SOH CAH TOA only works for right-angled triangles (triangles containing a 90° angle).

When is the Sine Rule used to solve a triangle?

The Sine Rule is used when you have a "matching pair" consisting of a side and its opposite angle, plus one other piece of information.

What is the Sine Rule formula for finding a missing side?

$\frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C}$

Which formula is used to find a third side when given two sides and the included angle (SAS)?

The Cosine Rule for sides: $a^2 = b^2 + c^2 - 2bc \cos A$.

How do you find an angle in a triangle if you know the lengths of all three sides (SSS)?

Use the rearranged Cosine Rule: $\cos A = \frac{b^2 + c^2 - a^2}{2bc}$.

What is the trigonometric formula for the area of a triangle?

$Area = \frac{1}{2}ab \sin C$, where $a$ and $b$ are sides and $C$ is the angle between them.

What is the Quotient Identity linking tangent, sine, and cosine?

$\tan \theta \equiv \frac{\sin \theta}{\cos \theta}$

What is the Pythagorean Identity for sine and cosine?

$\sin^2 \theta + \cos^2 \theta \equiv 1$

What are the periods of the sine, cosine, and tangent functions?

Sine and cosine have a period of 360°, while tangent has a period of 180°.

Which quadrants in the CAST diagram indicate where sine, cosine, and tangent are positive?

Q1 (A): All positive; Q2 (S): Sine only; Q3 (T): Tangent only; Q4 (C): Cosine only.

How do you find the "base angle" (α) when solving a trigonometric equation?

Calculate $\alpha = \text{trig}^{-1}(\text{positive value})$, always ignoring any negative signs from the original ratio.

What are the formulas for finding second solutions in quadrants II, III, and IV based on base angle α?

Q2: $180^\circ - \alpha$; Q3: $180^\circ + \alpha$; Q4: $360^\circ - \alpha$.

If an equation involves kθ (e.g., sin 2θ), what adjustment must be made to the solution process?

You must multiply the boundaries of the range by k and find all solutions for kθ within that expanded range before dividing by k.

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