Trigonometry

Q: What is the Sine Rule and when should you use it?

The Sine Rule states \(\frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C}\) (or inverted for angles), used when you know an angle-side opposite pair.

Q: What is the ambiguous case of the Sine Rule and how do you check for a second solution?

When given two sides and a non-included angle (SSA), a second possible angle is \(180^\circ - \theta\); it exists if adding it to the given angle leaves a total strictly less than \(180^\circ\).

Q: What is the Cosine Rule and under what triangle conditions is it used?

It is \(a^2 = b^2 + c^2 - 2bc \cos A\) (or \(\cos A = \frac{b^2 + c^2 - a^2}{2bc}\)), used when given two sides and the included angle (SAS) or all three sides (SSS).

Q: What is the formula for the area of any non-right-angled triangle?

\(\text{Area} = \frac{1}{2}ab \sin C\), where \(a\) and \(b\) are two side lengths and \(C\) is the included angle between them.

Q: What are the key properties of the graph \(y = \sin \theta\)?

It has a period of \(360^\circ\), amplitude bounded between \(-1\) and \(1\), passes through the origin \((0^\circ, 0)\), and satisfies \(\sin(-\theta) = -\sin \theta\).

Q: What are the key properties of the graph \(y = \cos \theta\)?

It has a period of \(360^\circ\), amplitude bounded between \(-1\) and \(1\), starts at \((0^\circ, 1)\), and has reflection symmetry across the \(y\)-axis (\(\cos(-\theta) = \cos \theta\)).

Q: What are the key features and period of the graph \(y = \tan \theta\)?

It has a period of \(180^\circ\), an infinite range from \(-\infty\) to \(+\infty\), and vertical asymptotes at odd multiples of \(90^\circ\) (e.g. \(\theta = \pm 90^\circ, \pm 270^\circ\)).

Q: How do you find the second related solution within \([0^\circ, 360^\circ]\) for basic trig equations?

For sine: \(180^\circ - \text{PV}\); for cosine: \(360^\circ - \text{PV}\); for tangent: \(\text{PV} + 180^\circ\), where \(\text{PV}\) is the principal value.

Q: What does the CAST diagram represent and which functions are positive in each quadrant?

It shows positive quadrants: Q1 (0°–90°): A ll; Q2 (90°–180°): S ine; Q3 (180°–270°): T angent; Q4 (270°–360°): C osine.

Mathematics 2210 · CCEA A Level · 8 min read · Updated: 31 Aug 2026

Welcome to AS Pure Mathematics: Trigonometry

Welcome to one of the most useful and rewarding chapters in A-Level Mathematics! You have likely met trigonometry before when working with right-angled triangles using SOH CAH TOA. In AS 1 Pure Mathematics, we are going to expand your toolkit so you can work with any triangle, understand the wave-like graphs of trigonometric functions, use powerful identities to simplify expressions, and solve trigonometric equations over any given range.

Don't worry if this seems like a big step up at first. We will break every concept down into simple, manageable steps with clear examples and memory tricks along the way.

1. Non-Right-Angled Triangles: Sine Rule, Cosine Rule, and Area

In standard triangle notation, we label the vertices (angles) with capital letters \(A\), \(B\), and \(C\), and the sides opposite those angles with lowercase letters \(a\), \(b\), and \(c\).

The Sine Rule

The Sine Rule relates the ratio of the sine of an angle to the length of its opposite side:

Finding a side: \(\frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C}\)

Finding an angle: \(\frac{\sin A}{a} = \frac{\sin B}{b} = \frac{\sin C}{c}\)

Top Tip: Put the unknown quantity in the numerator (on the top of the fraction) to make rearranging much easier!

The Ambiguous Case of the Sine Rule

Did you know that when you are given two sides and a non-included angle (SSA), there might be two possible triangles? Because \(\sin(180^\circ - \theta) = \sin \theta\), your calculator will only give you the acute angle \(\theta\). The second possible angle is always \(180^\circ - \theta\). Always check if this obtuse angle, when added to the given angle, leaves room for a third angle (i.e. total is less than \(180^\circ\)).

The Cosine Rule

The Cosine Rule is an extension of Pythagoras' Theorem for any triangle:

Finding a side (when given two sides and the included angle):
\(a^2 = b^2 + c^2 - 2bc \cos A\)

Finding an angle (when given all three sides):
\(\cos A = \frac{b^2 + c^2 - a^2}{2bc}\)

Area of Any Triangle

When you know two sides and the included angle (the angle trapped between them):

\(\text{Area} = \frac{1}{2}ab \sin C\)

Key Takeaway for Triangles:
• Use the Sine Rule when you have an "angle-side opposite pair".
• Use the Cosine Rule when you have "SAS" (two sides and the included angle) or "SSS" (all three sides).
• Use \(\text{Area} = \frac{1}{2}ab\sin C\) whenever you have two sides and the angle between them.

2. Trigonometric Functions and Their Graphs

In A-Level Mathematics, we extend angles beyond \(90^\circ\) and even into negative values. Let's look at how the three primary trigonometric functions behave.

1. The Sine Graph: \(y = \sin \theta\)

• Period: Repeats every \(360^\circ\)
• Amplitude: Maximum value is \(1\), minimum value is \(-1\)
• Key features: Passes through \((0^\circ, 0)\), reaches peak at \((90^\circ, 1)\), crosses the axis at \((180^\circ, 0)\), troughs at \((270^\circ, -1)\), and returns to \((360^\circ, 0)\).
• Symmetry: Rotational symmetry about the origin; \(\sin(-\theta) = -\sin \theta\).

2. The Cosine Graph: \(y = \cos \theta\)

• Period: Repeats every \(360^\circ\)
• Amplitude: Maximum value is \(1\), minimum value is \(-1\)
• Key features: Starts at the peak \((0^\circ, 1)\), crosses at \((90^\circ, 0)\), troughs at \((180^\circ, -1)\), crosses at \((270^\circ, 0)\), and returns to the peak at \((360^\circ, 1)\).
• Symmetry: Reflection symmetry across the \(y\)-axis; \(\cos(-\theta) = \cos \theta\).

3. The Tangent Graph: \(y = \tan \theta\)

• Period: Repeats every \(180^\circ\) (notice this is shorter!)
• Range: \(\tan \theta\) goes from \(-\infty\) to \(+\infty\) (no maximum or minimum values).
• Asymptotes: Vertical dotted lines where the function is undefined at \(\theta = \pm 90^\circ, \pm 270^\circ, \dots\)

Key Takeaway: Sine and Cosine are waves bounded between \(-1\) and \(1\) with period \(360^\circ\). Tangent has vertical asymptotes and a period of \(180^\circ\).

3. Essential Trigonometric Identities

An identity is an equation that is true for all values of the variable. You must memorize and be confident applying two foundational identities in AS Pure Mathematics:

Identity 1: The Ratio Identity

\(\tan \theta \equiv \frac{\sin \theta}{\cos \theta}\) (for \(\cos \theta \neq 0\))

Identity 2: The Pythagorean Identity

\(\sin^2 \theta + \cos^2 \theta \equiv 1\)

Note on notation: \(\sin^2 \theta\) means \((\sin \theta)^2\). Writing \(\sin \theta^2\) means something completely different (taking the sine of the angle squared)!

Rearranging the Pythagorean Identity

You will frequently need to replace one term with another when solving equations:
• \(\sin^2 \theta = 1 - \cos^2 \theta\)
• \(\cos^2 \theta = 1 - \sin^2 \theta\)

Worked Example: Simplifying an Expression

Simplify \(\frac{1 - \cos^2 \theta}{\sin \theta}\):
Step 1: Replace \(1 - \cos^2 \theta\) with \(\sin^2 \theta\).
\(\frac{\sin^2 \theta}{\sin \theta}\)
Step 2: Cancel common factors.
\(= \sin \theta\)

Key Takeaway: When an equation has both \(\sin^2 \theta\) and \(\cos \theta\) (or vice versa), use \(\sin^2 \theta + \cos^2 \theta = 1\) to rewrite everything in terms of a single trigonometric function.

4. Solving Trigonometric Equations

Because trigonometric graphs repeat infinitely, equations like \(\sin \theta = 0.5\) have infinitely many answers! That is why exam questions always specify an interval (e.g. \(0^\circ \le \theta \le 360^\circ\)).

Step-by-Step Method for Basic Equations

Step 1: Find the Principal Value. Use the inverse function on your calculator (e.g. \(\theta = \sin^{-1}(\dots)\)).
Step 2: Find the Second Related Solution in the first cycle \([0^\circ, 360^\circ]\):
• For Sine: Second angle is \(180^\circ - \text{PV}\)
• For Cosine: Second angle is \(360^\circ - \text{PV}\)
• For Tangent: Second angle is \(\text{PV} + 180^\circ\)
Step 3: Find all other solutions in the required range by adding or subtracting multiples of the period (\(360^\circ\) for \(\sin\) and \(\cos\); \(180^\circ\) for \(\tan\)).

The CAST Diagram (A Handy Memory Aid)

The CAST diagram shows which trigonometric functions are positive in each quadrant (starting in Quadrant 1 where \(0^\circ < \theta < 90^\circ\) and moving anti-clockwise):
• Quadrant 1 (0° to 90°): All are positive (\(\sin, \cos, \tan\)).
• Quadrant 2 (90° to 180°): Sine only is positive.
• Quadrant 3 (180° to 270°): Tangent only is positive.
• Quadrant 4 (270° to 360°): Cosine only is positive.
Memory trick: Going counter-clockwise starting from the bottom-right quadrant spells C - A - S - T!

Worked Example 1: Solving a Linear Trig Equation

Problem: Solve \(2\sin \theta + 1 = 0\) for \(0^\circ \le \theta \le 360^\circ\).

Solution:
1. Rearrange for \(\sin \theta\):
\(2\sin \theta = -1 \implies \sin \theta = -0.5\)
2. Find the principal value using a calculator:
\(\sin^{-1}(-0.5) = -30^\circ\)
3. Use the sine symmetry rule (\(180^\circ - \theta\)):
Second angle \(= 180^\circ - (-30^\circ) = 210^\circ\)
4. Bring \(-30^\circ\) into our target interval \([0^\circ, 360^\circ]\) by adding \(360^\circ\):
\(-30^\circ + 360^\circ = 330^\circ\)
5. Check solutions: \(\theta = 210^\circ, 330^\circ\). Both are inside \([0^\circ, 360^\circ]\).

Worked Example 2: Quadratic Equations in Trigonometry

Problem: Solve \(2\cos^2 \theta - \sin \theta - 1 = 0\) for \(0^\circ \le \theta \le 360^\circ\).

Solution:
Step 1: We have a mix of \(\cos^2 \theta\) and \(\sin \theta\). Substitute \(\cos^2 \theta = 1 - \sin^2 \theta\):
\(2(1 - \sin^2 \theta) - \sin \theta - 1 = 0\)
\(2 - 2\sin^2 \theta - \sin \theta - 1 = 0\)
\(-2\sin^2 \theta - \sin \theta + 1 = 0\)
Multiply through by \(-1\) to make the leading term positive:
\(2\sin^2 \theta + \sin \theta - 1 = 0\)

Step 2: Factorise like a normal quadratic (let \(y = \sin \theta\)):
\((2\sin \theta - 1)(\sin \theta + 1) = 0\)

Step 3: Solve each branch separately:
Branch 1: \(2\sin \theta - 1 = 0 \implies \sin \theta = 0.5\)
• \(\theta = \sin^{-1}(0.5) = 30^\circ\)
• Other angle \(= 180^\circ - 30^\circ = 150^\circ\)

Branch 2: \(\sin \theta + 1 = 0 \implies \sin \theta = -1\)
• \(\theta = \sin^{-1}(-1) = -90^\circ\)
• In range: \(-90^\circ + 360^\circ = 270^\circ\)

Final Answer: \(\theta = 30^\circ, 150^\circ, 270^\circ\)

Worked Example 3: Multiple Angles

Problem: Solve \(\cos(2\theta) = 0.5\) for \(0^\circ \le \theta \le 360^\circ\).

Solution:
Step 1: Adjust the interval for the angle argument!
If \(0^\circ \le \theta \le 360^\circ\), then for \(2\theta\):
\(0^\circ \le 2\theta \le 720^\circ\)
Step 2: Find all solutions for \(2\theta\) within this wider interval:
• Principal value: \(2\theta = \cos^{-1}(0.5) = 60^\circ\)
• Second value: \(2\theta = 360^\circ - 60^\circ = 300^\circ\)
• Add \(360^\circ\) to each to find values up to \(720^\circ\):
\(2\theta = 60^\circ + 360^\circ = 420^\circ\)
\(2\theta = 300^\circ + 360^\circ = 660^\circ\)
So, \(2\theta = 60^\circ, 300^\circ, 420^\circ, 660^\circ\)
Step 3: Divide every solution by \(2\) to get \(\theta\):
\(\theta = 30^\circ, 150^\circ, 210^\circ, 330^\circ\)

Key Takeaway for Equations: Always adjust your search interval first when dealing with multiple angles like \(2\theta\) or \((\theta - 30^\circ)\), find all intermediate solutions, and only rearrange for \(\theta\) as the very last step!

5. Common Mistakes to Avoid

• Dividing by \(\sin \theta\) or \(\cos \theta\): Never divide an equation like \(\sin \theta \cos \theta = \sin \theta\) by \(\sin \theta\)! You will lose solutions where \(\sin \theta = 0\). Instead, move everything to one side and factorise: \(\sin \theta (\cos \theta - 1) = 0\).
• Calculator in wrong angle mode: Ensure your calculator is set to Degrees (D) mode for AS 1 questions.
• Forgetting the second solution: Always remember to find the partner angle using the symmetry rules before adding or subtracting periods.
• Rounding too early: Keep intermediate values exact or stored in your calculator memory until the final line to avoid rounding errors.

Quick Summary Checklist

Before moving on to past paper practice, make sure you can:
• Select and apply the Sine Rule, Cosine Rule, and Triangle Area formula.
• Sketch \(y = \sin \theta\), \(y = \cos \theta\), and \(y = \tan \theta\) from memory with key coordinates and asymptotes.
• Use \(\tan \theta \equiv \frac{\sin \theta}{\cos \theta}\) and \(\sin^2 \theta + \cos^2 \theta \equiv 1\) to simplify expressions and transform equations.
• Solve linear, quadratic, and multiple-angle trigonometric equations within any given range.

Quick check

Can you answer these now?

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

What is the Sine Rule and when should you use it?

The Sine Rule states \(\frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C}\) (or inverted for angles), used when you know an angle-side opposite pair.

What is the ambiguous case of the Sine Rule and how do you check for a second solution?

When given two sides and a non-included angle (SSA), a second possible angle is \(180^\circ - \theta\); it exists if adding it to the given angle leaves a total strictly less than \(180^\circ\).

What is the Cosine Rule and under what triangle conditions is it used?

It is \(a^2 = b^2 + c^2 - 2bc \cos A\) (or \(\cos A = \frac{b^2 + c^2 - a^2}{2bc}\)), used when given two sides and the included angle (SAS) or all three sides (SSS).

What is the formula for the area of any non-right-angled triangle?

\(\text{Area} = \frac{1}{2}ab \sin C\), where \(a\) and \(b\) are two side lengths and \(C\) is the included angle between them.

What are the key properties of the graph \(y = \sin \theta\)?

It has a period of \(360^\circ\), amplitude bounded between \(-1\) and \(1\), passes through the origin \((0^\circ, 0)\), and satisfies \(\sin(-\theta) = -\sin \theta\).

What are the key properties of the graph \(y = \cos \theta\)?

It has a period of \(360^\circ\), amplitude bounded between \(-1\) and \(1\), starts at \((0^\circ, 1)\), and has reflection symmetry across the \(y\)-axis (\(\cos(-\theta) = \cos \theta\)).

What are the key features and period of the graph \(y = \tan \theta\)?

It has a period of \(180^\circ\), an infinite range from \(-\infty\) to \(+\infty\), and vertical asymptotes at odd multiples of \(90^\circ\) (e.g. \(\theta = \pm 90^\circ, \pm 270^\circ\)).

What are the two fundamental trigonometric identities required in AS Mathematics?

\(\tan \theta \equiv \frac{\sin \theta}{\cos \theta}\) (for \(\cos \theta \neq 0\)) and \(\sin^2 \theta + \cos^2 \theta \equiv 1\).

How do you find the second related solution within \([0^\circ, 360^\circ]\) for basic trig equations?

For sine: \(180^\circ - \text{PV}\); for cosine: \(360^\circ - \text{PV}\); for tangent: \(\text{PV} + 180^\circ\), where \(\text{PV}\) is the principal value.

What does the CAST diagram represent and which functions are positive in each quadrant?

It shows positive quadrants: Q1 (0°–90°): All; Q2 (90°–180°): Sine; Q3 (180°–270°): Tangent; Q4 (270°–360°): Cosine.

What is the crucial first step when solving a trigonometric equation with a multiple angle such as \(\cos(2\theta) = k\)?

Adjust the interval for the composite argument (e.g., if \(0^\circ \le \theta \le 360^\circ\), solve for \(0^\circ \le 2\theta \le 720^\circ\)) before dividing to find \(\theta\).

Why should you factorise rather than divide by \(\sin \theta\) or \(\cos \theta\) when solving equations like \(\sin \theta \cos \theta = \sin \theta\)?

Dividing by a trigonometric function eliminates roots where that function equals zero, causing you to lose valid solutions.

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