Solving trigonometric equations in a given interval

Q: What is the first step you should take when using your calculator to solve a trigonometric equation?

Always check if your calculator is in Degree mode or Radian mode based on the interval given in the question.

Q: Which two fundamental trigonometric identities are often required to simplify equations?

The two key identities are \(\tan \theta = \frac{\sin \theta}{\cos \theta}\) and \(\sin^2 \theta + \cos^2 \theta = 1\).

Q: Why is it necessary to adjust the search interval if the angle in the equation is not simply \(x\)?

You must adjust the interval to ensure you find all possible solutions for the transformed angle (e.g., \(2x\)) before solving for \(x\).

Q: What is the "Principal Value" (PV) in the context of solving a trig equation?

The Principal Value is the first angle your calculator gives you when you perform an inverse trigonometric function (e.g., \(\sin^{-1}\)).

Q: How do you find the second value for a Sine equation within one full rotation (\[360^{\circ}\] or \[2\pi\])?

For sine, the second value is found by calculating \(180^{\circ} - \theta\) (or \(\pi - \theta\) in radians).

Q: When solving an equation like \(\sin(x - \frac{\pi}{2}) = k\), when should you perform the addition of \(\frac{\pi}{2}\)?

You should add \(\frac{\pi}{2}\) as the very last step , after finding all possible values for the entire bracket.

Q: How should you approach a trigonometric equation that contains both \(\cos^2 x\) and \(\sin x\)?

Substitute \(\cos^2 x\) with \(1 - \sin^2 x\) to create a quadratic equation in terms of \(\sin x\) only.

Q: Why should you never divide both sides of a trigonometric equation by a trigonometric function like \(\sin x\)?

Dividing by a trig function can result in losing valid solutions where that function equals zero.

Further Pure Mathematics · Pearson Edexcel IGCSE · 4 min read · Updated: 7 Sept 2026

Introduction to Solving Trigonometric Equations

In this chapter, we learn how to find the values of an angle (usually called \(x\) or \(\theta\)) that make a trigonometric equation true. Think of a trigonometric function like a wave—because waves repeat themselves, there is often more than one answer! Our job is to find all the correct answers within a specific "search area" called an interval (for example, between \(0^{\circ}\) and \(360^{\circ}\)).

Whether you are aiming for a Grade 4 or a Grade 9, mastering this topic is about following a clear, logical set of steps. Don't worry if it feels a bit "loopy" at first; once you see the pattern, it becomes much easier!

1. Essential Tools for Your Toolkit

Before we start solving, you need to be comfortable with a few things from previous chapters:

Radian vs. Degree Mode: Always check your calculator! If the interval is \(0 \leq x < 360^{\circ}\), use degrees. If it is \(0 \leq x < 2\pi\), use radians.
The Identities:
1. \(\tan \theta = \frac{\sin \theta}{\cos \theta}\)
2. \(\sin^2 \theta + \cos^2 \theta = 1\) (Very useful for turning everything into the same "type" of trig function).
The Unit Circle (CAST) or Graphs: You must have a method to find the "secondary" angles. Whether you use the CAST diagram or the symmetries of the graphs, consistency is key.

2. Solving Simple Equations: The Step-by-Step Method

Let’s look at a basic example: Solve \(\tan 2x = 1\) for \(0^{\circ} \leq x \leq 360^{\circ}\).

Step 1: Adjust the Interval

If the equation has \(2x\) instead of \(x\), you must multiply your interval boundaries by 2.
Original interval: \(0^{\circ} \leq x \leq 360^{\circ}\)
New search interval for \(2x\): \(0^{\circ} \leq 2x \leq 720^{\circ}\)

Step 2: Find the Principal Value (PV)

Use your calculator to find the first angle:
\(2x = \tan^{-1}(1) = 45^{\circ}\)

Step 3: Find All Other Values in the Range

Using the properties of the tangent graph (which repeats every \(180^{\circ}\)):
\(2x = 45^{\circ}, 225^{\circ}, 405^{\circ}, 585^{\circ}\)

Step 4: Solve for \(x\)

Now, divide everything by 2 to get your final answers:
\(x = 22.5^{\circ}, 112.5^{\circ}, 202.5^{\circ}, 292.5^{\circ}\)

Key Takeaway: Always "deal with the trig" first, and "deal with the algebra" (like the 2 in \(2x\)) last!

3. Handling More Complex Arguments

Sometimes the angle inside the bracket is more complicated, like \(\sin(x - \frac{\pi}{2}) = \frac{3}{4}\) for \(0 < x < 2\pi\).

Don't panic! Just treat the whole bracket \((x - \frac{\pi}{2})\) as a single block (let's call it \(X\)).

Adjust your interval: Since \(0 < x < 2\pi\), your search area for \(X\) is \(-\frac{\pi}{2} < X < \frac{3\pi}{2}\).
Find the Principal Value: \(X = \sin^{-1}(\frac{3}{4}) \approx 0.848\) radians.
Find the second value: For sine, the other value is \(\pi - 0.848 = 2.29\) radians.
Finally, solve for \(x\) by adding \(\frac{\pi}{2}\) to your values.

4. Equations Involving Quadratics

In Further Pure Maths, you will often see equations like:
\(6\cos^2 x + \sin x - 5 = 0\) for \(0 \leq x < 360^{\circ}\)

The problem here is that we have both \(\cos\) and \(\sin\). We need them to match!

How to solve it:

1. Substitute: Use \(\cos^2 x = 1 - \sin^2 x\).
\(6(1 - \sin^2 x) + \sin x - 5 = 0\)

2. Expand and Simplify:
\(6 - 6\sin^2 x + \sin x - 5 = 0\)
\(-6\sin^2 x + \sin x + 1 = 0\)
\(6\sin^2 x - \sin x - 1 = 0\)

3. Factorise: (Treat this like \(6y^2 - y - 1 = 0\))
\((3\sin x + 1)(2\sin x - 1) = 0\)

4. Split into two equations:
\(\sin x = -\frac{1}{3}\) or \(\sin x = \frac{1}{2}\)

5. Find all solutions: Solve both equations within the given interval to find all possible values for \(x\).

5. Common Pitfalls to Avoid

Dividing by a trig function: If you have \(\sin x \cos x = \sin x\), never divide by \(\sin x\). You will lose solutions! Instead, rearrange to \(\sin x \cos x - \sin x = 0\) and factorise to \(\sin x(\cos x - 1) = 0\).
Ignoring the range: Always check if your final answers fall within the original interval. If you are asked for \(-180^{\circ}\) to \(180^{\circ}\), an answer of \(210^{\circ}\) is wrong, even if it is mathematically a root!
Rounding too early: Keep 4 decimal places in your calculator until the very end to avoid accuracy errors.

Quick Review: Finding "The Other" Angles

If \(\theta\) is your calculator's answer (the Principal Value):

For Sine: Other values are \((180^{\circ} - \theta)\) or \((360^{\circ} + \theta)\).
For Cosine: Other values are \((360^{\circ} - \theta)\) or \((-\theta)\).
For Tangent: Other values are \((180^{\circ} + \theta)\) or \((180^{\circ} - \theta)\) depending on the quadrant. (Actually, tan just repeats every \(180^{\circ}\)).

Summary Checklist

✓ Check the mode (Degrees/Radians).
✓ Adjust the interval if the angle is not just "\(x\)".
✓ Use identities to get one type of trig function if necessary.
✓ Find the Principal Value using \(\sin^{-1}, \cos^{-1},\) or \(\tan^{-1}\).
✓ Use symmetry (CAST/Graphs) to find all other values in the range.
✓ Finish the algebra to find \(x\).

Quick check

Can you answer these now?

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

What is the first step you should take when using your calculator to solve a trigonometric equation?

Always check if your calculator is in Degree mode or Radian mode based on the interval given in the question.

Which two fundamental trigonometric identities are often required to simplify equations?

The two key identities are \(\tan \theta = \frac{\sin \theta}{\cos \theta}\) and \(\sin^2 \theta + \cos^2 \theta = 1\).

Why is it necessary to adjust the search interval if the angle in the equation is not simply \(x\)?

You must adjust the interval to ensure you find all possible solutions for the transformed angle (e.g., \(2x\)) before solving for \(x\).

What is the "Principal Value" (PV) in the context of solving a trig equation?

The Principal Value is the first angle your calculator gives you when you perform an inverse trigonometric function (e.g., \(\sin^{-1}\)).

How do you find the second value for a Sine equation within one full rotation (\[360^{\circ}\] or \[2\pi\])?

For sine, the second value is found by calculating \(180^{\circ} - \theta\) (or \(\pi - \theta\) in radians).

When solving an equation like \(\sin(x - \frac{\pi}{2}) = k\), when should you perform the addition of \(\frac{\pi}{2}\)?

You should add \(\frac{\pi}{2}\) as the very last step, after finding all possible values for the entire bracket.

How should you approach a trigonometric equation that contains both \(\cos^2 x\) and \(\sin x\)?

Substitute \(\cos^2 x\) with \(1 - \sin^2 x\) to create a quadratic equation in terms of \(\sin x\) only.

Why should you never divide both sides of a trigonometric equation by a trigonometric function like \(\sin x\)?

Dividing by a trig function can result in losing valid solutions where that function equals zero.

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