Trigonometry

Q: What are the key characteristics of the sine graph \(y = \sin x\)?

The sine graph is a wave starting at \(0\), reaching a maximum of \(1\) at \(90^\circ\), and returning to \(0\) at \(180^\circ\).

Q: How does the cosine graph \(y = \cos x\) differ from the sine graph?

The cosine graph has the same wave shape as the sine graph but starts at its peak value of \(1\) when \(x = 0^\circ\).

Q: What is a vertical asymptote in the context of the tangent graph \(y = \tan x\)?

A vertical asymptote is a line that the graph approaches but never touches, occurring where the function is undefined, such as at \(90^\circ\) and \(270^\circ\).

Q: How do you convert degrees into radians?

To convert degrees to radians, multiply the angle by \(\frac{\pi}{180}\).

Q: What are the exact values for \(\sin 30^\circ\) and \(\cos 60^\circ\)?

Both \(\sin 30^\circ\) and \(\cos 60^\circ\) are exactly equal to \(\frac{1}{2}\).

Q: Why is it necessary to look beyond the calculator answer when solving trig equations?

The calculator only provides the "Principal Value," but because trig functions are periodic, there are usually multiple solutions within a given range.

Q: What is the "Basic Angle" (\(\alpha\)) and how is it used?

The basic angle is the acute angle found by taking the inverse trig function of the positive ratio; it is then used with the CAST diagram to find actual solutions.

Q: What extra step is required when solving an equation with a multiplier, such as \(\sin 2x = k\)?

You must first solve for the inner bracket (e.g., find values for \(2x\)) and then divide those results by the multiplier to find \(x\).

Mathematics (9709) · Cambridge International AS Level · 5 min read · Updated: Mar 28, 2026

Welcome to the World of Trigonometry!

Trigonometry might sound like a mouthful, but it’s simply the study of how the sides and angles of triangles relate to one another. It is one of the most useful tools in mathematics! Whether you are an aspiring architect, a game developer, or a musician looking at sound waves, trigonometry is the language you’ll use. In this chapter, we are going to master the "waves" of math and learn how to solve equations that look scary but are actually quite logical once you know the tricks.

1. Meeting the Functions: Sine, Cosine, and Tangent

You’ve probably met Sine (\(\sin\)), Cosine (\(\cos\)), and Tangent (\(\tan\)) before using right-angled triangles (remember SOH CAH TOA?). In AS Level, we move beyond just triangles and look at these as functions that repeat forever.

The Shapes of the Graphs

Imagine a person on a Ferris wheel. Their height goes up, reaches a peak, goes down, and then repeats. This is exactly what a trigonometric graph does!

The Sine Graph (\(y = \sin x\)): Starts at \(0\), reaches its peak at \(90^\circ\) (or \(\frac{\pi}{2}\)), and returns to \(0\) at \(180^\circ\) (\(\pi\)). It looks like a smooth, balanced wave.
The Cosine Graph (\(y = \cos x\)): Looks just like the sine wave but starts at its peak (\(1\)) when \(x = 0\). Think of it as a sine wave that has been shifted to the left.
The Tangent Graph (\(y = \tan x\)): This one is the "rebel." Instead of a wave, it has vertical lines called asymptotes where the graph disappears and shoots to infinity. These happen at \(90^\circ, 270^\circ\), and so on.

Did you know? Everything that repeats in a cycle—like the seasons, your heartbeat, or the tides—can be modeled using these waves!

Quick Review: Degrees vs. Radians

In this chapter, you will use both degrees (\(^\circ\)) and radians (\(rad\)).
Remember: \(180^\circ = \pi\) radians.
Memory Trick: To turn degrees into radians, just multiply by \(\frac{\pi}{180}\). To go the other way, multiply by \(\frac{180}{\pi}\).

Key Takeaway: \(\sin\) and \(\cos\) are waves that stay between \(1\) and \(-1\). \(\tan\) is a series of curves that repeat every \(180^\circ\) (\(\pi\)).

2. The "Must-Know" Exact Values

Sometimes, your calculator will give you a long, messy decimal. However, for certain angles (\(30^\circ, 45^\circ,\) and \(60^\circ\)), we use exact values involving square roots. You need to know these by heart!

The Table of Champions:
- \(\sin 30^\circ = \frac{1}{2}\)
- \(\cos 60^\circ = \frac{1}{2}\)
- \(\sin 45^\circ = \cos 45^\circ = \frac{1}{\sqrt{2}}\)
- \(\tan 45^\circ = 1\)

Don't worry if this seems like a lot to memorize! You can always sketch a small "Special Triangle" (an isosceles right triangle with sides \(1, 1, \sqrt{2}\) or a half-equilateral triangle with sides \(1, \sqrt{3}, 2\)) to find these values if you forget them.

Key Takeaway: Examiners love using these values. If you see a \(\sqrt{3}\) or a \(\frac{1}{2}\) in a question, it's a hint that you might be dealing with these special angles.

3. Trigonometric Identities: The "Aliases"

In math, an identity is like a nickname. It’s a different way of writing the exact same thing. We use these to simplify complex-looking equations.

Identity 1: The Tangent Rule

\(\tan \theta \equiv \frac{\sin \theta}{\cos \theta}\)
Analogy: Imagine \(\tan\) is a person whose last name is \(\sin\) and first name is \(\cos\). They are the same person!

Identity 2: The Pythagorean Identity

\(\sin^2 \theta + \cos^2 \theta \equiv 1\)
This is based on Pythagoras' Theorem (\(a^2 + b^2 = c^2\)). No matter what the angle \(\theta\) is, if you square the sine and cosine and add them, you will always get \(1\).

Common Mistake: Be careful with notation! \(\sin^2 \theta\) means \((\sin \theta) \times (\sin \theta)\). It is not the same as \(\sin (\theta^2)\).

Key Takeaway: Use these identities to turn an equation with both \(\sin\) and \(\cos\) into an equation with just one of them. This makes solving much easier!

4. Inverse Functions: Finding the Angle

When we know the ratio but want to find the angle, we use inverse functions: \(\sin^{-1} x\), \(\cos^{-1} x\), and \(\tan^{-1} x\). These are also called "Principal Values."

Your calculator will only give you one answer for an inverse function. However, because trig graphs are waves, there are actually many different angles that give the same result! We use the CAST diagram or the symmetry of the graphs to find these other "hidden" answers.

Key Takeaway: The calculator answer is just the starting point. Always check the range requested in the question (e.g., \(0^\circ \le \theta \le 360^\circ\)) to see if you need more answers.

5. Solving Trig Equations: Step-by-Step

Let's look at how to solve something like: \(3 \sin 2x + 1 = 0\) for \(-\pi < x < \pi\).

Step 1: Isolate the Trig Function
Move everything else to the other side.
\(3 \sin 2x = -1\)
\(\sin 2x = -\frac{1}{3}\)

Step 2: Find the Basic Angle (\(\alpha\))
Ignore the negative sign for a moment and use your calculator:
\(\alpha = \sin^{-1}(\frac{1}{3}) \approx 0.34\) rad.

Step 3: Find all values for the inner bracket (\(2x\))
Using the CAST diagram or graph symmetry, find where \(\sin\) is negative (the 3rd and 4th quadrants).
The values for \(2x\) would be \(\pi + 0.34\) and \(2\pi - 0.34\). Also, remember to look in the negative direction if the interval requires it!

Step 4: Solve for \(x\)
Finally, divide your answers by \(2\) to get the final values for \(x\).

Quick Review Box:
1. Simplify the equation.
2. Find the basic acute angle (\(\alpha\)).
3. Use the CAST diagram to find other angles.
4. Adjust for any multipliers (like the \(2\) in \(2x\)).

Key Takeaway: Always check your final answers by plugging them back into the original equation. If they don't make the equation true, something went wrong in your quadrants!

Summary & Final Encouragement

Trigonometry in Pure Mathematics 1 is all about patterns. Once you recognize the shape of the waves and master the two main identities, you have all the tools you need.

Important points to remember:
- \(\sin^2 \theta + \cos^2 \theta = 1\)
- \(\tan \theta = \frac{\sin \theta}{\cos \theta}\)
- Use the CAST diagram for multiple solutions.
- Watch your calculator mode (Deg vs Rad)!

Keep practicing your sketches—it’s much harder to get lost when you can see the wave you’re working on! You've got this!

References

Cambridge International AS & A Level Mathematics (9709) Syllabus for 2026 and 2027 — Cambridge University Press & Assessment
Cambridge International AS & A Level Mathematics (9709) Syllabus for 2028, 2029 and 2030 — Cambridge University Press & Assessment

Quick check

Can you answer these now?

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

What are the key characteristics of the sine graph \(y = \sin x\)?

The sine graph is a wave starting at \(0\), reaching a maximum of \(1\) at \(90^\circ\), and returning to \(0\) at \(180^\circ\).

How does the cosine graph \(y = \cos x\) differ from the sine graph?

The cosine graph has the same wave shape as the sine graph but starts at its peak value of \(1\) when \(x = 0^\circ\).

What is a vertical asymptote in the context of the tangent graph \(y = \tan x\)?

A vertical asymptote is a line that the graph approaches but never touches, occurring where the function is undefined, such as at \(90^\circ\) and \(270^\circ\).

How do you convert degrees into radians?

To convert degrees to radians, multiply the angle by \(\frac{\pi}{180}\).

What are the exact values for \(\sin 30^\circ\) and \(\cos 60^\circ\)?

Both \(\sin 30^\circ\) and \(\cos 60^\circ\) are exactly equal to \(\frac{1}{2}\).

State the identity relating tangent, sine, and cosine.

\(\tan \theta \equiv \frac{\sin \theta}{\cos \theta}\)

State the Pythagorean trigonometric identity.

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

Why is it necessary to look beyond the calculator answer when solving trig equations?

The calculator only provides the "Principal Value," but because trig functions are periodic, there are usually multiple solutions within a given range.

What is the "Basic Angle" (\(\alpha\)) and how is it used?

The basic angle is the acute angle found by taking the inverse trig function of the positive ratio; it is then used with the CAST diagram to find actual solutions.

What extra step is required when solving an equation with a multiplier, such as \(\sin 2x = k\)?

You must first solve for the inner bracket (e.g., find values for \(2x\)) and then divide those results by the multiplier to find \(x\).

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