Trigonometric ratios of any magnitude and their graphs

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

Introduction to Trigonometry Beyond 90°

In your earlier math studies, trigonometry was probably all about right-angled triangles. But what happens if an angle is \(120^\circ\), \(270^\circ\), or even negative? In Further Pure Mathematics, we treat trigonometry as the study of rotation. This allows us to find the sine, cosine, and tangent of any angle imaginable!

Understanding these "angles of any magnitude" is essential for sketching graphs and solving the complex equations you will meet later in this course. Don't worry if it feels a bit abstract at first; once you master the Unit Circle and the CAST diagram, it will all click into place.

1. The Four Quadrants and the CAST Diagram

To find the trigonometric ratio of any angle, we place the angle on a set of axes \((x, y)\). We always start measuring from the positive x-axis and rotate anticlockwise.

The graph is divided into four sections called quadrants:

1st Quadrant (\(0^\circ\) to \(90^\circ\)): All ratios (\(\sin\), \(\cos\), \(\tan\)) are positive.
2nd Quadrant (\(90^\circ\) to \(180^\circ\)): Only \(\sin\) is positive.
3rd Quadrant (\(180^\circ\) to \(270^\circ\)): Only \(\tan\) is positive.
4th Quadrant (\(270^\circ\) to \(360^\circ\)): Only \(\cos\) is positive.

Memory Aid: Use the mnemonic "All Stations To Central" or "Add Sugar To Coffee" to remember which ratio is positive in each quadrant (A-S-T-C).

Key Takeaway:

The principal angle (let's call it \(\alpha\)) is the acute angle made with the x-axis. To find the ratio of any large angle \(\theta\):
1. Identify the quadrant.
2. Find the acute angle \(\alpha\) made with the x-axis.
3. Use the CAST diagram to decide if the result is positive or negative.

2. Exact Values You Must Know

The Edexcel syllabus requires you to know the exact values for \(30^\circ\), \(45^\circ\), and \(60^\circ\). These often appear in "non-calculator" style questions (even though you can use a calculator, you must show the exact surd form!).

For \(45^\circ\) (or \(\frac{\pi}{4}\) radians):
\(\sin 45^\circ = \frac{1}{\sqrt{2}}\)
\(\cos 45^\circ = \frac{1}{\sqrt{2}}\)
\(\tan 45^\circ = 1\)

For \(30^\circ\) (or \(\frac{\pi}{6}\) radians):
\(\sin 30^\circ = \frac{1}{2}\)
\(\cos 30^\circ = \frac{\sqrt{3}}{2}\)
\(\tan 30^\circ = \frac{1}{\sqrt{3}}\)

For \(60^\circ\) (or \(\frac{\pi}{3}\) radians):
\(\sin 60^\circ = \frac{\sqrt{3}}{2}\)
\(\cos 60^\circ = \frac{1}{2}\)
\(\tan 60^\circ = \sqrt{3}\)

Quick Review: If a question asks for \(\cos 120^\circ\), notice that \(120^\circ\) is in the 2nd quadrant. The acute angle with the x-axis is \(180^\circ - 120^\circ = 60^\circ\). Since only \(\sin\) is positive in the 2nd quadrant, \(\cos 120^\circ = -\cos 60^\circ = -\frac{1}{2}\).

3. Trigonometric Graphs

Visualizing the functions helps you understand how they behave over long periods. You should be able to recognize and sketch these three primary graphs.

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

Starts at \((0, 0)\).
Peak: \(1\) at \(90^\circ\) (\(\frac{\pi}{2}\)).
Trough: \(-1\) at \(270^\circ\) (\(\frac{3\pi}{2}\)).
Period: \(360^\circ\) (\(2\pi\)) - this means the wave repeats every \(360^\circ\).
Symmetry: \(\sin \theta = \sin(180^\circ - \theta)\).

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

Starts at its maximum point \((0, 1)\).
Intercepts: Crosses the \(\theta\)-axis at \(90^\circ\) and \(270^\circ\).
Period: \(360^\circ\) (\(2\pi\)).
Symmetry: \(\cos \theta = \cos(360^\circ - \theta)\) and \(\cos(-\theta) = \cos \theta\).

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

Does not have a wave shape; it has asymptotes.
Asymptotes: These are vertical lines where the function is undefined (at \(90^\circ, 270^\circ, \dots\)). The graph gets closer and closer to these lines but never touches them.
Period: \(180^\circ\) (\(\pi\)) - it repeats twice as often as sine and cosine!

4. Related Angles and Symmetry

One of the most useful skills is finding related values. For any acute angle \(\theta\):

Quadrant 2: Angles look like \((180^\circ - \theta)\). Example: \(150^\circ = 180^\circ - 30^\circ\).
Quadrant 3: Angles look like \((180^\circ + \theta)\). Example: \(210^\circ = 180^\circ + 30^\circ\).
Quadrant 4: Angles look like \((360^\circ - \theta)\). Example: \(330^\circ = 360^\circ - 30^\circ\).

Step-by-Step Example: Find the exact value of \(\tan 300^\circ\)
1. Identify Quadrant: \(300^\circ\) is between \(270^\circ\) and \(360^\circ\), so it's in the 4th Quadrant.
2. Find the Acute Angle: \(360^\circ - 300^\circ = 60^\circ\).
3. Check Sign: In the 4th Quadrant (C), only \(\cos\) is positive. So \(\tan\) must be negative.
4. Combine: \(\tan 300^\circ = -\tan 60^\circ = -\sqrt{3}\).

5. Common Pitfalls to Avoid

Calculator Mode: Always check if your calculator is in Degrees (D) or Radians (R). The syllabus uses both!
Measuring from the Y-axis: Never find your acute angle from the vertical y-axis. Always find the difference between your angle and the horizontal x-axis (\(180^\circ\) or \(360^\circ\)).
Tangent Asymptotes: Remember that \(\tan 90^\circ\) is undefined. If you get a "Math Error" on your calculator, it's likely an asymptote!

Summary Key Takeaways:

Use the CAST diagram to determine the sign of the ratio.
Memorize the exact surd values for \(30^\circ, 45^\circ,\) and \(60^\circ\).
Understand that sine and cosine graphs are periodic waves with a range of \([-1, 1]\).
The tangent graph repeats every \(180^\circ\) and has breaks (asymptotes) at \(90^\circ\) intervals.

Quick check

Can you answer these now?

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

In trigonometry as rotation, how are the quadrants and angle measurements defined?

Angles are measured starting from the positive \(x\)-axis rotating anticlockwise, with the axes divided into four quadrants (\(0^\circ-90^\circ\), \(90^\circ-180^\circ\), \(180^\circ-270^\circ\), and \(270^\circ-360^\circ\)).

What is the CAST diagram used for?

The CAST diagram helps determine which trigonometric ratios (Sine, Cosine, or Tangent) are positive in each of the four quadrants.

What is a "principal angle" (or reference angle) and how is it used?

The principal angle \(\alpha\) is the acute angle made between the terminal side of the angle and the horizontal x-axis.

What are the exact values for \(\sin, \cos,\) and \(\tan\) of \(45^\circ\)?

\(\sin 45^\circ = \frac{1}{\sqrt{2}}\), \(\cos 45^\circ = \frac{1}{\sqrt{2}}\), and \(\tan 45^\circ = 1\).

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

\(\sin 30^\circ = \frac{1}{2}\), \(\cos 30^\circ = \frac{\sqrt{3}}{2}\), and \(\tan 30^\circ = \frac{1}{\sqrt{3}}\).

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

\(\sin 60^\circ = \frac{\sqrt{3}}{2}\), \(\cos 60^\circ = \frac{1}{2}\), and \(\tan 60^\circ = \sqrt{3}\).

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

It is a periodic wave starting at \((0,0)\), reaching a maximum of \(1\) at \(90^\circ\) and a minimum of \(-1\) at \(270^\circ\), with a period of \(360^\circ\).

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

It is a periodic wave starting at \((0,1)\), crossing the axis at \(90^\circ\) and \(270^\circ\), with a period of \(360^\circ\).

Why does the tangent graph have asymptotes?

The tangent graph has vertical asymptotes at \(90^\circ, 270^\circ, \dots\) because the function is undefined at those points (where cosine is zero).

How do you calculate the ratio for an angle in the 4th quadrant?

Identify the acute angle \(\alpha = 360^\circ - \theta\) and apply the sign from the CAST diagram (only cosine is positive).

What is a common mistake when finding the principal angle \(\alpha\)?

Measuring the angle from the vertical y-axis instead of the horizontal x-axis.

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