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How do the sine, cosine and tangent ratios extend beyond right-angled triangles using the unit circle?

Define sine, cosine and tangent for any angle using the unit circle, and find exact values and signs in each quadrant

A focused answer to the N(A)-Level Additional Mathematics outcome on trigonometric ratios. The unit-circle definitions of sine, cosine and tangent, the sign of each ratio in the four quadrants, and exact values for standard angles.

Generated by Claude Opus 4.88 min answer

Reviewed by: AI editorial process; not yet individually human-reviewed

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  1. What this dot point is asking
  2. The answer
  3. Examples in context
  4. Try this

What this dot point is asking

SEAB wants you to extend sine, cosine and tangent beyond right-angled triangles to any angle, using the unit circle. You should know the sign of each ratio in each of the four quadrants, recall the exact values of the standard angles (0∘,30∘,45∘,60∘,90∘0^\circ, 30^\circ, 45^\circ, 60^\circ, 90^\circ), and use a related acute angle to evaluate ratios of larger angles. This is the foundation for trigonometric identities and equations.

The answer

The unit-circle definition

Draw a circle of radius 11 centred at the origin. For an angle θ\theta measured anticlockwise from the positive xx-axis, the point where the radius meets the circle has coordinates (cos⁡θ, sin⁡θ)(\cos\theta,\ \sin\theta). So:

cos⁡θ=x-coordinate,sin⁡θ=y-coordinate,tan⁡θ=sin⁡θcos⁡θ\cos\theta = x\text{-coordinate}, \qquad \sin\theta = y\text{-coordinate}, \qquad \tan\theta = \frac{\sin\theta}{\cos\theta}

This definition works for any angle, including those greater than 90∘90^\circ and negative angles, because the point keeps moving around the circle.

Signs in the four quadrants

The signs of xx and yy change from quadrant to quadrant, so the ratios do too. The memory aid is "All, Sine, Tangent, Cosine" (sometimes "All Students Take Cake"), read anticlockwise from the first quadrant:

  • First quadrant (0∘0^\circ to 90∘90^\circ): All ratios positive.
  • Second quadrant (90∘90^\circ to 180∘180^\circ): Sine positive only.
  • Third quadrant (180∘180^\circ to 270∘270^\circ): Tangent positive only.
  • Fourth quadrant (270∘270^\circ to 360∘360^\circ): Cosine positive only.

Exact values of standard angles

These come from the 4545-4545-9090 and 3030-6060-9090 triangles and should be memorised:

sin⁡30∘=12,cos⁡30∘=32,tan⁡30∘=13\sin 30^\circ = \tfrac{1}{2}, \quad \cos 30^\circ = \tfrac{\sqrt{3}}{2}, \quad \tan 30^\circ = \tfrac{1}{\sqrt{3}}

sin⁡45∘=cos⁡45∘=12,tan⁡45∘=1\sin 45^\circ = \cos 45^\circ = \tfrac{1}{\sqrt{2}}, \quad \tan 45^\circ = 1

sin⁡60∘=32,cos⁡60∘=12,tan⁡60∘=3\sin 60^\circ = \tfrac{\sqrt{3}}{2}, \quad \cos 60^\circ = \tfrac{1}{2}, \quad \tan 60^\circ = \sqrt{3}

Using a related acute angle

For an angle outside the first quadrant, find the acute angle it makes with the xx-axis (the basic or reference angle), evaluate the ratio for that acute angle, then attach the sign from the correct quadrant. For example cos⁡150∘=−cos⁡30∘=−32\cos 150^\circ = -\cos 30^\circ = -\tfrac{\sqrt{3}}{2}.

Examples in context

Example 1. Reading a graph of sine. The unit-circle definition explains the shape of the y=sin⁡θy = \sin\theta curve: as the angle sweeps round, the yy-coordinate rises to 11, falls through 00 to −1-1, and returns, giving the familiar wave. Understanding the circle makes the graph predictable rather than memorised.

Example 2. Bearings and directions. Angles greater than 90∘90^\circ appear constantly in bearings and navigation. The unit circle is what lets you take the sine or cosine of, say, a bearing of 200∘200^\circ and get the correct signed component, which is essential in applied trigonometry.

Try this

Q1. State the quadrant of 120∘120^\circ and the sign of cos⁡120∘\cos 120^\circ. [2 marks]

  • Cue. Second quadrant; cosine is negative there, so cos⁡120∘<0\cos 120^\circ < 0.

Q2. Write down the exact value of tan⁡45∘\tan 45^\circ. [1 mark]

  • Cue. tan⁡45∘=1\tan 45^\circ = 1.

Q3. Find the exact value of cos⁡300∘\cos 300^\circ. [2 marks]

  • Cue. Fourth quadrant, reference angle 60∘60^\circ, cosine positive: cos⁡300∘=cos⁡60∘=12\cos 300^\circ = \cos 60^\circ = \tfrac{1}{2}.

Exam-style practice questions

Practice questions written in the style of SEAB exam questions on this dot point, with worked answer explainers. The year tag is the paper they imitate, not the source.

Original3 marksState the sign of (a) sin⁡200∘\sin 200^\circ, (b) cos⁡150∘\cos 150^\circ, and (c) tan⁡300∘\tan 300^\circ, giving the quadrant in each case.
Show worked answer →

(a) 200∘200^\circ is in the third quadrant, where sine is negative, so sin⁡200∘\sin 200^\circ is negative.

(b) 150∘150^\circ is in the second quadrant, where cosine is negative, so cos⁡150∘\cos 150^\circ is negative.

(c) 300∘300^\circ is in the fourth quadrant, where tangent is negative, so tan⁡300∘\tan 300^\circ is negative.

What markers reward: identifying the correct quadrant for each angle and applying the "all, sine, tangent, cosine" sign rule to state each sign.

Original2 marksWrite down the exact values of sin⁡30∘\sin 30^\circ and cos⁡60∘\cos 60^\circ.
Show worked answer →

From the standard 30-60-90 triangle, sin⁡30∘=12\sin 30^\circ = \dfrac{1}{2} and cos⁡60∘=12\cos 60^\circ = \dfrac{1}{2}.

What markers reward: recalling the exact value 12\tfrac{1}{2} for both, recognising that sin⁡30∘=cos⁡60∘\sin 30^\circ = \cos 60^\circ because the angles are complementary.

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