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10.1PureCore

Angles in All Four Quadrants

Trig ratios are defined for angles of any size, not just those in a right-angled triangle. The CAST diagram is the tool that tells you the sign of sin, cos and tan in each of the four quadrants — essential for solving equations later.

30 min Trigonometric Identities and Equations
Edexcel AS Level Maths: 10.1 Angles in all Four Quadrants (CAST)Curated by Kepler, not produced by us. Watch it before the notes below.
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What you'll be able to do

  • Understand how angles are measured around the four quadrants
  • Use the CAST diagram to find where each ratio is positive
  • Find the related acute angle for any angle
  • Determine the sign of a trig ratio for any angle
1

The four quadrants

Measuring anticlockwise from the positive xx-axis, angles fall into four quadrants: 0∘0^\circ–90∘90^\circ, 90∘90^\circ–180∘180^\circ, 180∘180^\circ–270∘270^\circ and 270∘270^\circ–360∘360^\circ. Each quadrant decides whether sin⁡\sin, cos⁡\cos and tan⁡\tan come out positive or negative.

2

The CAST diagram

CAST records which ratios are positive in each quadrant, read anticlockwise from the 4th: C\textbf{C} (4th: only cos⁡\cos), A\textbf{A} (1st: a\textbf{a}ll), S\textbf{S} (2nd: only sin⁡\sin), T\textbf{T} (3rd: only tan⁡\tan).

1st: all+2nd: sin⁡+3rd: tan⁡+4th: cos⁡+\text{1st: all} + \quad \text{2nd: } \sin + \quad \text{3rd: } \tan + \quad \text{4th: } \cos +
Everything else in that quadrant is negative.
Example 1Is cos⁡200∘\cos 200^\circ positive or negative?
1200∘200^\circ is in the 3rd quadrant.
2There only tan⁡\tan is positive, so cos⁡\cos is negative.
Answernegative

Tip — Remember the order anticlockwise from bottom-right: CAST. “All Students Take Calculus” read the other way also works.

3

The related acute angle

For any angle, the related acute angle\textbf{related acute angle} is its distance to the nearest part of the xx-axis. The size of the trig ratio matches that acute angle; the CAST diagram supplies the sign.

Example 1Express sin⁡150∘\sin 150^\circ using its related acute angle.
1150∘150^\circ is in the 2nd quadrant; related acute angle is 180∘−150∘=30∘180^\circ - 150^\circ = 30^\circ.
22nd quadrant: sin⁡\sin is positive.
Answersin⁡150∘=+sin⁡30∘=0.5\sin 150^\circ = +\sin 30^\circ = 0.5

Formula recap

CAST: cos⁡,all,sin⁡,tan⁡ positive\text{CAST: } \cos, \text{all}, \sin, \tan \text{ positive}
Signs by quadrant (4th→1st→2nd→3rd).
related acute angle\text{related acute angle}
Distance to the x-axis sets the value.
sin⁡(180∘−θ)=sin⁡θ\sin(180^\circ - \theta) = \sin\theta
Example symmetry from CAST.

Common mistakes to avoid

Assuming trig ratios are always positive.
Only in the 1st quadrant are all positive; CAST gives the sign elsewhere.
Measuring the related angle from the y-axis.
The related acute angle is measured to the nearest x-axis.

Key takeaways

  • Angles run anticlockwise through four quadrants.
  • CAST shows which ratio is positive: all / sin / tan / cos.
  • The related acute angle gives the size; CAST gives the sign.

Test yourself

Answer a few questions, then review anything you missed.

Next step

Keep practising

Start with the quiz above to check the key ideas. You can also choose another topic from the practice library.

Browse practice topics

Related lessons

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Put the methods into practice: see how A-Level maths duels work. An account is required to play; the overview and sample are public.