Ratios and Graphs of Trigonometric Functions
How to Find trig ratios for angles beyond 90°
Use this to find sine, cosine or tangent for an angle between 0° and 360°, including where the value is negative.
Before you start
- Finding sin, cos and tan of acute angles on a calculator
- The idea of the four quadrants on a coordinate grid
- Handling positive and negative signs
When to use it
Use this to find sine, cosine or tangent for an angle between 0° and 360°, including where the value is negative.
The steps
- Sketch the four quadrants and mark which ratios are positive in each.
- Find the quadrant the angle falls in.
- Work out the basic (acute) reference angle to the horizontal axis.
- Find the ratio of the reference angle from the calculator.
- Apply the correct sign for that quadrant.
Worked example
Find the value of cos 210°, giving your answer in surd form and as a decimal to 3 significant figures.
- Quadrant signs (ASTC): All positive in Q1, only Sine in Q2, only Tangent in Q3, only Cosine in Q4.
- 210° lies between 180° and 270°, so it is in the third quadrant (Q3).
- Reference angle to the horizontal axis = 210° − 180° = 30°.
- From the calculator, cos 30° = √3/2 ≈ 0.866.
- In Q3 cosine is negative, so cos 210° = −cos 30° = −√3/2 ≈ −0.866.
A second example, with a twist
It uses tangent in the fourth quadrant, where the answer is negative and the reference angle is 360° minus the angle. Find the exact value of tan 315°.
- Quadrant signs (ASTC): All in Q1, Sine in Q2, Tangent in Q3, Cosine in Q4.
- 315° lies between 270° and 360°, so it is in the fourth quadrant (Q4).
- Reference angle to the horizontal axis = 360° − 315° = 45°.
- From the calculator, tan 45° = 1.
- In Q4 only cosine is positive, so tangent is negative: tan 315° = −tan 45° = −1.
Formula pages
Practise this in a KBAT problem
Frequently asked questions
What does ASTC actually mean?
It is a memory aid for which ratios are positive in each quadrant, read anticlockwise from Q1: All, Sine, Tangent, Cosine. In Q1 all three are positive, in Q2 only sine, in Q3 only tangent, in Q4 only cosine; every other ratio in that quadrant is negative.
How do I find the reference angle each time?
Measure the acute angle to the horizontal x-axis, never the vertical one. In Q2 use 180° − θ, in Q3 use θ − 180°, in Q4 use 360° − θ.
The calculator ratio of this reference angle is always positive, and the quadrant then decides the sign.
If the calculator gives the right sign, why learn the quadrants?
For a single value the calculator is enough, but SPM awards working marks for showing the quadrant and reference angle. You also need the method to solve equations, where several angles between 0° and 360° share one ratio.
Learn it properly so you do not lose those marks.