Ratios and Graphs of Trigonometric Functions · 6.1.2

Finding trig values by quadrant

Q: Do I need a calculator to find these values?

Not always. For common angles like 30°, 45° and 60°, use the known exact ratios together with the correct sign from the quadrant. For other angles, a calculator can be used directly, but always check that the sign matches the quadrant.

Q: What if the angle is negative or more than 360°?

First add or subtract multiples of 360° until the angle lies between 0° and 360°. Then determine its quadrant and reference angle as usual before finding the value.

Q: Why do sine, cosine and tangent have different signs in the same quadrant?

Because they are defined differently on the unit circle, sine from the y-coordinate, cosine from the x-coordinate, and tangent as their ratio, so each ratio's sign depends separately on whether x and/or y is positive or negative in that quadrant.

Building on the sign conjecture, students calculate the actual value of sin, cos or tan for any given angle between 90° and 360°, by first finding its quadrant and reference angle, then applying the correct sign to the ratio of the reference angle.

Written by the spmmath.com.my editorial team.

The official learning standard (6.1.2)

“Determine the value of sine, cosine and tangent for angles in quadrants II, III and IV based on the corresponding reference angle.”

What it means

How it is examined

Paper 1 regularly asks for the exact or calculator value of sin, cos or tan for a specific angle in this range (such as 210° or 300°). Paper 2 uses these values within larger trigonometric equations or graph-related questions.

Worked example

Find the value of tan300°, showing the quadrant, reference angle and sign used.

300° lies in quadrant IV (270° < 300° < 360°).
Reference angle = 360° − 300° = 60°.
In quadrant IV, tangent is negative.
tan300° = −tan60° = −√3.

Source:DSKP KSSM Mathematics Form 4 and 5 (Versi English)

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Frequently asked questions

Do I need a calculator to find these values?

Not always. For common angles like 30°, 45° and 60°, use the known exact ratios together with the correct sign from the quadrant.

For other angles, a calculator can be used directly, but always check that the sign matches the quadrant.

What if the angle is negative or more than 360°?

First add or subtract multiples of 360° until the angle lies between 0° and 360°. Then determine its quadrant and reference angle as usual before finding the value.

Why do sine, cosine and tangent have different signs in the same quadrant?

Because they are defined differently on the unit circle, sine from the y-coordinate, cosine from the x-coordinate, and tangent as their ratio, so each ratio's sign depends separately on whether x and/or y is positive or negative in that quadrant.