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Trigonometry with any angle

Determine the sign of trigonometric ratios in each quadrant, evaluate ratios at multiples of 90°, and apply supplementary and reflection identities to find exact values for any angle.

Worked examples

Determining the sign of a trig ratio from the quadrant

Straightforward

Problem

Without using a calculator, determine the sign of sin⁡250°\sin 250° and state which quadrant the angle lies in.

Using the supplementary angle identity for an obtuse angle

Moderate

Problem

Find the exact value of cos⁡135°\cos 135° using the supplementary angle identity.

Simplifying an expression using reflection identities

Challenging

Problem

Use the identities sin⁡(360°−θ)=−sin⁡θ\sin(360° - \theta) = -\sin\theta and cos⁡(−θ)=cos⁡θ\cos(-\theta) = \cos\theta to simplify and evaluate sin⁡(360°−30°)×cos⁡(−60°)\sin(360° - 30°) \times \cos(-60°).

Practise

Q1·Straightforward
What is the sign of sin⁡θ\sin\theta for an angle θ\theta in the second quadrant?
Q2·Straightforward
What is the sign of cos⁡θ\cos\theta for an angle θ\theta in the third quadrant?
Q3·Straightforward
What is the sign of tan⁡θ\tan\theta for an angle θ\theta in the fourth quadrant?
Q4·Straightforward
Find the exact value of cos⁡270°\cos 270°.
Q5·Moderate
Find the exact value of sin⁡150°\sin 150° using the supplementary angle identity sin⁡(180°−θ)=sin⁡θ\sin(180° - \theta) = \sin\theta. Give your answer as a fraction.
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Q6·Moderate
Find the exact value of cos⁡120°\cos 120° using the supplementary angle identity cos⁡(180°−θ)=−cos⁡θ\cos(180° - \theta) = -\cos\theta. Give your answer as a fraction.
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Q7·Moderate
Use the identity tan⁡(180°−θ)=−tan⁡θ\tan(180° - \theta) = -\tan\theta to find the exact value of tan⁡135°\tan 135°.
Q8·Moderate
Use the supplementary angle identity to find the value of cos⁡150°\cos 150°. Give your answer to 2 decimal places.
Q9·Challenging
Use the identity sin⁡(−θ)=−sin⁡θ\sin(-\theta) = -\sin\theta to simplify sin⁡(−30°)\sin(-30°) and state its exact value as a fraction.
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Q10·Challenging
Use the identity cos⁡(360°−θ)=cos⁡θ\cos(360° - \theta) = \cos\theta to find the exact value of cos⁡300°\cos 300°. Give your answer as a fraction.
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Q11·Challenging
Use the identity tan⁡(180°+θ)=tan⁡θ\tan(180° + \theta) = \tan\theta to find the exact value of tan⁡225°\tan 225°.
Q12·Challenging
Use the identities sin⁡(180°+θ)=−sin⁡θ\sin(180° + \theta) = -\sin\theta and cos⁡(−θ)=cos⁡θ\cos(-\theta) = \cos\theta to evaluate sin⁡(225°)+cos⁡(−45°)\sin(225°) + \cos(-45°).