Grade 12 · Mathematics

Use the fundamental trigonometric identity

The identity sin²θ + cos²θ = 1 is true for every angle θ. It lets Grade 12 Mathematics learners rewrite expressions and find one trigonometric ratio from another. When taking a square root, the quadrant determines the sign. A positive squared value alone does not tell you whether the original sine or cosine is positive or negative.

Original explanation prepared with AI assistance.

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What you will practise

Apply sin²θ + cos²θ = 1 and choose a ratio’s sign from its quadrant.

The method

Rearrange the identity, substitute the known ratio, and use the quadrant before choosing a square-root sign.

Worked example

If sin θ = 3/5 and θ is in quadrant II, find cos θ.

cos²θ = 1 − sin²θ = 1 − 9/25 = 16/25.
Thus cos θ = ±4/5 before using the quadrant.
Cosine is negative in quadrant II.
Therefore cos θ = −4/5. Check: 9/25 + 16/25 = 1.

Common mistake

Taking only the positive square root ignores the quadrant.

Try it yourself

If cos θ = 12/13 and θ is in quadrant IV, find sin θ.

Show the explained solution

sin²θ = 1 − 144/169 = 25/169. Sine is negative in quadrant IV, so sin θ = −5/13.

Curriculum and sources

Use the official CAPS subject document and your teacher’s annual plan to confirm the sequence and scope for your class. The link below is an official reference, not evidence of DBE approval of this explanation.

DBE: CAPS for the FET phase

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