Trigonometric identities and proof
Use sin2θ + cos2θ = 1 and tanθ = sinθcosθ to find one ratio from another, spot the faulty line in a ‘proof’, and learn what counts as a proof of an identity.
- UK: Year 12
- Germany: grade 10
- Iran: grade 10
- Afghanistan: grade 11
- Syria: grade 9
- Palestine: grade 9
- Ukraine: grade 8
- Russia: grade 10
Guide
For Afghan Grade 11 (chapter 2: trigonometric identities), Iran Grade 10–11 and A-level. In the picture a circle of radius 3 squares stands for radius 1, and P is at (cosθ,sinθ) = (0.6,0.8): the 3-4-5 triangle.
An identity is true for every angle for which both sides are defined. The basic ones are sin2θ + cos2θ = 1 (Pythagoras in the triangle with hypotenuse 1) and tanθ = sinθcosθ. Dividing the first by cos2θ gives 1 + tan2θ = 1cos2θ.
To prove an identity, work on one side only (usually the more complicated one), with steps that are always true, until it becomes the other side. Checking one angle is not a proof, and neither is starting from the result you want.