Introduction to Trigonometry
Trigonometric identities
Prove trigonometric identities with sin²A + cos²A = 1.
An identity is an equation that is true for every angle where both sides make sense. Example: (a + b)² = a² + 2ab + b² is true for every a and b.
All three trig identities come from Pythagoras. In right triangle ABC, right-angled at B: AB² + BC² = AC². Now divide by each side squared:
| Divide by | You get |
|---|---|
| AC² | cos²A + sin²A = 1 (every angle from 0° to 90°) |
| AB² | 1 + tan²A = sec²A (not at 90°, where tan and sec aren't defined) |
| BC² | cot²A + 1 = cosec²A (not at 0°) |
Learn three, not nine. Move a term across to get the other forms: sin²A = 1 – cos²A, sec²A – tan²A = 1, and so on.
They work for any angle, even ones you don't know. You don't know sin 50°, but 2sin²50° + 2cos²50° = 2(1) = 2.
Changing one ratio into another: cos A = √(1 – sin²A). Take the positive root, since A is acute.
How to prove an identity:
- Work on one side (usually the left) until it becomes the other side.
- Look at the other side for a hint. If it is in sin and cos, change everything to sin and cos.
- Stuck? Change everything to sin and cos. It nearly always works.
Useful moves: combine fractions over one denominator (LCM); use (1 – sin A)(1 + sin A) = 1 – sin²A = cos²A; replace a '1' with sin² + cos² or sec² – tan².
Don't cancel what you can't. You can only cancel a common factor, never part of a sum: in (sin A + 1)/sin A you can't cross out the sin A.
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