Squared, then added
Show on a big screenA 5 m ladder reaches 4 m up a wall. Its foot is 3 m out. Call the angle at the foot A. Square sin A and cos A. Add them.
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- A right triangle with sides a, b, c. Side c is the hypotenuse.
- a² + b² = c², by Pythagoras.
- Divide every term by c².
- That gives sin²A + cos²A = 1.
The proof in three moves
Write sin A as opposite ÷ hypotenuse. Write cos A as adjacent ÷ hypotenuse. Square each one:
sin²A = opposite² ÷ hypotenuse²
cos²A = adjacent² ÷ hypotenuse²
The two bottoms match. So add the tops over that bottom:
sin²A + cos²A = (opposite² + adjacent²) ÷ hypotenuse²
Pythagoras says opposite² + adjacent² = hypotenuse². So the fraction is hypotenuse² ÷ hypotenuse², which is 1.
Rule: sin²A + cos²A = 1 for every acute angle A.
Such an equation, true for every acute angle, is an identity.
For the ladder, sin A = 4/5 and cos A = 3/5. The squares are 16/25 and 9/25. Their sum is 25/25 = 1.
Watch out!
Sin²A means (sin A)². Sin²A is not sin(A²). And sin A + cos A is not 1. For the ladder, sin A + cos A is 7/5.
Try the lab to finish this lesson.