Divide Pythagoras by AC²

The idea

cos² A + sin² A = 1 for all A with 0° ≤ A ≤ 90°, obtained by dividing AB² + BC² = AC² in a triangle right-angled at B by AC².

The 3-4-5 triangle again

In the 3-4-5 triangle, 3² + 4² = 5². That is 9 + 16 = 25.

Now divide every term by 5², which is 25. The right side becomes 25/25 = 1.

What do the two terms on the left add up to now: 9/25 + 16/25?

• 1
• 7/25
• 25

Divide every term

The squares on the three sides of a right triangle are drawn to scale. Press the button to divide every term by AC². Then change the angle A and divide again.

Pythagoras becomes an identity

cos² A + sin² A = 1 for all A with 0° ≤ A ≤ 90°.

In triangle ABC, right-angled at B: AB² + BC² = AC². Divide each term by AC²: (AB/AC)² + (BC/AC)² = (AC/AC)² i.e. (cos A)² + (sin A)² = 1. It holds for every angle from 0° to 90°, so it is a trigonometric identity.

Notes

cos² A + sin² A = 1 for all A with 0° ≤ A ≤ 90°. It is Pythagoras, AB² + BC² = AC², divided by AC².

Check yourself

sin A = 0.6 for an acute angle A. Use cos² A + sin² A = 1 to find cos² A.

Answer: 0.64

sin² A = 0.36, so cos² A = 1 − 0.36 = 0.64. (Then cos A = 0.8.)

Triangle ACB is right-angled at C with AB = 29, BC = 21 and AC = 20. For the angle θ at B, sin θ = 20/29 and cos θ = 21/29. Find cos² θ − sin² θ as a decimal, to three places.

Answer: 0.049

cos² θ − sin² θ = 441/841 − 400/841 = 41/841, which is about 0.049. (The sum cos² θ + sin² θ = 841/841 = 1.)

Which statement is true for every acute angle A?

Why do we divide AB² + BC² = AC² by AC²?

• cos A + sin A = 1. It fails for the 3-4-5 triangle: 4/5 + 3/5 = 7/5. The identity needs squares.
• cos² A + sin² A = 1 — correct. Yes! It comes from AB² + BC² = AC² divided by AC².
• cos² A − sin² A = 1. It fails at 30°: 3/4 − 1/4 = 1/2. Only the sum of the squares is 1.
• So that each term becomes the square of a ratio: (cos A)², (sin A)² and 1. — correct. Yes! (AB/AC)² = cos² A, (BC/AC)² = sin² A and (AC/AC)² = 1.
• Because AC is the shortest side.. AC is the hypotenuse, the longest side.
• To change the angle A.. Dividing every term by the same number does not change the angle. The identity holds for the same angle A.