One ratio tells you all

The idea

If one trigonometric ratio of an acute angle is known, take the two sides it relates as multiples of a positive number k, find the third side by the Pythagoras theorem, and write the remaining ratios.

A ratio is a recipe

A right triangle has sin A = 1/3. That is BC/AC = 1/3.

This does not give the sides themselves. It tells us how the two sides compare.

What can you say about BC and AC?

• BC is one third of AC
• BC = 1 and AC = 3 exactly
• BC is three times AC

Grow it, find the third side

Do the three steps for each ratio. Step 1: slide k. Step 2: pick the third side. Step 3: tap each ratio.

The k method

If one ratio of an acute angle is known: take the two sides it relates as multiples of k, find the third side by the Pythagoras theorem, then write the other ratios.

k is a positive number (a length cannot be negative). In every ratio the k cancels, so the answers do not depend on k: tan A = 4/3 gives sin A = 4/5 and cos A = 3/5 whatever size the triangle is.

Notes

Know one ratio? Write the two sides it relates as multiples of k, get the third side from the Pythagoras theorem, and read off the other ratios. The k always cancels.

Check yourself

sin A = 5/13. Take BC = 5k and AC = 13k. How many k is the side AB? (AB = ? k)

Answer: 12

AB² = (13k)² − (5k)² = 169k² − 25k² = 144k², so AB = 12k.

In the same triangle (BC = 5k, AB = 12k, AC = 13k), what is cos A? Give a decimal.

Answer: 0.92

cos A = 12k/13k = 12/13, which is about 0.92.

tan A = 4/3. What is cos A? (Take BC = 4k, AB = 3k, so AC = 5k.)

cos A = 4/5. Take AB = 4k and AC = 5k, so BC = 3k. What is tan A? Give a decimal.

Answer: 0.75

tan A = BC/AB = 3k/4k = 3/4 = 0.75.

• 4/5. 4/5 is sin A = BC/AC. cos A = AB/AC = 3k/5k.
• 3/4. 3/4 is cot A = AB/BC. cos A has the hypotenuse underneath.
• 3/5 — correct. Yes! cos A = AB/AC = 3k/5k = 3/5.