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Study guides › Level 9 · Trigonometry

Finding cos from sin (and back)

Level 9 · Trigonometry · for questions like “sin x = 3/5 and x is between 90° and 180°. What is cos x?”

💡 The idea

sin²x + cos²x = 1, which is Pythagoras on the unit circle. It gives the size. The quadrant gives the sign.

A right triangle with sides 3 and 5, third side unknown

🪜 The steps

  1. Draw a right triangle: opposite 3 and hypotenuse 5, so the third side is 4.
  2. Read the size: cos = 4/5.
  3. Fix the sign: between 90° and 180° is quadrant II, where cos is negative. So −4/5.

✏️ Worked examples

1. sin x = 3/5, quadrant II: cos x = −4/5, and tan x = sin ÷ cos = −3/4.

2. cos x = 5/13, quadrant IV: sin x = −12/13.

⚠️ Watch out

🎯 Try these

  1. sin x = 8/17, quadrant I: cos x?
  2. cos x = −3/5, quadrant III: sin x?
  3. sin x = 5/13, quadrant II: tan x?

Answers: 1) 15/17 2) −4/5 3) −5/12

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