If tan x = $\frac{4}{3}$, 0$^o$ < x < 90$^o$, find the value of sin x - cos x

Analyze the question: We're given tan x = 4/3 with x in the first quadrant (so all ratios are positive), and we need sin x − cos x.

Concept: tan x = opposite/adjacent. Use a right-angled triangle and Pythagoras to find the hypotenuse, then read off sin x and cos x.

Solution:

• Opposite = 4, adjacent = 3
• Hypotenuse = √(4² + 3²) = √25 = 5 (a 3-4-5 triangle)

So:

• sin x = opposite/hypotenuse = 4/5
• cos x = adjacent/hypotenuse = 3/5

Then:

sin x − cos x = 4/5 − 3/5 = 1/5

Tip: Whenever you see tan x as a ratio like 4/3, 3/4, 5/12 or 8/15, think of Pythagorean triples (3-4-5, 5-12-13, 8-15-17). They let you find the hypotenuse instantly and save time in the exam. Also, check the quadrant: here 0° < x < 90°, so sin and cos are both positive.