To find the value of [tex]\(\sin x - \cos x\)[/tex] given that [tex]\(\tan x = \frac{4}{3}\)[/tex] and [tex]\(0 \leq x \leq 90^\circ\)[/tex], we can proceed as follows:
1. Understanding [tex]\(\tan x\)[/tex]:
- We know that [tex]\(\tan x = \frac{\sin x}{\cos x}\)[/tex].
- Given [tex]\(\tan x = \frac{4}{3}\)[/tex], we can write:
[tex]\[
\frac{\sin x}{\cos x} = \frac{4}{3}
\][/tex]
- This means [tex]\(\sin x = 4k\)[/tex] and [tex]\(\cos x = 3k\)[/tex] for some constant [tex]\(k\)[/tex].
2. Normalize [tex]\(\sin x\)[/tex] and [tex]\(\cos x\)[/tex]:
- The Pythagorean identity states that [tex]\(\sin^2 x + \cos^2 x = 1\)[/tex].
- Substitute [tex]\(\sin x = 4k\)[/tex] and [tex]\(\cos x = 3k\)[/tex]:
[tex]\[
(4k)^2 + (3k)^2 = 1
\][/tex]
- Simplifying, we get:
[tex]\[
16k^2 + 9k^2 = 1 \implies 25k^2 = 1 \implies k^2 = \frac{1}{25} \implies k = \frac{1}{5}
\][/tex]
3. Find [tex]\(\sin x\)[/tex] and [tex]\(\cos x\)[/tex]:
- Using [tex]\(k = \frac{1}{5}\)[/tex]:
[tex]\[
\sin x = 4k = 4 \times \frac{1}{5} = \frac{4}{5} = 0.8
\][/tex]
[tex]\[
\cos x = 3k = 3 \times \frac{1}{5} = \frac{3}{5} = 0.6
\][/tex]
4. Calculate [tex]\(\sin x - \cos x\)[/tex]:
- Now we have [tex]\(\sin x = 0.8\)[/tex] and [tex]\(\cos x = 0.6\)[/tex]:
[tex]\[
\sin x - \cos x = 0.8 - 0.6 = 0.2
\][/tex]
Thus, the value of [tex]\(\sin x - \cos x\)[/tex] is [tex]\(0.20000000000000007\)[/tex] (or simply [tex]\(0.2\)[/tex] considering the precision).
