KCET · Maths · Trigonometric Equations
The general solution of \(\sin x-\cos x=\sqrt{2}\), for any integer \(n\) is
- A \(2 n \pi+\frac{3 \pi}{4}\)
- B \(n \pi\)
- C \((2 n+1) \pi\)
- D \(2 n \pi\)
Answer & Solution
Correct Answer
(A) \(2 n \pi+\frac{3 \pi}{4}\)
Step-by-step Solution
Detailed explanation
\(\begin{array}{ll}\text { Given, } & \sin x-\cos x=\sqrt{2} \\ \Rightarrow & \frac{1}{\sqrt{2}} \sin x-\frac{1}{\sqrt{2}} \cos x=1 \\ \Rightarrow & \sin x \cdot \sin \frac{\pi}{4}-\cos x \cdot \cos \frac{\pi}{4}=1 \\ \Rightarrow & -\cos \left(x+\frac{\pi}{4}\right)=+1 \\ \Rightarrow & \cos \left(x+\frac{\pi}{4}\right)=\cos \pi \\ \Rightarrow & x+\frac{\pi}{4}=2 n \pi+\pi \\ \therefore & x=2 n \pi+\frac{3 \pi}{4}\end{array}\)
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