Question : If $3 \sin x + 4 \cos x = 2$ , then the value of $3 \cos x - 4 \sin x$ is equal to:
Option 1: $\sqrt{23}$
Option 2: $\sqrt{21}$
Option 3: $\sqrt{29}$
Option 4: $21$

Question

Question : If $\cos x + \sin x = \sqrt{2} \cos x$ , what is the value of $(\cos x - \sin x)^{2} + (\cos x + \sin x)^{2}$ ?
Option 1: $2$
Option 2: $1$
Option 3: $0$
Option 4: $\frac{1}{\sqrt{2}}$

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Answer 1

Correct Answer: $\sqrt{21}$

Solution : Given: $3 \sin x + 4 \cos x = 2$ (equation 1)
Let $3 \cos x - 4 \sin x = k$ (equation 2)
We know the trigonometric identity, $\sin^{2} x + \cos^{2} x = 1$ .
Squaring the equation and adding them together,
$(3 \cos x - 4 \sin x)^{2} + (3 \sin x + 4 \cos x)^{2} = k^{2} + 2^{2}$
⇒ $9 \cos^{2} x + 16 \sin^{2} x - 24 \sin x \cos x + 9 \sin^{2} x + 16 \cos^{2} x + 24 \sin x \cos x = k^{2} + 4$
⇒ $9 (\cos^{2} x + \sin^{2} x) - 24 \sin x \cos x + 16 (\sin^{2} x + \cos^{2} x) + 24 \sin x \cos x = k^{2} + 4$
⇒ $9 + 16 = k^{2} + 4$
⇒ $25 = k^{2} + 4$
⇒ $25 - 4 = k^{2}$
⇒ $k^{2} = 21$
⇒ $k = \sqrt{21}$
The value of $3 \cos x - 4 \sin x = \sqrt{21}$ .
Hence, the correct answer is $\sqrt{21}$ .

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Question : If 3sin⁡x+4cos⁡x=2, then the value of 3cos⁡x–4sin⁡x is equal to:Option 1: 23Option 2: 21Option 3: 29Option 4: 21

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Question : If $3 \sin x + 4 \cos x = 2$ , then the value of $3 \cos x - 4 \sin x$ is equal to:
Option 1: $\sqrt{23}$
Option 2: $\sqrt{21}$
Option 3: $\sqrt{29}$
Option 4: $21$