Gujarat BoardEnglish MediumSTD 11 ScienceMATHSTrigonometric Equations1 Mark
MCQ
The equation $3\cos\text{x}+4\sin\text{x}=6$ has .... solution.
A
Finite
B
Infinite
C
One
✓
None of these
✓
Answer
Correct option: D.
None of these
Given equation:
$3\cos\text{x}+4\sin\text{x}=6\ .....(1)$
Thus, the equation is of the form $\text{a}\cos\text{x}+\text{b}\sin\text{x}=\text{c},$ where and $\text{c}=6.$
Let:
$\text{a}=3=\text{r}\cos\alpha$ and $\text{b}=4=\text{r}\sin\alpha$
Now,
$\tan\alpha=\frac{\text{b}}{\text{a}}=\frac{4}{3}$
$\Rightarrow\alpha=\tan^{-1}\Big(\frac{4}{3}\Big)$
Also,
$\text{r}=\sqrt{\text{a}^2+\text{b}^2}=\sqrt{9+16}=\sqrt{25}=5$
On putting $\text{a}=3=\text{r}\cos\alpha$ and $\text{b}=4=\text{r}\sin\alpha$ in equation (i), we get:
$\Rightarrow\text{r}\cos(\theta-\alpha)=6$
$\Rightarrow\text{5}\cos(\theta-\alpha)=6$
$\Rightarrow\text{}\cos(\theta-\alpha)=\frac{6}{5}$
From here, we cannot find the value of $\theta.$
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