Trigonometric Equations — MATHS STD 11 Science — Question

4. None of these

Solution:

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 $\theta.$