Question

Given that 4cos& + 3sin& =5. Find the value of sin &

Answer 1

To solve the equation \(4\cos\theta + 3\sin\theta = 5\), we can use a method involving the amplitude-phase form of a sinusoidal function.

1. Rewrite the equation in a standard form. We can express \(4\cos\theta + 3\sin\theta\) in the form \( R\sin(\theta + \phi) \), where:
\[
R = \sqrt{4^2 + 3^2} = \sqrt{16 + 9} = \sqrt{25} = 5.
\]

2. Next, we find the angle \(\phi\):
\[
\cos\phi = \frac{4}{5}, \quad \sin\phi = \frac{3}{5}.
\]
(Here, \(\cos\phi\) aligns with the coefficient of \(\cos\theta\), and \(\sin\phi\) aligns with the coefficient of \(\sin\theta\)).

3. Thus, we can rewrite the original equation as:
\[
5\sin(\theta + \phi) = 5.
\]
This simplifies to:
\[
\sin(\theta + \phi) = 1.
\]

4. The solution to \(\sin(\theta + \phi) = 1\) is:
\[
\theta + \phi = \frac{\pi}{2} + 2n\pi \quad (n \in \mathbb{Z}).
\]

5. Hence,
\[
\theta = \frac{\pi}{2} - \phi + 2n\pi.
\]

6. We need to find \(\sin\theta\). Since \(\theta + \phi = \frac{\pi}{2}\), we have:
\[
\sin\theta = \cos\phi = \frac{4}{5}.
\]

Thus, the value of \(\sin\theta\) is \(\frac{4}{5}\).

In summary, the required value is:
\[
\sin\theta = \frac{4}{5}.
\]