May 2, 2018

Find the minimum value of $4 \cos \theta + 3 \sin \theta .$

The linear combination is a phase shifted and scaled sine wave, the scale determined by the magnitude of the coefficients in polar form, $\setminus \sqrt{{3}^{2} + {4}^{2}} = 5 ,$ so a minimum of $- 5$.

#### Explanation:

Find the minimum value of $4 \cos \theta + 3 \sin \theta$

The linear combination of sine and cosine of the same angle is a phase shift and a scaling. We recognize the Pythagorean Triple ${3}^{2} + {4}^{2} = {5}^{2.}$

Let $\phi$ be the angle such that $\cos \phi = \frac{4}{5}$ and $\sin \phi = \frac{3}{5}$. The angle $\phi$ is the principal value of $\arctan \left(\frac{3}{4}\right)$ but that doesn't really matter to us. What matters to us is we can rewrite our constants: $4 = 5 \cos \phi$ and $3 = 5 \sin \phi$. So

$4 \cos \theta + 3 \sin \theta$

$= 5 \left(\cos \phi \cos \theta + \sin \phi \sin \theta\right)$

$= 5 \cos \left(\theta - \phi\right)$

so has a minimum of $- 5$.

May 2, 2018

$- 5$ is the required minimum value.

#### Explanation:

Divide the equation $3 \sin x + 4 \cos x$ by $\sqrt{{a}^{2} + {b}^{2}}$ to reduce it to the form $\sin \left(x \pm \alpha\right) \mathmr{and} \cos \left(x \pm \alpha\right)$ where $a$ and $b$
are the coefficients of $\sin x$ and $\cos x$ respectively.

$\rightarrow 3 \sin x + 4 \cos x$

$= 5 \left[\sin x \cdot \left(\frac{3}{5}\right) + \cos x \cdot \left(\frac{4}{5}\right)\right]$

Let $\cos \alpha = \frac{3}{5}$ then $\sin \alpha = \frac{4}{5}$

Now, $3 \sin x + 4 \cos x$

$= 5 \left[\sin x \cdot \cos \alpha + \cos x \cdot \sin \alpha\right]$

$= 5 \sin \left(x + \alpha\right) = 5 \sin \left(x + \alpha\right)$

The value of $5 \sin \left(x + \alpha\right)$ will be minimum when sin(x+alpha) is minimum and the minimum value of $\sin \left(x + \alpha\right)$ is $- 1$.

So, the minimum value of $5 \sin \left(x + \alpha\right) = - 5$