1. ## Rsin(x+a) problem

Hi
I am stuck on the last bit were I am asked to state the values of x, their maximum and where the occur

do i just set the other side to -1 and multiply out to solve?

2. Hi

$a \sin x + b \cos x = \sqrt{a^2+b^2} \cdot \left(\frac{a}{\sqrt{a^2+b^2}} \sin x + \frac{b}{\sqrt{a^2+b^2}} \cos x \right)$

And since $\left(\frac{a}{\sqrt{a^2+b^2}}\right)^2 + \left(\frac{b}{\sqrt{a^2+b^2}}\right)^2 = 1$

there exists $\phi$ such that

$\frac{a}{\sqrt{a^2+b^2}} = \cos \phi$

$\frac{b}{\sqrt{a^2+b^2}} = \sin \phi$

Therefore $a \sin x + b \cos x = \sqrt{a^2+b^2} \cdot \left(\sin x \cos \phi + \cos x \sin \phi \right) = \sqrt{a^2+b^2} \cdot \sin\left(x+\phi\right)$

For instance

$5 \sin x + 12 \cos x = 13 \sin\left(x+\phi\right)$

The minimum is -13 and the maximum is 13

3. Thanks for the lengthy response
Im cool with that but not where that 30 has come from at the end?

4. Well the minimum of $5 \sin x + 12 \cos x$ is -13 and the maximum is 13

Therefore
the minimum of $5 \sin x + 12 \cos x +17$ is -13+17=4 and the maximum is 13+17=30

the minimum of $\frac{1}{5 \sin x + 12 \cos x +17}$ is 1/30 and the maximum is 1/4

the minimum of $\frac{30}{5 \sin x + 12 \cos x +17}$ is 30/30 = 1 and the maximum is 30/4 = 7.5

5. great! makes sense now