Question

an electric current, ( i ), in amps, is given by

( i=cos (w t)+sqrt{8} sin (w t) ),

where ( w
eq 0 ) is a constant. what are the maximum and minimum values of ( i )?

minimum ( i ): amps

maximum ( i ): amps

Explanation:

Step1: Use the formula \(a\cos x + b\sin x = R\cos(x - \alpha)\)

For \(I=\cos(wt)+\sqrt{8}\sin(wt)\), we have \(a = 1\), \(b=\sqrt{8}\). Then \(R=\sqrt{a^{2}+b^{2}}\).

$$R=\sqrt{1^{2}+(\sqrt{8})^{2}}=\sqrt{1 + 8}=\sqrt{9}=3$$

So \(I = 3\cos(wt-\alpha)\) (where \(\tan\alpha=\frac{b}{a}=\sqrt{8}\))

Step2: Find the maximum and minimum values

Since \(- 1\leqslant\cos(wt-\alpha)\leqslant1\)
When \(\cos(wt-\alpha)=1\), \(I_{max}=3\times1 = 3\)
When \(\cos(wt-\alpha)=-1\), \(I_{min}=3\times(-1)=- 3\)

Answer:

Minimum \(I\): \(-3\) amps
Maximum \(I\): \(3\) amps