the function is periodic but not a sinusoid. find the period graphically and sketch a graph showing one…
the function is periodic but not a sinusoid. find the period graphically and sketch a graph showing one period. y = cos 3x - 4 sin 4x the period of the function is . (type an exact answer, using π as needed.) graph one period of the function. choose the correct graph below. oa. 0,2π by -6,6 ob. 0, π/2 by -6,6 oc. 0,2π by -6,6 od. -2π,2π by -6,6
Answer
Explanation:
Step1: Recall period - formula for trigonometric functions
The period of $y = A\cos(Bx)$ is $T_1=\frac{2\pi}{|B|}$ and for $y = A\sin(Bx)$ is $T_2=\frac{2\pi}{|B|}$. For $y=\cos(3x)$, $B = 3$, so $T_1=\frac{2\pi}{3}$. For $y=-4\sin(4x)$, $B = 4$, so $T_2=\frac{2\pi}{4}=\frac{\pi}{2}$.
Step2: Find the least - common multiple of the periods
We need to find the least - common multiple of $\frac{2\pi}{3}$ and $\frac{\pi}{2}$. Rewrite $\frac{2\pi}{3}$ and $\frac{\pi}{2}$ with a common denominator. $\frac{2\pi}{3}=\frac{4\pi}{6}$ and $\frac{\pi}{2}=\frac{3\pi}{6}$. The least - common multiple of the numerators 4 and 3 is 12. So the least - common multiple of $\frac{2\pi}{3}$ and $\frac{\pi}{2}$ is $2\pi$.
Answer:
$2\pi$