Question

multiple choice 1 point
write the equation for the sine function shown below.

( y = - 5 sin 2 \theta )
( y = 3 sin 4 \theta )
( y = - 5 sin \theta )
( y = 5 sin 2 \theta )

Explanation:

Step1: Recall the general form of sine function

The general form of a sine function is \(y = A\sin(B\theta)\), where \(|A|\) is the amplitude and \(T=\frac{2\pi}{B}\) is the period.

Step2: Determine the amplitude \(A\)

The amplitude is the maximum distance from the mid - line. Looking at the graph, the maximum value (above the mid - line \(y = 0\)) is \(5\) and the minimum is \(- 5\). So \(|A| = 5\). Also, since the graph starts at the origin and goes down first (compared to \(y=\sin\theta\) which starts at the origin and goes up first), \(A=-5\).

Step3: Determine the period \(T\) and find \(B\)

The period \(T\) is the length of one full cycle. From the graph, if we assume \(\theta\) is on the \(x\) - axis, one full cycle occurs over an interval of \(\pi\). Using the formula \(T=\frac{2\pi}{B}\), and since \(T = \pi\), we solve \(\pi=\frac{2\pi}{B}\) for \(B\). Cross - multiply gives \(B\pi=2\pi\), so \(B = 2\).

Answer:

\(y=-5\sin(2\theta)\) (the first option \(y = - 5\sin(2\theta)\))