Question

write the equation of a sine function with amplitude = 2 and period = π. type the equation in the form y = a sin(ωx) or y = a cos(ωx) with ω > 0. select the correct choice and, if necessary, fill in the answer box(es) to complete your choice. a. there are two equations; the equation when a < 0 is y = □ and the equation when a > 0 is y = □. (simplify your answers. type exact answers, using π as needed. use integers or fractions for any numbers in the expressions.) b. there is one equation. it is y = □. (simplify your answer. type an exact answer, using π as needed. use integers or fractions for any numbers in the expression.) c. there are no equations with these characteristics.

Explanation:

Step1: Recall the formula for amplitude and period

The general form of a sine function is \(y = A\sin(\omega x)\) or \(y=A\cos(\omega x)\). The amplitude is \(|A|\) and the period \(T=\frac{2\pi}{\omega}\).
Given that the amplitude \(|A| = 2\), so \(A=\pm2\). Given that the period \(T = \pi\).

Step2: Solve for \(\omega\)

Since \(T=\frac{2\pi}{\omega}\) and \(T = \pi\), we substitute \(T\) into the formula: \(\pi=\frac{2\pi}{\omega}\).
Cross - multiply gives \(\pi\omega=2\pi\). Divide both sides by \(\pi\) (since \(\pi
eq0\)), we get \(\omega = 2\).

Answer:

A. There are two equations; the equation when \(A<0\) is \(y=- 2\sin(2x)\) (or \(y = - 2\cos(2x)\)) and the equation when \(A>0\) is \(y = 2\sin(2x)\) (or \(y=2\cos(2x)\))