write the equation of a sine function with amplitude = 8 and period = 4π. type the equation in the form y =…
write the equation of a sine function with amplitude = 8 and period = 4π. 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 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.) b. 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.) c. there are no equations with these characteristics.
Answer
Explanation:
Step1: Recall the formula for the amplitude and period of a sine function
The general form of a sine function is (y = A\sin(\omega x)). The amplitude is (|A|) and the period is (T=\frac{2\pi}{\omega}). Given that the amplitude (|A| = 8), so (A=\pm8). Given that the period (T = 4\pi).
Step2: Solve for (\omega)
Using the period formula (T=\frac{2\pi}{\omega}), substitute (T = 4\pi) into it: [ \begin{align*} 4\pi&=\frac{2\pi}{\omega}\ \omega&=\frac{2\pi}{4\pi}\ \omega&=\frac{1}{2} \end{align*} ]
Step3: Write the equations
When (A = 8), the equation is (y = 8\sin(\frac{1}{2}x)). When (A=- 8), the equation is (y=-8\sin(\frac{1}{2}x))
Answer:
B. There are two equations; the equation when (A < 0) is (y=-8\sin(\frac{1}{2}x)) and the equation when (A>0) is (y = 8\sin(\frac{1}{2}x))