Question Type 3: Formulating sinusoidal models through descriptions Exercises

Express $\sin(5x+9\pi)$ in the form $\cos\bigl(5(x-c)\bigr)$.

Write $\sin\bigl(3x-\tfrac{\pi}{4}\bigr)$ as $\cos\bigl(3(x-c)\bigr)$.

Express $\cos\bigl(4x+\tfrac{\pi}{6}\bigr)$ in the form $\sin\bigl(4(x+c)\bigr)$.

Given an amplitude of 4, period 3 and principal axis 3, find a sinusoidal model of the form $f(x)=3+4\\sin\bigl(bx+\phi\bigr)$ such that $f(0)=3$.

Given amplitude 5, period $\frac{2\pi}{3}$ and principal axis 1, find a model of the form $f(x)=1+5\cos(bx+\phi)$ satisfying $f(0)=6$.

Express $\cos\bigl(2x-\tfrac{\pi}{3}\bigr)$ in the form $\sin\bigl(2(x-c)\bigr)$.

Amplitude 3, period $\pi$ and principal axis 2. Find a model of the form $f(x)=2+3\cos(bx+\phi)$ such that $f\bigl(\tfrac{\pi}{2}\bigr)=2-3\,$ a minimum.

Amplitude 3, period 4 and midline 1. At $x=2$ the function attains its maximum and is decreasing. Find $f(x)$.

Amplitude 2, period 4 and principal axis -1. Find a model of the form $f(x)=-1+2\sin\bigl(b(x-c)\bigr)$ so that $f(1)=-1$.

Tidal height (m) follows a sinusoid with amplitude 2.5, period 12 h, midline 3 m. At t=4 h it is at a minimum and rising. Find a model $H(t)$.

A daily temperature (°C) is modeled by a sinusoid with amplitude 10, period 24 h and midline 15 °C. The maximum temperature occurs at 3 pm (t=15 h). Write a model $T(t)$.

Given amplitude 6, period $\pi$, midline $-2$ and $f\bigl(\tfrac{\pi}{4}\bigr)=-2+\frac{6}{\sqrt2}$ with function increasing there, find $f(x)$ in the form $-2+6\sin(bx+\phi)$.