Question Type 4: Converting between sinusoidal model representations Exercises

Simplify $y = \sin\left(6x + \frac{17\pi}{3}\right)$ and express it in the form $\cos\left(6(x-i)\right)$. Find a value of $i$ in the interval $[0, 2\pi)$.

Derive the general formula for writing $\sin(\omega x + \alpha)$ in the form $\cos\bigl(\omega(x - c)\bigr)$, expressing $c$ in terms of $\alpha$ and $\omega$.

Express $y=4\cos\bigl(5x-\frac{7\pi}{3}\bigr)$ in the form $y=4\sin\bigl(5(x+q)\bigr)$, where $q$ is the smallest positive constant.

Rewrite $y = 5 \cos \left( 2\theta + \frac{\pi}{3} \right)$ in the form $A \sin(2(\theta + e))$ and find the values of $A$ and $e$.

Express $y = 3\sin\left(4x - \frac{7\pi}{3}\right)$ in the form $y = 3\cos(4(x - c))$ and find the smallest positive value of $c$.

Express $y=4\sin\left(3x-\frac{\pi}{4}\right)$ in the form $4\cos\bigl(3(x-d)\bigr)$ and find the value of $d$.

Express $y=7\cos\left(2\phi+\frac{3\pi}{4}\right)$ in the form $y=7\sin\bigl(2(\phi+h)\bigr)$ and find the value of $h$.

Express $y = -5\sin\left(3t - \frac{5\pi}{6}\right)$ in the form $5\cos(3(t - g))$ and determine the value of $g$.

Convert $y = \sin(5x + 9\pi)$ into the form $\cos(5(x - c))$ and give a value for $c$ in the interval $0 \le c < 2\pi$.

Write $y = \sin \left( 2x + \frac{\pi}{6} \right)$ in the form $\cos (2(x - c))$ and determine the value of $c$.

Rewrite $y = 3\cos\left(2x + \frac{5\pi}{6}\right)$ in the form $3\sin\left(2(x - c)\right)$ and determine the value of $c$ for $c \in [0, \pi)$.

Write $y=2\cos\bigl(4t-\pi\bigr)$ in the form $2\sin\bigl(4(t-f)\bigr)$ and find $f$.