Question Type 4: Converting between sinusoidal model representations Exercises

Write $y=\sin\bigl(2x+\tfrac{\pi}{6}\bigr)$ in the form $\cos\bigl(2(x-c)\bigr)$ and determine $c$.

Express $y=4\sin\bigl(3x-\tfrac{\pi}{4}\bigr)$ as $\cos\bigl(3(x-d)\bigr)$ and find $d$.

Rewrite $y=5\cos\bigl(2\theta+\tfrac{\pi}{3}\bigr)$ in the form $A\sin\bigl(2(\theta+e)\bigr)$ and find $A$ and $e$.

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

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

Rewrite $y=7\cos\bigl(2\phi+\tfrac{3\pi}{4}\bigr)$ as $\sin\bigl(2(\phi+h)\bigr)$ and find $h$.

Express $y=-5\sin\bigl(3t-\tfrac{5\pi}{6}\bigr)$ as $\cos\bigl(3(t-g)\bigr)$ and determine $g$.

Express $y=4\cos\bigl(5x-\tfrac{7\pi}{3}\bigr)$ as $\sin\bigl(5(x+q)\bigr)$ and find $q$ in $[0,2\pi)$.

Rewrite $y=3\cos\bigl(2x+\tfrac{5\pi}{6}\bigr)$ in the form $3\sin\bigl(2(x-c)\bigr)$ and determine $c$ in $[0,\pi)$.

Simplify $y=\sin\bigl(6x+\tfrac{17\pi}{3}\bigr)$ and express it as $\cos\bigl(6(x-i)\bigr)$. Find $i$ in $[0,2\pi)$.

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 $\omega$.

Convert $y=3\sin\bigl(4x-\tfrac{7\pi}{3}\bigr)$ into the form $\cos\bigl(4(x-c)\bigr)$ and give the smallest positive $c$.