Chapter 10.6, Problem 44PS

Solution

(a)

To find: Find the values of $\sin Cand\cos C$ .

The value of $\sin C=\frac{3}{5},\cos C=\frac{4}{5}$ .

Given information:

Given that $C={\tan }^{-1}\left(\frac{3}{4}\right)$ .

Calculation:

  $C={\tan }^{-1}\left(\frac{3}{4}\right)$ can be rewritten as:

  $\tan C=\left(\frac{3}{4}\right)$

By trigonometric formula it is known that:

  $\tan \theta =\frac{perpendicular}{base}$

Therefore $base=4,perpendicular=3$ .

It is required to find hypotenuse.

Apply the Pythagoras theorem:

  $Hypotenus{e}^2=bas{e}^2+perpendicula{r}^2$

  $\begin{array}{l}Hypotenus{e}^2=4^2+3^2\\ Hypotenus{e}^2=16+9\\ Hypotenus{e}^2=25\\ Hypotenuse=\sqrt{25}\\ Hypotenuse=5\end{array}$

Apply the trigonometric formula for $\sin \theta$ :

  $\begin{array}{l}\sin C=\frac{Perpendicular}{Hypotenuse}\\ \sin C=\frac{3}{5}\end{array}$

Apply the trigonometric formula for $\cos \theta$ :

  $\begin{array}{l}\cos C=\frac{Base}{Hypotenuse}\\ \cos C=\frac{4}{5}\end{array}$

The value of $\sin C=\frac{3}{5},\cos C=\frac{4}{5}$ .

(b)

To find: Use sum formula to show that $y=5\sin \left(2t+C\right)$ can be written as $y=4\sin 2t+3\cos 2t$

The result is shown.

Given information:

Given that $C={\tan }^{-1}\left(\frac{3}{4}\right)$ and $y=5\sin \left(2t+C\right)$ .

Calculation:

Apply the formula $\sin \left(A+B\right)=\sin A\cos B+\cos A\sin B$ :

  $y=5\sin \left(2t+C\right)$ can be written as:

  $y=5\sin 2t\cos C+5\cos 2t\sin C$

Substitute the value of $\cos C=\frac{4}{5},\sin C=\frac{3}{5}$ :

  $\begin{array}{l}y=5\sin 2t\cos C+5\cos 2t\sin C\\ y=5\sin 2t\left(\frac{4}{5}\right)+5\cos 2t\left(\frac{3}{5}\right)\\ y=4\sin 2t+3\cos 2t\end{array}$

This is the required result.

The value of $\sin C=\frac{3}{5},\cos C=\frac{4}{5}$ .

(c)

To find: Graph the function $y=4\sin 2t+3\cos 2t$ and find its maximum value. What this maximum value represents.

The graph is shown.

Given information:

The function is given as $y=4\sin 2t+3\cos 2t$

In order to graph this function substitute different values of $t$ and find corresponding values of $y$ then plot these points on the graph.

Let $t=0$ substitute this value in the given equation:

  $\begin{array}{l}y=4\sin 2\left(0\right)+3\cos 2(0)\\ y=0+3\cdot 1\cos (0)=1\\ y=3\end{array}$

Let $t=\frac{\pi }{2}$ substitute this value in the given equation:

  $\begin{array}{l}y=4\sin 2\left(\frac{\pi }{2}\right)+3\cos 2\left(\frac{\pi }{2}\right)\\ y=4\sin \pi +3\cos (\pi )\\ y=0-3\\ y=-3\end{array}$

Now substitute $t=\frac{\pi }{4}$ :

  $\begin{array}{l}y=4\sin 2\left(\frac{\pi }{4}\right)+3\cos 2\left(\frac{\pi }{2}\right)\\ y=4\sin \left(\frac{\pi }{2}\right)+3\cos \left(\frac{\pi }{2}\right)\\ y=4+3\cdot 0\\ y=4\end{array}$

Similarly, more value can be found to plot the graph. The graph is below:

  

The maximum value of the function $y=4\sin 2t+3\cos 2t$ is at $x=3.06,y=5$ .

This value represents that the function attains its maximum value at this point after that is start decreasing.