Chapter 0.6, Problem 56E

(a)

To determine

To graph : The function and explain it.

Explanation of Solution

Given information :

The function $y=f(x)$ , where $f\left(x\right)=\sin x+\cos x$

Graph :

The graph of the given function is shown below:

  

Interpretation:

The graph is a sine/cosine type graph with amplitude 1.

To determine

To identify : Amplitude, period, horizontal shift and vertical shift using the graph.

Answer to Problem 56E

Amplitude $=1.414$ ,

Period $=6.28\left(i.e2\pi \right)$ ,

Horizontal shift $=-0.785\left(\text{which is}-\frac{\pi }{4}\right)$ and

Vertical shift $=0$ .

Explanation of Solution

Given information :

The function $y=f(x)$ , where $f\left(x\right)=\sin x+\cos x$

From the graph, we measure that

Amplitude $=1.414$

Period $=6.28\left(i.e2\pi \right)$

Horizontal shift $=-0.785\left(\text{which is}-\frac{\pi }{4}\right)$

Vertical shift $=0$

(c)

To determine

To calculate : The sine of the sum of two angles using the given formula.

Answer to Problem 56E

The sine of the sum of two angles is $\sqrt{2}\sin \left(x+\frac{\pi }{4}\right)$ .

Explanation of Solution

Given information :

The function $y=f(x)$ , where $f\left(x\right)=\sin x+\cos x$ and the given formula is $\sin \alpha \cos \beta +\cos \alpha \sin \beta =\sin \left(\alpha +\beta \right)$ .

Formula used :

  $\sin \alpha \cos \beta +\cos \alpha \sin \beta =\sin \left(\alpha +\beta \right)$

Calculation :

Consider

  $\begin{array}{l}f\left(x\right)=\sin x+\cos x\\ =\sqrt{2}\left[\left(\sin x\right)\left(\frac{1}{\sqrt{2}}\right)+\left(\cos x\right)\left(\frac{1}{\sqrt{2}}\right)\right]\\ =\sqrt{2}\left[\left(\sin x\right)\left(\cos \frac{\pi }{4}\right)+\left(\cos x\right)\left(\sin \frac{\pi }{4}\right)\right]\\ =\sqrt{2}\sin \left(x+\frac{\pi }{4}\right)\end{array}$

Therefore, the sine of the sum of two angles is $\sqrt{2}\sin \left(x+\frac{\pi }{4}\right)$ .