Chapter 1.6, Problem 57E

(a)

To determine

To find: The inverse trigonometric function.

Answer to Problem 57E

The function as a sinusoid is $y=\sqrt{2}\sin \left(ax+\frac{\pi }{4}\right)$ .

Explanation of Solution

Given information:

The given function is $y=\sin \left(ax\right)+\cos \left(ax\right)$ .

Calculation:

Calculate the function.

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

Therefore, the function as a sinusoid is $y=\sqrt{2}\sin \left(ax+\frac{\pi }{4}\right)$ .

(b)

To determine

To find: The positive integer value of $n$ .

Answer to Problem 57E

The positive value of $n$ is $f\left(x\right)=\sqrt{2}\sin \left(nx+\frac{\pi }{4}\right)$ .

Explanation of Solution

Given information:

The given function is $y=\sin \left(ax\right)+\cos \left(ax\right)$ .

Calculation:

For conjecture another formula substitute n or $a$ in the formula $y=\sqrt{2}\sin \left(ax+\frac{\pi }{4}\right)$ .

  $f\left(x\right)=\sqrt{2}\sin \left(nx+\frac{\pi }{4}\right)$

Therefore, the positive integer value of $f\left(x\right)=\sqrt{2}\sin \left(nx+\frac{\pi }{4}\right)$ .

(c)

To determine

To check: The conjecture with the help of CAS.

Answer to Problem 57E

The conjecture formed is true for positive value of $a$ .

Explanation of Solution

Given information:

The given function is $y=\sin \left(ax\right)+\cos \left(ax\right)$ .

The conjecture work with a computer algebraic system for all positive value of $a$ .

Hence, the conjecture formed is true for positive value of $a$ .

(d)

To determine

To check: The conjecture by the formula for sine of the sum of two angles.

Answer to Problem 57E

The conjecture is true by the formula for sine of the sum of two angles.

Explanation of Solution

Given information:

The given function is $y=\sin \left(ax\right)+\cos \left(ax\right)$ .

Calculation:

Check the conjecture by the sum of two angles of sine function as:

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

Therefore, the conjecture is true by the formula for sine of the sum of two angles.