Chapter 1.6, Problem 58E

(a)

To determine

To find: The sinusoid wave pair of the value.

Answer to Problem 58E

One possible sinusoid wave pair of the value is $f\left(x\right)=5\sin \left(x+{\tan }^{-1}\left(\frac{4}{3}\right)\right)$ .

Explanation of Solution

Given information:

The given function is $y=a\sin x+b\cos x$ .

Calculation:

Calculate the function.

  $\begin{array}{c}y=a\sin x+b\cos x\\ =\sqrt{a^2+b^2}\left[\left(\sin x\right)\left(\cos x\right)+\left(\cos x\right)\left(\sin x\right)\right]\\ =\sqrt{a^2+b^2}\sin \left(x+\theta \right)\end{array}$

Here, $\sin \theta =\frac{a}{\sqrt{a^2+b^2}}$ and $\cos \theta =\frac{b}{\sqrt{a^2+b^2}}$ . So,

  $\begin{array}{c}\tan \theta =\frac{b}{a}\\ \theta ={\tan }^{-1}\left(\frac{b}{a}\right)\end{array}$

The sinusoid function is:

  $f\left(x\right)=\sqrt{a^2+b^2}\sin \left(x+{\tan }^{-1}\left(\frac{b}{a}\right)\right)$

For pair of values $a=2$ and $b=1$ . The sinusoid function is:

  $f\left(x\right)=\sqrt{5}\sin \left(x+{\tan }^{-1}\left(\frac{1}{2}\right)\right)$

For pair of values $a=1$ and $b=2$ . The sinusoid function is:

  $f\left(x\right)=\sqrt{5}\sin \left(x+{\tan }^{-1}2\right)$

For pair of values $a=5$ and $b=2$ . The sinusoid function is:

  $f\left(x\right)=\sqrt{29}\sin \left(x+{\tan }^{-1}\left(\frac{2}{5}\right)\right)$

For pair of values $a=2$ and $b=5$ . The sinusoid function is:

  $f\left(x\right)=\sqrt{29}\sin \left(x+{\tan }^{-1}\left(\frac{5}{2}\right)\right)$

For pair of values $a=3$ and $b=4$ . The sinusoid function is:

  $f\left(x\right)=5\sin \left(x+{\tan }^{-1}\left(\frac{4}{3}\right)\right)$

Therefore, one possible sinusoid wave pair is $f\left(x\right)=5\sin \left(x+{\tan }^{-1}\left(\frac{4}{3}\right)\right)$ .

(b)

To determine

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

Answer to Problem 58E

The positive integer value of $n$ is $f\left(x\right)=\sqrt{a^2+b^2}\sin \left(x+{\tan }^{-1}\left(\frac{b}{a}\right)\right)$ .

Explanation of Solution

Given information:

The given function is $y=a\sin x+b\cos x$ .

Calculation:

For conjecture another formula substitute n or $a$ in the formula calculated in part (a).

  $f\left(x\right)=\sqrt{a^2+b^2}\sin \left(x+{\tan }^{-1}\left(\frac{b}{a}\right)\right)$

Here, $n$ is positive integer.

Therefore, the positive integer value of $n$ is $f\left(x\right)=\sqrt{a^2+b^2}\sin \left(x+{\tan }^{-1}\left(\frac{b}{a}\right)\right)$ .

(c)

To determine

To check: The conjecture with the help of CAS.

Answer to Problem 58E

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

Explanation of Solution

Given information:

The given function is $y=a\sin x+b\cos x$ .

The computed conjecture works with a computer algebraic system for all positive values 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 58E

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

Explanation of Solution

Given information:

The given function is $y=a\sin x+b\cos x$ .

Calculation:

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

  $\begin{array}{c}\sin \left(x+{\tan }^{-1}\left(\frac{b}{a}\right)\right)=\left(\sin x\right)\left(\cos \left({\tan }^{-1}\left(\frac{b}{a}\right)\right)\right)+\left(\cos x\right)\left(\sin \left({\tan }^{-1}\left(\frac{b}{a}\right)\right)\right)\\ =\sin x\left(\frac{a}{\sqrt{a^2+b^2}}\right)+\cos x\left(\frac{b}{\sqrt{a^2+b^2}}\right)\\ =\left(\frac{1}{\sqrt{a^2+b^2}}\right)\left(a\sin x+b\cos x\right)\end{array}$

It can be observed that conjecture is not true for negative values of a .

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