Chapter 0.6, Problem 58E

(a)

To determine

To calculate : Using the symbolic manipulator of a computer algebra system express $y$ as a sinusoid for the given pair of values. $a=2,b=1;a=1,b=2;a=5,b=2;a=2,b=5;a=3,b=4$ .

Answer to Problem 58E

The required formula for $y$ is $\sqrt{a^2+b^2}\sin \left(x+{\tan }^{-1}\left(\frac{b}{a}\right)\right)$ .

Explanation of Solution

Given information :

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

Calculation:

Consider $y=f\left(x\right)$

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

Where $\begin{array}{l}\sin \theta =\frac{a}{\sqrt{a^2+b^2}}\text{ and cos}\theta =\frac{b}{\sqrt{a^2+b^2}}\\ \tan \theta =\frac{b}{a}\\ \Rightarrow \theta ={\tan }^{-1}\left(\frac{b}{a}\right)\end{array}$

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

For $a=2,b=1$ ,

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

For $a=1,b=2$ ,

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

For $a=5,b=2$ ,

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

For $a=2,b=5$ ,

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

For $a=3,b=4$ ,

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

(b)

To determine

To calculate : Using the symbolic manipulator of a computer algebra system conjecture another formula for $y$ for any pair of positive integers.

Answer to Problem 58E

The required formula for $y$ is $\sqrt{a^2+b^2}\sin \left(x+{\tan }^{-1}\left(\frac{b}{a}\right)\right)$ where $n$ is a positive integer.

Explanation of Solution

Given information :

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

Calculation:

Conjecture another formula when $a=n$ can be written as

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

(c)

To determine

To check : The conjecture using CAS.

Answer to Problem 58E

The conjecture works with a Computer Algebraic System for all positive values of $a$ .

Explanation of Solution

Given information :

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

The conjecture works with a Computer Algebraic System for all positive values of $a$ .

(c)

To determine

To calculate : The sine and cosine of a sum or a difference of two angles using the given formula.

Answer to Problem 58E

The sine and cosine of a sum or a difference of two angles is $\left(\frac{1}{\sqrt{a^2+b^2}}\right)\left(a\sin x+b\cos x\right)$ .

Explanation of Solution

Given information :

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

Formula used :

  $\begin{array}{l}\sin \alpha \cos \beta \pm \cos \alpha \sin \beta =\sin \left(\alpha \pm \beta \right)\\ \cos \alpha \cos \beta \pm \sin \alpha \sin \beta =\cos \left(\alpha \stackrel{-}{+}\beta \right)\end{array}$

Calculation :

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