Chapter 4.4, Problem 92E

Solution

a.

To find: The general rule for the values of $a$ and $b$ .

The general rule for the values of $a$ and $b$ is $a=b+1$ .

Given information:

  $y=\cos \left(ax\right)\cos \left(bx\right)+\sin \left(ax\right)\sin \left(bx\right)$ .

Calculation:

Considering the equation,

  $y=\cos \left(ax\right)\cos \left(bx\right)+\sin \left(ax\right)\sin \left(bx\right)$

Some values of $a$ and $b$ in the graph is given by,

  $y=\cos x$

Now for cosine reversing the angle addition formula to get,

  $y=\cos \left(\left(a-b\right)x\right)$

From the above equation it is known that,

  $\cos \left(\left(a-b\right)x\right)=\cos x$ , when $a-b=1$

Thus,

  $\begin{array}{c}a-b=1\\ a=b+1\end{array}$

Therefore, The general rule for the values of $a$ and $b$ is $a=b+1$ .

b.

To find: The general rule for the values of $a$ and $b$ .

The general rule for the values of $a$ and $b$ is $a=b+2$ .

Given information:

  $y=\cos \left(ax\right)\cos \left(bx\right)+\sin \left(ax\right)\sin \left(bx\right)$ .

Calculation:

Considering the equation,

  $y=\cos \left(ax\right)\cos \left(bx\right)+\sin \left(ax\right)\sin \left(bx\right)$

Some values of $a$ and $b$ in the graph is given by,

  $y=\cos 2x$

Now for cosine reversing the angle addition formula to get,

  $y=\cos \left(\left(a-b\right)x\right)$

From the above equation it is known that,

  $\cos \left(\left(a-b\right)x\right)=\cos 2x$ , when $a-b=2$

Thus,

  $\begin{array}{c}a-b=2\\ a=b+2\end{array}$

Therefore, the general rule for the values of $a$ and $b$ is $a=b+2$ .

c.

To find: The general rule for an arbitrary integer $k$ .

The general rule for an arbitrary integer is $a=b+k$ .

Given information:

  $y=\cos \left(ax\right)\cos \left(bx\right)+\sin \left(ax\right)\sin \left(bx\right)$ .

Calculation:

Considering the equation,

  $y=\cos \left(ax\right)\cos \left(bx\right)+\sin \left(ax\right)\sin \left(bx\right)$

Some values of $a$ and $b$ in the graph is given by,

  $y=\cos kx$

Now for cosine reversing the angle addition formula to get,

  $y=\cos \left(\left(a-b\right)x\right)$

From the above equation it is known that,

  $\cos \left(\left(a-b\right)x\right)=\cos kx$ , when $a-b=k$

Thus,

  $\begin{array}{c}a-b=k\\ a=b+k\end{array}$

Therefore, the general rule for an arbitrary integer is $a=b+k$ .