1. Draw the graphs of \(\cos x\), \(\sin x\), and \(a\cos x + b\sin x\).

[Since

\(a\cos x + b\sin x = \beta\cos(x – \alpha)\), where

\(\beta = \sqrt{a^{2} + b^{2}}\), and

\(\alpha\) is an angle whose cosine and sine are

\(a/\sqrt{a^{2} + b^{2}}\) and

\(b/\sqrt{a^{2} + b^{2}}\), the graphs of these three functions are similar in character.]

2. Draw the graphs of \(\cos^{2} x\), \(\sin^{2} x\), \(a\cos^{2} x + b\sin^{2} x\).

3. Suppose the graphs of \(f(x)\) and \(F(x)\) drawn. Then the graph of \[f(x)\cos^{2} x + F(x)\sin^{2} x\] is a wavy curve which oscillates between the curves \(y = f(x)\), \(y = F(x)\). Draw the graph when \(f(x) = x\), \(F(x) = x^{2}\).

4. Show that the graph of \(\cos px + \cos qx\) lies between those of \(2\cos\frac{1}{2}(p – q)x\) and \(-2\cos\frac{1}{2}(p + q)x\), touching each in turn. Sketch the graph when \((p – q)/(p + q)\) is small.

5. Draw the graphs of \(x + \sin x\), \((1/x) + \sin x\), \(x\sin x\), \((\sin x)/x\).

6. Draw the graph of \(\sin(1/x)\).

[If

\(y = \sin(1/x)\), then

\(y = 0\) when

\(x = 1/m\pi\), where

\(m\) is any integer. Similarly

\(y = 1\) when

\(x = 1/(2m + \frac{1}{2})\pi\) and

\(y = -1\) when

\(x = 1/(2m – \frac{1}{2})\pi\). The curve is entirely comprised between the lines

\(y = -1\) and

\(y = 1\) (

fig. 13). It oscillates up and down, the rapidity of the oscillations becoming greater and greater as

\(x\) approaches

\(0\). For

\(x = 0\) the function is undefined. When

\(x\) is large

\(y\) is small.

The negative half of the curve is similar in character to the positive half.]

7. Draw the graph of \(x\sin(1/x)\).

[This curve is comprised between the lines

\(y = -x\) and

\(y = x\) just as the last curve is comprised between the lines

\(y = -1\) and

\(y = 1\) (

fig. 14).]

8. Draw the graphs of \(x^{2}\sin(1/x)\), \((1/x)\sin(1/x)\), \(\sin^{2}(1/x)\), \(\{x\sin(1/x)\}^{2}\), \(a\cos^{2}(1/x) + b\sin^{2}(1/x)\), \(\sin x + \sin(1/x)\), \(\sin x\sin(1/x)\).

9. Draw the graphs of \(\cos x^{2}\), \(\sin x^{2}\), \(a\cos x^{2} + b\sin x^{2}\).

10. Draw the graphs of \(\arccos x\) and \(\arcsin x\).

[If

\(y = \arccos x\),

\(x = \cos y\). This enables us to draw the graph of

\(x\), considered as a function of

\(y\), and the same curve shows

\(y\) as a function of

\(x\). It is clear that

\(y\) is only defined for

\(-1 \leq x \leq 1\), and is infinitely many-valued for these values of

\(x\). As the reader no doubt remembers, there is, when

\(-1 < x < 1\), a value of

\(y\) between

\(0\) and

\(\pi\), say

\(\alpha\), and the other values of

\(y\) are given by the formula

\(2n\pi \pm \alpha\), where

\(n\) is any integer, positive or negative.]

11. Draw the graphs of \[\tan x,\quad \cot x,\quad \sec x,\quad \csc x,\quad \tan^{2} x,\quad \cot^{2} x,\quad \sec^{2} x,\quad \csc^{2} x.\]

12. Draw the graphs of \(\arctan x\), \(\operatorname{arccot} x\), \(\operatorname{arcsec} x\), \(\operatorname{arccsc} x\). Give formulae (as in Ex. 10) expressing all the values of each of these functions in terms of any particular value.

13. Draw the graphs of \(\tan(1/x)\), \(\cot(1/x)\), \(\sec(1/x)\), \(\csc(1/x)\).

14. Show that \(\cos x\) and \(\sin x\) are not rational functions of \(x\).

[A function is said to be

*periodic*, with period

\(a\), if

\(f(x) = f(x + a)\) for all values of

\(x\) for which

\(f(x)\) is defined. Thus

\(\cos x\) and

\(\sin x\) have the period

\(2\pi\). It is easy to see that no periodic function can be a rational function, unless it is a constant. For suppose that

\[f(x) = P(x)/Q(x),\] where

\(P\) and

\(Q\) are polynomials, and that

\(f(x) = f(x + a)\), each of these equations holding for all values of

\(x\). Let

\(f(0) = k\). Then the equation

\(P(x) – kQ(x) = 0\) is satisfied by an infinite number of values of

\(x\), viz.

\(x = 0\),

\(a\),

\(2a\), etc., and therefore for all values of

\(x\). Thus

\(f(x) = k\) for all values of

\(x\),

\(f(x)\) is a constant.]

15. Show, more generally, that no function with a period can be an algebraical function of \(x\).

[Let the equation which defines the algebraical function be

\[\begin{equation*}y^{m} + R_{1}y^{m-1} + \dots + R_{m} = 0 \tag{1}\end{equation*}\] where

\(R_{1}\), … are rational functions of

\(x\). This may be put in the form

\[P_{0}y^{m} + P_{1}y^{m-1} + \dots + P_{m} = 0,\] where

\(P_{0}\),

\(P_{1}\), … are polynomials in

\(x\). Arguing as above, we see that

\[P_{0}k^{m} + P_{1}k^{m-1} + \dots + P_{m} = 0\] for all values of

\(x\). Hence

\(y = k\) satisfies the equation for all values of

\(x\), and one set of values of our algebraical function reduces to a constant.

Now divide (1) by \(y – k\) and repeat the argument. Our final conclusion is that our algebraical function has, for any value of \(x\), the same set of values \(k\), \(k’\), …; it is composed of a certain number of constants.]

16. The inverse sine and inverse cosine are not rational or algebraical functions. [This follows from the fact that, for any value of \(x\) between \(-1\) and \(+1\), \(\arcsin x\) and \(\arccos x\) have infinitely many values.]