Investigating trigonometric functions

Calculus, grade 11 (5 units) Lesson 23 of 26Learning

Investigating trigonometric functions

The steps of an investigation

Investigating a trigonometric function goes through the same steps as any investigation, and starts by asking whether the function is periodic:

The period: if $\displaystyle f$ is periodic, investigate it on an interval of one period, for example $\displaystyle 0\leq x\leq2\pi$ , and repeat the graph. If not – on the interval the question gives.
The domain: $\displaystyle \tan x$ , a denominator and a root leave out points or intervals.
The derivative: solve $\displaystyle {f'(x)}={0}$ in the interval – a trigonometric equation, whose solutions in the interval are a finite number of points.
A table of signs of $\displaystyle f'$ : the extremum points and where the function rises and falls.
The second derivative: the sign of $\displaystyle f''$ gives the intervals of concavity, and a point where it changes is an inflection point.
A sketch: the points you found, then repeated by the period.

Example:

\displaystyle x+2\sin x

The example below investigates $\displaystyle {f(x)}={x+2\sin x}$ on the interval $\displaystyle 0\leq x\leq2\pi$ . The function is not periodic, because of the term $\displaystyle x$ .

$\displaystyle {f'(x)}={1+2\cos x}$ . $\displaystyle {f'(x)}={0}$ where $\displaystyle {\cos x}={-\frac{1}{2}}$ , and in the interval: $\displaystyle {x}={\frac{2\pi}{3}}$ or $\displaystyle {x}={\frac{4\pi}{3}}$ .

$\displaystyle x$	$\displaystyle 0<x<\frac{2\pi}{3}$	$\displaystyle \frac{2\pi}{3}$	$\displaystyle \frac{2\pi}{3}<x<\frac{4\pi}{3}$	$\displaystyle \frac{4\pi}{3}$	$\displaystyle \frac{4\pi}{3}<x<2\pi$
$\displaystyle f'(x)$	+	0	−	0	+
$\displaystyle f(x)$	rises	maximum	falls	minimum	rises

$\displaystyle {f\left(\frac{2\pi}{3}\right)}={\frac{2\pi}{3}+2\cdot\frac{\sqrt{3}}{2}}={\frac{2\pi}{3}+\sqrt{3}\approx3.83}$ ; $\displaystyle {f\left(\frac{4\pi}{3}\right)}={\frac{4\pi}{3}-\sqrt{3}\approx2.46}$ .
$\displaystyle {f''(x)}={-2\sin x}$ : negative where $\displaystyle 0<x<\pi$ – concave down; positive where $\displaystyle \pi<x<2\pi$ – concave up.
At $\displaystyle {x}={\pi}$ the concavity changes: an inflection point $\displaystyle (\pi,\pi)$ .
At the ends of the interval: $\displaystyle {f(0)}={0}$ – a minimum at an end, $\displaystyle {f(2\pi)}={2\pi}$ – a maximum at an end.

\displaystyle f(x)=x+2\sin x

\displaystyle y=x

\displaystyle M=(\frac{2\pi}{3},\frac{2\pi}{3}+\sqrt{3})

\displaystyle N=(\frac{4\pi}{3},\frac{4\pi}{3}-\sqrt{3})

\displaystyle I=(\pi,\pi)

The graph winds around the line $\displaystyle {y}={x}$ (dashed): $\displaystyle 2\sin x$ adds up to 2 to it and takes up to 2 from it. It crosses the line where $\displaystyle {\sin x}={0}$ – at the inflection point and at the ends of the interval.

Example:

\displaystyle \sqrt{\sin x}

The example below investigates $\displaystyle {f(x)}={\sqrt{\sin x}}$ – the root of a trigonometric function:

The domain: $\displaystyle \sin x\geq0$ , that is $\displaystyle 0\leq x\leq\pi$ , and in every such period: $\displaystyle 2\pi k\leq x\leq\pi+2\pi k$ . Between these intervals the function has no values.
The period is $\displaystyle 2\pi$ , so it is enough to investigate the interval $\displaystyle 0\leq x\leq\pi$ .
$\displaystyle {f'(x)}={\frac{\cos x}{2\sqrt{\sin x}}}$ , and inside the interval $\displaystyle {f'(x)}={0}$ where $\displaystyle {\cos x}={0}$ , that is $\displaystyle {x}={\frac{\pi}{2}}$ .
$\displaystyle f'$ is positive where $\displaystyle 0<x<\frac{\pi}{2}$ and negative where $\displaystyle \frac{\pi}{2}<x<\pi$ : a maximum $\displaystyle \left(\frac{\pi}{2},1\right)$ .

\displaystyle f(x)=\sqrt{\sin x}

At the ends of the interval, $\displaystyle (0,0)$ and $\displaystyle (\pi,0)$ , the function reaches its least value: minimum points at the ends of the domain.
There the denominator of $\displaystyle f'$ is 0 and the derivative is not defined: the graph leaves the ends with a vertical tangent, like $\displaystyle \sqrt{x}$ at 0.

The family

\displaystyle {y}={\sin x+ax}

All the functions of the family $\displaystyle {f(x)}={\sin x+ax}$ have the same inflection points, but only some of them have extremum points:

$\displaystyle {f''(x)}={-\sin x}$ , with no $\displaystyle a$ : the concavity changes at every $\displaystyle {x}={\pi k}$ , for every value of $\displaystyle a$ .
$\displaystyle {f'(x)}={\cos x+a}$ , and $\displaystyle {f'(x)}={0}$ where $\displaystyle {\cos x}={-a}$ . This equation has a solution only when $\displaystyle -1\leq a\leq1$ .
When $\displaystyle -1<a<1$ the derivative changes sign, and there are extremum points; when $\displaystyle a>1$ or $\displaystyle a<-1$ – there are none.

On the board: $\displaystyle {a}={0.5}$ , $\displaystyle {a}={1}$ and $\displaystyle {a}={2}$ .

\displaystyle f(x)=\sin x+0.5x

\displaystyle g(x)=\sin x+x

\displaystyle h(x)=\sin x+2x

$\displaystyle {a}={0.5}$ : the graph rises and falls, with extremum points.
$\displaystyle {a}={1}$ : $\displaystyle {f'(x)}={\cos x+1\geq0}$ , and the function rises. At the points $\displaystyle {x}={\pi+2\pi k}$ the derivative is 0 but does not change sign – these are horizontal inflection points.
$\displaystyle {a}={2}$ : $\displaystyle f'(x)\geq1$ , and the function rises with no extremum points; the inflection points stayed at the same $\displaystyle x$ .

The graph of

\displaystyle x^2\sin x

You can sketch the graph of $\displaystyle {f(x)}={x^2\sin x}$ even before differentiating, from the properties of its two factors:

$\displaystyle -1\leq\sin x\leq1$ , so $\displaystyle -x^2\leq f(x)\leq x^2$ : the graph is caught between the parabolas $\displaystyle {y}={x^2}$ and $\displaystyle {y}={-x^2}$ – its “envelope”.
$\displaystyle {f(x)}={x^2}$ where $\displaystyle {\sin x}={1}$ , and $\displaystyle {f(x)}={-x^2}$ where $\displaystyle {\sin x}={-1}$ : there the graph touches the envelope.
The zeros are the zeros of $\displaystyle \sin x$ , $\displaystyle {x}={\pi k}$ .
$\displaystyle {f(-x)}={(-x)^2\sin(-x)}={-f(x)}$ : the function is odd.

\displaystyle f(x)=x^{2}\sin x

\displaystyle y=x^{2}

\displaystyle y=-x^{2}

The function is not periodic: the factor $\displaystyle \sin x$ repeats itself, but the factor $\displaystyle x^2$ makes the waves bigger the further you go from 0. In $\displaystyle \sin x+x^2$ , by contrast, $\displaystyle \sin x$ adds waves of a fixed height to the parabola.

Exercise 1

Investigate the function $\displaystyle {f(x)}={\sin x+\cos x}$ on the interval $\displaystyle 0\leq x\leq2\pi$ .

Find $\displaystyle f'(x)$ .

$\displaystyle f(x)=\sin x+\cos x$

/ – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – check and a new line · * – times · ^ – power, ↓ – end of the power

/ – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – a new line · * – times · ^ – power, ↓ – end of the power

Solve the equation $\displaystyle {f'(x)}={0}$ in the interval.

Find $\displaystyle f''(x)$ .

$\displaystyle f(x)=\sin x+\cos x$

/ – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – a new line · * – times · ^ – power, ↓ – end of the power

Solve the equation $\displaystyle {f''(x)}={0}$ in the interval.

Mark on the graph the extremum points and the inflection points on the board.

\displaystyle f(x)=\sin x+\cos x

Write where $\displaystyle f$ rises and falls and the intervals of concavity of $\displaystyle f$ in the interval.
What is the period of $\displaystyle f$ ?

Exercise 2

Investigate the function $\displaystyle {f(x)}={\sin^2x}$ on the interval $\displaystyle 0\leq x\leq\pi$ – one period of it.

Find $\displaystyle f'(x)$ , and write it with $\displaystyle \sin2x$ .

$\displaystyle f(x)=\sin^{2}x$

/ – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – a new line · * – times · ^ – power, ↓ – end of the power

Solve the equation $\displaystyle {f'(x)}={0}$ in the interval.

Find $\displaystyle f''(x)$ .

$\displaystyle f(x)=\sin^{2}x$

/ – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – a new line · * – times · ^ – power, ↓ – end of the power

Solve the equation $\displaystyle {f''(x)}={0}$ in the interval.

Mark on the graph the extremum points and the inflection points on the board.

\displaystyle f(x)=\sin^{2}x

Exercise 3

Investigate the function $\displaystyle {f(x)}={\frac{\sin x}{2+\cos x}}$ on the interval $\displaystyle 0\leq x\leq2\pi$ .

Find $\displaystyle f'(x)$ .

$\displaystyle f(x)=\frac{\sin x}{2+\cos x}$

/ – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – a new line · * – times · ^ – power, ↓ – end of the power

Solve the equation $\displaystyle {f'(x)}={0}$ in the interval.

Work out $\displaystyle f\left(\frac{2\pi}{3}\right)$ .

$\displaystyle f(x)=\frac{\sin x}{2+\cos x}$

/ – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – a new line · * – times · ^ – power, ↓ – end of the power

Mark the extremum points on the graph.

\displaystyle f(x)=\frac{\sin x}{2+\cos x}

Why is the function defined for every $\displaystyle x$ ?
Show that $\displaystyle f$ is odd, and explain how this fits the extremum points you found.

Exercise 4

Investigate the function $\displaystyle {f(x)}={2\sin x-x}$ on the interval $\displaystyle 0\leq x\leq2\pi$ .

Find $\displaystyle f'(x)$ .

$\displaystyle f(x)=2\sin x-x$

/ – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – a new line · * – times · ^ – power, ↓ – end of the power

Solve the equation $\displaystyle {f'(x)}={0}$ in the interval.

Find $\displaystyle f''(x)$ .

$\displaystyle f(x)=2\sin x-x$

/ – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – a new line · * – times · ^ – power, ↓ – end of the power

Mark on the graph the extremum points and the inflection points on the board.

\displaystyle f(x)=2\sin x-x

Write the kind of each extremum point, and the greatest and the least value of $\displaystyle f$ on the interval.
In the family $\displaystyle {y}={\sin x+ax}$ of the example, $\displaystyle \frac{1}{2}f(x)$ belongs to which $\displaystyle a$ ? How does this explain why $\displaystyle f$ has extremum points?

\displaystyle a

/ – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – a new line · * – times · ^ – power, ↓ – end of the power

Formulas: calculus, grade 11 (5 units)

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Rules of differentiation and the tangent line

The derivative of a power
$\displaystyle (x^n)'$ $\displaystyle =$ $\displaystyle nx^{n-1}$
The derivative of a product
$\displaystyle [f(x)\cdot g(x)]'$ $\displaystyle =$ $\displaystyle f'(x)\cdot g(x)+f(x)\cdot g'(x)$
The derivative of a quotient
$\displaystyle \left[\frac{f(x)}{g(x)}\right]'$ $\displaystyle =$ $\displaystyle \frac{f'(x)\cdot g(x)-f(x)\cdot g'(x)}{[g(x)]^2}$
The derivative of a composite function – the outer derivative times the inner derivative
$\displaystyle [f(u(x))]'$ $\displaystyle =$ $\displaystyle f'(u)\cdot u'(x)$
The derivative of $\displaystyle \frac{1}{x}$
$\displaystyle \left(\frac{1}{x}\right)'$ $\displaystyle =$ $\displaystyle -\frac{1}{x^2}$
The tangent line at the point $\displaystyle (x_1,y_1)$ on the graph
$\displaystyle y-y_1=f'(x_1)(x-x_1)$

The second derivative

The second derivative – the derivative of the derivative
$\displaystyle f''(x)$ $\displaystyle =$ $\displaystyle \left(f'(x)\right)'$
The function is concave up ( $\displaystyle \cup$ )
$\displaystyle f''(x)>0$
The function is concave down ( $\displaystyle \cap$ )
$\displaystyle f''(x)<0$
A possible inflection point – check that the sign of $\displaystyle f''$ changes there
$\displaystyle f''(x_0)=0$
A minimum point
$\displaystyle f'(x_0)=0,\ f''(x_0)>0$
A maximum point
$\displaystyle f'(x_0)=0,\ f''(x_0)<0$
Acceleration – the derivative of the speed
$\displaystyle a(t)=v'(t)=s''(t)$

Asymptotes

A vertical asymptote – $\displaystyle f(x)$ grows or falls without end as $\displaystyle x$ approaches $\displaystyle a$
$\displaystyle x=a$
A horizontal asymptote – $\displaystyle f(x)$ approaches $\displaystyle b$ as $\displaystyle x\to\infty$ or as $\displaystyle x\to-\infty$
$\displaystyle y=b$

Roots

The root of a square
$\displaystyle \sqrt{x^2}$ $\displaystyle =$ $\displaystyle |x|$
The derivative of $\displaystyle \sqrt{x}$
$\displaystyle (\sqrt{x})'$ $\displaystyle =$ $\displaystyle \frac{1}{2\sqrt{x}}$ $\displaystyle x>0$
The derivative of the root of a function
$\displaystyle (\sqrt{f(x)})'$ $\displaystyle =$ $\displaystyle \frac{f'(x)}{2\sqrt{f(x)}}$ $\displaystyle f(x)>0$

Radians and periods

Half a turn in radians
$\displaystyle \pi$ $\displaystyle =$ $\displaystyle 180^{\circ}$
The length of an arc of $\displaystyle \alpha$ radians in a circle of radius $\displaystyle R$
$\displaystyle l$ $\displaystyle =$ $\displaystyle \alpha R$
The period of $\displaystyle \sin x$ and of $\displaystyle \cos x$ is $\displaystyle 2\pi$
$\displaystyle \sin(x+2\pi)$ $\displaystyle =$ $\displaystyle \sin x$
The period of $\displaystyle \tan x$ is $\displaystyle \pi$
$\displaystyle \tan(x+\pi)$ $\displaystyle =$ $\displaystyle \tan x$
The limit behind the derivative of $\displaystyle \sin x$ ( $\displaystyle x$ in radians)
$\displaystyle \lim_{x\to0}\frac{\sin x}{x}=1$

Trigonometric identities

The Pythagorean identity
$\displaystyle \sin^2\alpha+\cos^2\alpha$ $\displaystyle =$ $\displaystyle 1$
Tangent
$\displaystyle \tan\alpha$ $\displaystyle =$ $\displaystyle \frac{\sin\alpha}{\cos\alpha}$
Sine – an odd function
$\displaystyle \sin(-\alpha)$ $\displaystyle =$ $\displaystyle -\sin\alpha$
Cosine – an even function
$\displaystyle \cos(-\alpha)$ $\displaystyle =$ $\displaystyle \cos\alpha$
The sine of the supplement to $\displaystyle \pi$
$\displaystyle \sin(\pi-\alpha)$ $\displaystyle =$ $\displaystyle \sin\alpha$
The cosine of the complement to $\displaystyle \frac{\pi}{2}$
$\displaystyle \cos\left(\frac{\pi}{2}-\alpha\right)$ $\displaystyle =$ $\displaystyle \sin\alpha$
The sine of a double angle
$\displaystyle \sin2\alpha$ $\displaystyle =$ $\displaystyle 2\sin\alpha\cdot\cos\alpha$
The cosine of a double angle
$\displaystyle \cos2\alpha=\cos^2\alpha-\sin^2\alpha=2\cos^2\alpha-1=1-2\sin^2\alpha$
The difference of two sines
$\displaystyle \sin\alpha-\sin\beta$ $\displaystyle =$ $\displaystyle 2\cos\frac{\alpha+\beta}{2}\cdot\sin\frac{\alpha-\beta}{2}$
The sine of a sum and of a difference
$\displaystyle \sin(\alpha\pm\beta)$ $\displaystyle =$ $\displaystyle \sin\alpha\cdot\cos\beta\pm\cos\alpha\cdot\sin\beta$

Derivatives of the trigonometric functions

The derivative of sine ( $\displaystyle x$ in radians)
$\displaystyle (\sin x)'$ $\displaystyle =$ $\displaystyle \cos x$
The derivative of cosine
$\displaystyle (\cos x)'$ $\displaystyle =$ $\displaystyle -\sin x$
The derivative of tangent
$\displaystyle (\tan x)'$ $\displaystyle =$ $\displaystyle \frac{1}{\cos^2x}$