Trigonometric Functions

6.1 Antiderivatives of Functions

Antiderivatives of Specific Functions

We will be looking at antiderivatives of the polynomial, exponential, and trigonometric functions.
The formulas are as below:

i) Polynomial functions

$\int x^{n} d x=\frac{1}{n+1} x^{n+1}+c \quad,\ n \neq-1$

$\int x^{-1} d x=\int \frac{1}{x} d x=\ln |x|+c$

Note: Notice that in there is an absolute value over to make the result fit the domain for all logarithmic functions is $(0,\ +\infty)$ , and $x=0$ is not possible in the first place as it would make $\frac{1}{x}$ undefined.

ii) Exponential functions

Differentiation Rules for $\sin(x), \cos(x), \tan(x)$

The other rules regarding differentiation such as addition, subtraction, multiplication, division of functions, chain rule etc. applies too.
All of these rules can be proven via first principles, but it is not required.
The basic rules in this section are:

i) If $f(x)=\sin(x)$ , then $f^{\prime}(x)=\cos x$

ii) If $f(x)=\cos x$ , then $f^{\prime}(x)=-\sin x$

iii) If $f(x)=\tan x$ , than $f^{\prime}(x)=\sec^2x=\sec x \times \sec x$

It is possible to expand these further using chain rule:

Basic Trigonometric Equations (Sine and Cosine)

The most basic equations in polynomials would be linear equations, such as $ax+b=c$ (which gives you $x=\frac{c-b}{a}$ ). The trigonometric equivalent ones would be $\sin t =a$ or $\cos t =b$ , and you are supposed to solve for $t$ .

Be careful, as solving trigonometric equations are not as simple. Refer to the examples below.

For each such equations, unless a restriction on $x$ (or more commonly used in trigonometric, $\theta$ ), there will be infinite or no solutions for $x$ . This is because of the periodic and symmetric properties that they have.

3.4 Trigonometric Function Laws

Note: These are already covered in the notes 1.5 Trigonometric Functions Background Knowledge, when trigonometric functions are first introduced. Here, it will be simply a reiteration of the formulas.

Trigonometric Function Laws

Sine, cosine, tangent can be understood as follows:

$\sin (\theta)=\frac{\text { opposite }}{\text { hypotenuse }}$

$\cos (\theta)=\frac{\text { adjacent }}{\text { hypotenuse }}$

$\tan (\theta)=\frac{\text { opposite }}{\text { adjacent }}$

and in particular we have

Applications of Circular Functions

A sinusoidal function has a rule of the form $y=a\sin (nt+k)+c$ , or, equivalently, of the form $y=a\cos (nt+k)+c$ . Such functions can be used to model periodic motion.

There are plenty of examples of periodic motions. For instance, the periodic switch between daytime and night time, circular motions of pendulum, yo-yo, and more.

Transformations with Sine and Cosine Graphs

We can see that the graph of $y=\cos x$ can simply be obtained by $y=\sin x$ shifted $\frac{\pi}{2}$ to the left. This is a translation, and the identity is not unfamiliar to us: $\sin x=\cos \left(x-\frac{\pi}{2}\right)$ .

Now we shall look at graphs of simple transformations of these graphs. We shall start with $y=a \sin (n t)$ and $y=b \cos (n t)$ . Naturally, $n$ would be a dilation from the y-axis, and $a$ (and $b$ ) is a dilation from the x-axis. The mechanism is similar, and is explained in A1 – Functions and Graphs/Relationship of Transformations . Here we will only provide a few simple examples.

Graphs of Sine and Cosine Functions

The graph of where $-\pi<x<3\pi$ are plotted below. Do note that it extends beyond the drawn range.

There are a few observations we can make, and we can tie it back to what we have already learnt:

i) The graph repeats itself after an interval of $2\pi$ units. We say that the graph has a period of $2\pi$ , and hence is called a periodic function. Previously, we have learnt that $\sin(x+2\pi)$ .

Degrees and Radians

All these while we are familiar with degrees. We know that a straight line is $180^{\circ}$ , and for a full circle it is $360^{\circ}$ .
There is another unit to represent degrees, it is known as radians.
The short form of radians can be written in several ways. Take 1 radian for example. The most formal way would be written as $1^{c}$ . It can be written as $1\ \mathrm{rad}$ as before, or the most common way, $1$ (without any units).
The term ‘radian’ actually originated when we wanted to find a neat expression for the angle when the corresponding arc length of a circle with 1 unit radius, for this angle, is also 1 unit. Using degrees, the number will not look nice, hence a new unit called radians are invented.
Radians are useful as it can be treated as numbers (without units).
Radians are defined as positive for angles moving anticlockwise. For the diagram at the right, that is a positive angle (1 rad). It is similar to how we define $90^{\circ}$ and $-90^{\circ}$ .
The conversion from radian to degrees is:

Derivatives

This tutorial covers material encountered in chapter 9 of the VCE Mathematical Methods Textbook, namely: The derivative of functions seen previously in tutorial worksheets 1-5… Read More »Derivatives