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Trigonometric Functions

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6.1 Antiderivatives of Functions

Antiderivatives of Specific Functions

  • We will be looking at antiderivatives of 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
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3.4 Differentiation Rules for Circular 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:
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5.3 Solution of Trigonometric Equations

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.
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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

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5.2 Applications of Circular Functions

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.
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3.8 Transformations of Trigonometric Functions

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.
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1.6 Trigonometric Functions

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).

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1.5 Trigonometric Functions Background Knowledge

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 as1^{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:
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