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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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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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1.4 Logarithmic Functions

The Logarithmic Function

  • The logarithmic function with base is defined as follows:

a^{x} = b is equivalent to log_{a}{b}=x, where a \in R^{+}\setminus \left \{ 1 \right \} .

Note:a \in R^{+}\setminus \left \{ 1 \right \}‘ means that could be any positive number, excluding 1.

  • The expression log_{a}{y}=x is read as: ‘the logarithm of y to the base a is equal to x’.


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1.3 Exponential Functions [Free]

The Exponential Function

  • The exponential function is defined as follows:

y=a^x, where a \in R^+\backslash {1}.

Note:a \in R^+\backslash {1}’ means that could be any positive number, excluding 1.


9=3^{2}=3 \times 3 is equivalent to log_{3}9=2.

8=16^{\frac{3}{4}}=\sqrt[4]{16^{3}}=\sqrt{\sqrt{16 \times 16 \times 16}}.

Graph & Properties

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1.2 Power Functions

Introduction to Power Functions

  • Power functions are functions with the form f(x)=x^{n}, where n is a rational number.
  • Some examples of power functions are:

f(x)=x^{2},\ f(x)=x^{4},\ f(x)=x^{\frac{1}{4}},\ f(x)=x^{-5},\ f(x)=x^{\frac{1}{3}}

  • The expression x^n is read as: ‘x to the power of n’.

Drawing Power Functions (for Positive Integers) and its Graph Properties

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1.1 Introduction to Graph Sketching

Introduction to Graph Sketching

  • In this section, 3 types of functions will be introduced accordingly:

i) Power functions, y=x^n, where is a number, n\neq 0.

This will be further extended into polynomial functions, which has the general form y=a_{n} x^{n}+a_{n-1} x^{n-1}+\ldots+a_{1} x+a_{0}, where a_i are real numbers, a_n\neq 0.

ii) Exponential functions, y=a^x and also logarithm functions, y=log_ax.

iii) Trigonometric functions or circular functions, which includes y=\sin x,\ y=\cos x,\ y=\tan x and more.

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