Intercepts

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.

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 $\sin(x+2\pi)$ .

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 $x$ ’.

Example

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.

Example

$9=3^{2}=3 \times 3$ 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

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

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.

Exponential and Logarithm Functions

This tutorial covers material encountered in chapter 5 of the VCE Mathematical Methods Textbook, namely: Exponential functions Index laws Log functions Log laws and change… Read More »Exponential and Logarithm Functions

Intercepts

5.3 Solution of Trigonometric Equations

Basic Trigonometric Equations (Sine and Cosine)

1.6 Trigonometric Functions

Graphs of Sine and Cosine Functions

1.4 Logarithmic Functions

The Logarithmic Function

1.3 Exponential Functions [Free]

The Exponential Function

1.2 Power Functions

Introduction to Power Functions

1.1 Introduction to Graph Sketching

Introduction to Graph Sketching

Exponential and Logarithm Functions