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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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5.2 Polynomial Functions

Solving Equations

  • Every now and then, you will be challenged with questions to solve .
  • Suppose we have y=f(x) and y=g(x) as two separate functions. If f(x)=g(x), it means that the graph of both functions intersect. Note that this tells nothing about the number of possible values for x.

Simultaneous Equations: Polynomials

  • We will first look at how this type of questions looks like, with polynomials first.
  • The steps are rather simple:
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4.1 Composite Functions

Composite Functions

  • A composite function is different from sum or product of functions; it is like a function acting on the result of another. If you picture functions as machines (with x as the input, and f(x) as the output), it is something like the diagram shown.

Note: Here, f(x)=3 x+2,\ g(x)=x^{2}, and x=3.

  • In comparison, (f+g)(x) is more like tow machines working separately with the same input x, and their two, separate outputs are placed together and packaged.
  • Just like how the processes of manufacturing has an order, so does composite functions. It is important which functions comes first, and which comes next. Using the example above where f is the first function), we write the composition of g with f (say we call it as h) as h=g \circ f (read as ‘composition of f followed by g‘). The rule of h(x) is given by
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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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3.3 Logarithm Laws [Free]

The Logarithm Function’s Algebraic Properties

  • There are four properties in logarithms to consider:

i) \log_{a}{mn}=\log_{a}{m}=\log_{a}{n}

ii) log_{a}{(\frac{m}{n})}=\log_{a}{m}-\log_{a}{n}

iii) log_{a}{(m^{p})}=p\cdot{\log_{a}{m}}

In particular when p=-1, we have log_{a}{(\frac{1}{m})}=-\log_{a}{m}

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3.2 Index Laws

The Exponential Function’s Algebraic Properties

  • There are six properties in exponentials to consider:

i) a^{x} \times a^{y} = a^{x+y}

ii) a^{x} {\div} a^{y} = a^{x-y}

iii) (a^{x})^{y}=a^{xy}

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3.1 Introduction to Functional Relations

Function Notation and Identities

  • Many of the properties which have been investigated for the functions introduced in the previous chapters may be expressed using function notation.
  • Below includes a few examples:

\ln x+\ln y=\ln (x y) becomes f(x)+f(y)=f(x y) where f(k)=\ln k

\ln x-\ln y=\ln \left(\frac{x}{y}\right) becomes f(x)-f(y)=f\left(\frac{x}{y}\right) where f(k)=\ln k

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2.2 Remainder Theorem, Factor Theorem and The Rational-Root Theorem

Remainder Theorem and Factor Theorem

Now we are going to set D(x)=x-a. Therefore, as \operatorname{deg}(R)<\operatorname{deg}(D), R(x) must be a constant, which we denote as R.

Rewriting the equation gives

P(x)=(x-a) Q(x)+R

  • Simply set x=a gives P(x)=(a-a) Q(x)+R=0+R=R. Thus, R=P(a).
  • Now we generalize D(x) into any linear polynomial, i.e. D(x)=a x+b, where a,\ b are integers. Therefore, now P(x)=(a x+b) Q(x)+R and substitute x=-\frac{b}{a} (to make D(x)=0) gives
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