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Algebraic Properties

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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