Revision:Sine and Cosine Formulae

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Revision:Sine and Cosine Formulae

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TSR Wiki > Study Help > Subjects and Revision > Revision Notes > Mathematics > Sine and Cosine Formulae

1 Sine and Cosine Formulae
2 The Sine Rule
3 The Cosine Rule
4 The area of a triangle
5 Comments

Sine and Cosine Formulae

$\displaystyle \sin x = \sin (180 - x)$

e.g. $\displaystyle \sin 130 = \sin (180 - 130) = \sin 50$

$\displaystyle \cos x = -\cos (180 - x)$

The Sine Rule

This works in any triangle:

$\displaystyle \frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C}$

alternatively:

$\displaystyle \frac{\sin A}{a} = \frac{\sin B}{b} = \frac{\sin C}{c}$

NOTE: the triangle is labelled as follows:

(diagram of the labelled triangle is missing)

The Cosine Rule

$\displaystyle c^2 = a^2 + b^2 - 2ab\cos C$

can also be written as:

$\displaystyle a^2 = b^2 + c^2 - 2bc\cos A$

This also works in any triangle.

The area of a triangle

This works out the area of any triangle.

$\displaystyle \mathsf{Area} = \frac{1}{2} ab\sin C$

(using the above notation)

This formula is useful if you don't know the height of a triangle (since you need to know the height for the $\frac{1}{2}\mathsf{base}\times\mathsf{height}$ formula).