# Trigonometry

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For many practical applications, trigonometry is concerned with triangles and the relationships between their side lengths and internal angles. For example, in the right-angled triangle shown in the figure below, the ratio of the length of the side opposite the indicated angle and the side adjacent to it is equal to $\tan\left(\ctext{angle}\right)$ ($\tan$ belongs to the same family of trigonometric functions as $\sin$ and $\cos$):$$\frac{\ctext{opposite side length}}{\ctext{adjacent side length}} = \tan(\ctext{angle}).$$

This formula is convenient because most scientific calculators have the standard trigonometric functions built in, and so, for example, an adjusted version of the above formula,

$$\ctext{opposite side length} = \ctext{adjacent side length} \times \tan\left(\ctext{angle}\right),$$can be used to calculate the length of the opposite side of a triangle when the length of the adjacent side and the angle between this side and the triangle's hypotenuse is given. ### Example Problem

If the pitch angle of the roof shown in the figure below is to be $30$ degrees, and the rafter's run is to be $4000$ mm, calculate the rafter's rise (in millimetres)? ### Solution

Using the formula above for calculating the opposite side length, we have
$$\ctext{opposite side length} = 4000\;\ctext{mm}\times \tan\left(30^\circ\right) \approx 4000\;\ctext{mm}\times0.57735 = 2309.4\;\ctext{mm}.$$