Introduction to Trigonometry

Pythagorean Theorem

The Pythagorean Theorem is an equation used on Right Triangles (triangles with one 90° angle and 2 acute angles) that determines the relation between the different sides of the triangle and identify missing side lengths.

This equation is expressed as:

\(\textcolor{red}{a}^2 + \textcolor{green}{b}^2 = \textcolor{blue}{c}^2\)

In this equation, \(\textcolor{blue}{c}\) represents the hypoteneuse, the longest side of the triangle, while \(\textcolor{red}{a}\) and \(\textcolor{green}{b}\) represent the shorter sides, referred to as opposite and adjacent respectively. Opposite refers to the side opposite of \(\angle A\) and adjacent refers to the side next to \(\angle A\):

Take note that the positions of the opposite and adjacent sides change in relation to the position of \(\angle A\) from triangle to triangle.
This equation can be rearranged to find one of the smaller side lengths:

\(\textcolor{blue}{c}^2 - \textcolor{green}{b}^2 = \textcolor{red}{a}^2\) or \(\textcolor{blue}{c}^2 - \textcolor{red}{a}^2 = \textcolor{green}{b}^2\)
Example

Identify the length of the hypoteneuse of the given triangle:

We can identify \(\textcolor{red}{a = 17}\) and \(\textcolor{green}{b = 39}\). Using these values, we can determine the hypoteneuse:

\(\textcolor{red}{a}^2 + \textcolor{green}{b}^2 = \textcolor{blue}{c}^2\)
\(\textcolor{red}{17}^2 + \textcolor{green}{39}^2 = \textcolor{blue}{c}^2\)
\(289 + 1521 = \textcolor{blue}{c}^2\)
\(\textcolor{blue}{c} = \sqrt{1810}\)
\(c = 42.54\;[m]\)

Therefore, we can determine that \(c = 42.54\;[m]\).

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Primary Trigonometric Ratios

Trigonometric Ratios are special measurements of a right triangle. They are used to help determine the missing sides and angles of a right triangle.
They are divided into 3 unique categories which take into account the different sides of a right triangle:

• \(\text{sine} (\theta) = \cfrac{\textcolor{red}{a}}{\textcolor{blue}{c}} = \cfrac{\textcolor{red}{\text{opposite}}}{\textcolor{blue}{\text{hypoteneuse}}}\)


• \(\text{cosine} (\theta) = \cfrac{\textcolor{green}{b}}{\textcolor{blue}{c}} = \cfrac{\textcolor{green}{\text{adjacent}}}{\textcolor{blue}{\text{hypoteneuse}}}\)


• \(\text{tangent} (\theta) = \cfrac{\textcolor{red}{a}}{\textcolor{green}{b}} = \cfrac{\textcolor{red}{\text{opposite}}}{\textcolor{green}{\text{adjacent}}}\)

An effective way of remembering these trigonometric ratios is by using the mnemonic device SOHCAHTOA. Each portion of the phrase outlines both the ratio and the sides used:

• SOH = Sine, Opposite, Hypoteneuse
• CAH = Cosine, Adjacent, Hypoteneuse
• TOA = Tangent, Opposite, Adjacent

In addition to these ratios, we can use solve for angles using the given values and vice versa using the trigonometric functions (sin, cos, and tan) on our calculators. Make sure to set the calculator mode to Degrees! (or DRG).

Example

Determine \(\cos \angle L\) if \(\angle L = 75°\).

We can simply plug the angle into the function to determine its value:

\(\cos \angle L = \cos(75°)\)
\(\cos \angle L = 0.26\)

Therefore, we can determine that \(\cos\angle L = 0.26\).

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Example

Solve for \(\angle P\) if \(\sin\angle P = 0.5\)

To solve for \(\angle P\), we would use the inverse sin (\(\sin⁻¹)\) to find the angle:

\(\angle P = \sin^{-1}(0.5)\)
\(\angle P = 30°\)

Therefore, we can determine that \(\angle P = 30°\).

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Example

Identify the missing side length of the given triangle using the correct trigonometric ratio:

We can identify \(\textcolor{blue}{c = 8}\) and \(\textcolor{seagreen}{\angle B = 21°}\). Using these values, we can determine the value of the adjacent side.
Since we are taking into account the hypoteneuse and adjacent sides, we can use \(cos\) as our trigonometric function. Then, we can solve for \(x\):

\(\cos\theta = \cfrac{\textcolor{green}{\text{adjacent}}}{\textcolor{blue}{\text{hypoteneuse}}}\)

\(\cos(\textcolor{seagreen}{21°}) = \cfrac{\textcolor{green}{x}}{\textcolor{blue}{8}}\)

\(\textcolor{green}{x} = (\textcolor{blue}{8})(\cos\textcolor{seagreen}{21°})\)
\(\textcolor{green}{x} = (\textcolor{blue}{8})(0.9336)\)
\(x = 7.47\;[m]\)

Therefore, when we round, we can determine that \(x = 7.47\;[m]\).

We can also use the inverse of a trigonometric function in order to determine the angle of a given value.
As an example, if \(\sin\)\(A = 0.8829\), we can determine the value of \(\textcolor{crimson}{\angle A}\):

\(\textcolor{crimson}{A} = \sin^{-1}(0.8829)\)
\(A = 61.994^\circ\)

Therefore, we can determine that \(A = 61.994^\circ\).

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