Definitions of Trigonometric Functions

Given a right triangle with an acute angle of $t$, the first three trigonometric functions are:

Right triangle

The sides of a right triangle in relation to angle $t$.

• Sine             $\displaystyle{ \sin{t} = \frac {opposite}{hypotenuse} }$
• Cosine        $\displaystyle{ \cos{t} = \frac {adjacent}{hypotenuse} }$
• Tangent      $\displaystyle{ \tan{t} = \frac {opposite}{adjacent} }$

A common mnemonic for remembering these relationships is SohCahToa, formed from the first letters of “Sine is opposite over hypotenuse (Soh), Cosine is adjacent over hypotenuse (Cah), Tangent is opposite over adjacent (Toa).”

Evaluating a Trigonometric Function of a Right Triangle

Example 1:   Given a right triangle with an acute angle of $62^{\circ}$ and an adjacent side of $45$ feet, solve for the opposite side length. (round to the nearest tenth)

Right triangle

Given a right triangle with an acute angle of $62$ degrees and an adjacent side of $45$ feet, solve for the opposite side length.

First, determine which trigonometric function to use when given an adjacent side, and you need to solve for the opposite side.  Always determine which side is given and which side is unknown from the acute angle ($62$ degrees).  Remembering the mnemonic, "SohCahToa", the sides given are opposite and adjacent or "o" and "a", which would use "T", meaning the tangent trigonometric function.

$\displaystyle{ \begin{aligned} \tan{t} &= \frac {opposite}{adjacent} \\ \tan{\left(62^{\circ}\right)} &=\frac{x}{45} \\ 45\cdot \tan{\left(62^{\circ}\right)} &=x \\ x &= 45\cdot \tan{\left(62^{\circ}\right)}\\ x &= 45\cdot \left( 1.8807\dots \right) \\ x &=84.6 \end{aligned} }$

Example 2:  A ladder with a length of $30~\mathrm{feet}$ is leaning against a building.  The angle the ladder makes with the ground is $32^{\circ}$.  How high up the building does the ladder reach? (round to the nearest tenth of a foot)

Right triangle

After sketching a picture of the problem, we have the triangle shown.

Determine which trigonometric function to use when given the hypotenuse, and you need to solve for the opposite side.  Remembering the mnemonic, "SohCahToa", the sides given are the hypotenuse and opposite or "h" and "o", which would use "S" or the sine trigonometric function.

$\displaystyle{ \begin{aligned} \sin{t} &= \frac {opposite}{hypotenuse} \\ \sin{ \left( 32^{\circ} \right) } & =\frac{x}{30} \\ 30\cdot \sin{ \left(32^{\circ}\right)} &=x \\ x &= 30\cdot \sin{ \left(32^{\circ}\right)}\\ x &= 30\cdot \left( 0.5299\dots \right) \\ x &= 15.9 ~\mathrm{feet} \end{aligned} }$