Trigonometric Functions

We can define the trigonometric functions in terms an angle $t$ and the lengths of the sides of the triangle. The adjacent side is the side closest to the angle. (Adjacent means “next to.”) The opposite side is the side across from the angle. The hypotenuse is the side of the triangle opposite the right angle, and it is the longest.

Right triangle

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

When solving for a missing side of a right triangle, but the only given information is an acute angle measurement and a side length, use the trigonometric functions listed below:

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

The trigonometric functions are equal to ratios that relate certain side lengths of a  right triangle.  When solving for a missing side, the first step is to identify what sides and what angle are given, and then select the appropriate function to use to solve the problem.

Evaluating a Trigonometric Function of a Right Triangle

Sometimes you know the length of one side of a triangle and an angle, and need to find other measurements.  Use one of the trigonometric functions ($\sin{}$, $\cos{}$, $\tan{}$), identify the sides and angle given, set up the equation and use the calculator and algebra to find the missing side length.

Example 1:   Given a right triangle with acute angle of $34^{\circ}$ and a hypotenuse length of $25$ feet, find the length of the side opposite the acute angle (round to the nearest tenth):

Right triangle

Given a right triangle with acute angle of $34$ degrees and a hypotenuse length of $25$ feet, find the opposite side length.

Looking at the figure, solve for the side opposite the acute angle of $34$ degrees.  The ratio of the sides would be the opposite side and the hypotenuse.  The ratio that relates those two sides is the sine function.

$\displaystyle{ \begin{aligned} \sin{t} &=\frac {opposite}{hypotenuse} \\ \sin{\left(34^{\circ}\right)} &=\frac{x}{25} \\ 25\cdot \sin{ \left(34^{\circ}\right)} &=x\\ x &= 25\cdot \sin{ \left(34^{\circ}\right)}\\ x &= 25 \cdot \left(0.559\dots\right)\\ x &=14.0 \end{aligned} }$

The side opposite the acute angle is $14.0$ feet.

Example 2: Given a right triangle with an acute angle of $83^{\circ}$ and a hypotenuse length of $300$ feet, find the hypotenuse length (round to the nearest tenth):

Right Triangle

Given a right triangle with an acute angle of $83$ degrees and a hypotenuse length of $300$ feet, find the hypotenuse length.

Looking at the figure, solve for the hypotenuse to the acute angle of $83$ degrees. The ratio of the sides would be the adjacent side and the hypotenuse.  The ratio that relates these two sides is the cosine function.

$\displaystyle{ \begin{aligned} \cos{t} &= \frac {adjacent}{hypotenuse} \\ \cos{ \left( 83 ^{\circ}\right)} &= \frac {300}{x} \\ x \cdot \cos{\left(83^{\circ}\right)} &=300 \\ x &=\frac{300}{\cos{\left(83^{\circ}\right)}} \\ x &= \frac{300}{\left(0.1218\dots\right)} \\ x &=2461.7~\mathrm{feet} \end{aligned} }$