Solving Unknown Sides

Since the trigonometric value of any two congruent angles are always the same (see above “What is a Trig Ratio”), that ratio can be used to help solve for unknown sides of the triangle.

Let’s look at one of our special triangles first.

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There are 3 questions to answer to solving for an unknown side\\

 

  1. Which side you’re solving for?
  2. Which angle are you going to use?
  3. Which trig function (proportion) to use?\\

 

\textbf{Use \sin(60^\circ)}

    \[ \begin{array}{c c c c l} &\sin(60^{\circ})    &        = &     \frac{\sqrt{3}}{2}  & \hspace{1cm}\text{from the special triangle} \vspace{1cm} \\ &\sin(60^{\circ})    &        = &     \frac{x}{8}      & \hspace{1cm}  \text{from the triangle in front of us}\vspace{1cm} \\ \Rightarrow &\frac{\sqrt{3}}{2}    &        = &        \frac{x}{8}         & \hspace{1cm} \text{set equal to each other -- solve proportion} \\ \end{array} \]

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Or \cos(30^\circ)

    \[ \begin{array}{c c c c l} &\cos(30^{\circ})    &        = &     \frac{\sqrt{3}}{2}  & \hspace{1cm}\text{from the special triangle} \vspace{1cm} \\ &\cos(30^{\circ})    &        = &     \frac{x}{8}      & \hspace{1cm} \text{from the triangle in front of us}\vspace{1cm} \\ \Rightarrow &\frac{\sqrt{3}}{2}    &        = &        \frac{x}{8}         & \hspace{1cm} \text{set equal to each other -- solve proportion} \\ \end{array} \]