Solving All Right Triangles

The calculator (or trig table) can determine a trig value for any angle. Therefore we can substitute the trig function with a number and determine how long an unknown side is.

 

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This triangle has a known side adjacent to 21^\circ, and since we want to know the value of x, opposite of 20^\circ we will use Tangent.

    \begin{align*} \tan(21^\circ)&=0.383864035...     &\text{from calculator}\\ \tan(21^\circ)&=\frac{x}{8}       &\text{from triangle}\\ 8\tan(21^\circ)&=x                    &\text{solved for } x\\ 8 \cdot 0.383864035... &= x        &\text{subtitute out}\tan(21^\circ) \end{align*}

    \[x \approx 3.0709\]

 

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    \begin{align*} \cos(36^\circ)&=0.809016994...     &\text{from calculator}\\ \cos(36^\circ)&=\frac{b}{10}       &\text{from triangle}\\ 10\cos(36^\circ)&=b                    &\text{solved for } b\\ 10 \cdot 0.809016994... &= b        &\text{subtitute out}\cos(36^\circ) \end{align*}

    \[b \approx 8.0902\]