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Trigonometry When Two Sides of a Right Angled Triangle Are Given in Terms of x and an Angle is Given

Suppose we have a right angled triangle, a given angle and sides given in terms of  
\[x\]
. We can find  
\[x\]
  using simple trigonometry. Consider the triangle below.

\[cos 30 = \frac{adjacent}{hypotenuse}\]

\[\frac{\sqrt{3}}{2} = \frac{3x+7}{5x+1}\]

\[\sqrt{3} (5x+1) = 2(3x+7)\]

\[5x \sqrt{3}+ \sqrt{3} = 6x+14\]

\[5x \sqrt{3}-6x = 14- \sqrt{3}\]

\[x(5 \sqrt{3}-6) = 14- \sqrt{3}\]

\[x = \frac{14- \sqrt{3}}{5 \sqrt{3}-6}= \frac{14- \sqrt{3}}{5 \sqrt{3}-6} \times \frac{5 \sqrt{3}+6}{5 \sqrt{3}+6}= \frac{69 +64 \sqrt{3}}{39}\]

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