Question - Solving for a Side Length in a Right Triangle Using Law of Sines
Solution:
To solve for $$ w $$, we'll use the Law of Sines, which states that in any triangle, the ratio of the length of a side to the sine of its opposite angle is the same for all three sides. Given triangle has angles 30°, 60°, and because the sum of angles in a triangle is always 180°, the remaining angle is 90°, making it a right triangle. The side opposite the 30° angle is half the hypotenuse in a right triangle, while the side opposite the 60° angle is equal to $$ \sqrt{3}/2 $$ times the hypotenuse.Since we have:$$\frac{w}{\sin(90°)} = \frac{10}{\sin(60°)}$$We know that $$ \sin(90°) = 1 $$ and $$ \sin(60°) = \sqrt{3}/2 $$, so we have:$$w = 10 \times \frac{1}{\sqrt{3}/2} = 10 \times \frac{2}{\sqrt{3}} = \frac{20}{\sqrt{3}}$$To express in simplest radical form, we rationalize the denominator:$$w = \frac{20}{\sqrt{3}} \times \frac{\sqrt{3}}{\sqrt{3}} = \frac{20\sqrt{3}}{3}$$Thus, $$ w = \frac{20\sqrt{3}}{3} $$ meters in simplest radical form.
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