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So first off I started with the pythagorean theorem to find the missing leg of the triangle. \begin{align*} 5^2 + b^2 ={}& 8^2 \\ 25 + b^2 ={}& 64 \\ 64 - 25 ={}& 39 \\ \text{missing leg}={}&\sqrt{39} = 6.244 \\ \end{align*}

Taking the 6.244 as the opposite leg of the triangle to the angle at the top of the triangle, using sin = opposite / hypotenuse \begin{align*} \sin (\text{angle}) ={}& 6.244 / 8 \\ \text{angle} ={}& \arcsin (0.7805) = 51.3064\text{˚}. \end{align*} If I'm on the right track, where do I go from here to find the unknown angle?

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  • $\begingroup$ The acute angles in a right triangle are complementary. Look at the largest of the three right triangles in the diagram. $\endgroup$ – N. F. Taussig Jul 6 '15 at 10:10
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Hint: $\sin(\theta)=\frac{5}{8}$ because

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HINT : The two small triangles are similar. (You don't need to find the lengths of other sides.)

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You are on track. Answer is complement of what you got, ~ 90-51.3 deg.

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