In right triangle $FGH$ with $\angle H = 90^\circ$, we have $FG = 17$ and $HG = 15$. Find $\sin G$.

 Aug 14, 2017

Here is a picture of your diagram:



In this diagram, I have labeled all of the given info on it. Our goal is to figure out what \(\sin\angle G\) is. In right-triangle trigonometry, the sine function compares, from the angle of reference, the opposite angle to the hypotenuse. Let's do that:


\(\frac{\sin m\angle G}{1}=\frac{FH}{17}\)Uh oh! We don't know what the length of the segment of FH. We can use Pythagorean theorem. If you are like me, I have memorized some Pythagorean triples (integer solutions to the Pythagorean theorem like 3,4, and 5), but I will solve for them anyway. 


Let's apply the Pythagorean Theorem to figure out the remaining side:


\(FH^2+15^2=17^2\)Simplify both sides of the equation before continuing.
\(FH^2+225=289\)Subtract 225 from both sides.
\(FH^2=64\)Take the square root of both sides to undo the exponent of 2.
\(|FH|=8\)Of course, the absolute value splits the possible answers to its positive and negative form.
\(FH=\pm8\)Of course, in the context of geometry, a negative side length is nonsensical, so we will should reject the negative answer.


Now that we know what FH is, we can plug it back into what we were originally solving for, \(\sin m\angle G\):


\(\sin m\angle G=\frac{8}{17}\)Technically, I should stop here, as the question asked for the value of \(\sin m\angle G\), but I will solve for G just in case that is actually what you are solving for. To solve for the angle measure, you must use the inverse function.
\(\sin^{-1}\left(\frac{8}{17}\right)=m\angle G\)Use a scientific calculator with trigonometric functions to approximate the measure of the angle. Since an angle is measured in degrees, the calculator should be in degree mode when the expression is inputted.
\(m\angle G=\sin^{-1}\left(\frac{8}{17}\right)\approx28.07^{\circ}\) 


We're done now!

 Aug 15, 2017

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