50 points what are the values of the variables in the triangle below? i need an explanation and all work shown.

because this is a right triangle, the rules of right-angle trigonometry apply here, namely

for an angle θ in a right-angle triangle.

clearly, it is the case that

sin(30°) = y / (34 √ 3) because opposite to 30 degrees is y and the hypotenuse is always opposite to the 90 degree angle.

solving for y, we get y = 34√3 * sin(30°). if you use a calculator or other knowledge, you find that sin(30°) = 1/2 so we get y = 34√3 * 1/2 = 17√3.

it is also the case that

cos(30°) = x / (34 √ 3), giving us x = 34√3 * cos(30°). using calculator or other methods, we find that cos(30°) = √3 / 2 so x = 34√3 * √3 / 2   = 34 * 3 / 2 = 17 * 3 = 51.

x = 51, y = 17√3.

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