Question

Please help! Geometry.

Answer 1

These are right triangles that will use either sin, cos, or tan, depending upon what you have to work with in regards to the reference angle. The first one has a reference angle of 51 with y being the side opposite it and 12 being the hypotenuse. The sin identity uses the side opposite over the hypotenuse as its formula:
$sin(51)= \frac{y}{12}$
and 12 sin(51) = y and y = 9.325
The second one has the reference angle as the unknown. You could use any of the identities here because you have all the sides of the triangle, but I will use sin again:
$sin \beta = \frac{12}{13}$
and $sin \beta =.9230769$
and $sin ^{-1}(.9230769)=67.38$
The next one has a referece angle of 13 with 24 being the side adjacent to it and the unknown being the side across from it.  You will use the tangent identity here:
$tan(13)= \frac{x}{24}$
and 24 tan(13) = x so x = 5.540
The last one has a reference angle of 20 with the hypotenuse as the unknown x, and the side across from it as 10. Use the sin identity again:
$sin(20)= \frac{10}{x}$ and
$x sin(20)=10$ and
$x= \frac{10}{sin(20)}$ with x = 29.238
Everything is in regards to the reference angle; you HAVE to be able to identify the reference angle and then how the given sides are related to it.