Physics: Area : With only 1 known side

Question

Given the following triangle, phi is 32 degrees and the length of side c is 36. What is the area of the triangle . (A=1/2bh)

Answer 1

I am making a few assumptions here - I am assuming that this is a right triangle and that side c is the hypotenuse of the triangle. If these assumptions are not correct, then my solution will not be correct. I am also going to make the assumption that the angle given is opposite the base of the triangle. This assumption does not affect the answer (swap b and h in the equations - nothing else changes).

Any right triangle is completely determined if you know one side and one angle in addition to the right angle or two sides. The values of the missing side(s) and angle(s) can be found from trig functions.

Using the assumptions above, the height, h is the leg of the right triangle adjacent to the angle phi and the base, b, is the leg of the triangle opposite the angle phi. Recall from your study of trigonometry, the cosine of an angle is the ratio of the side adjacent to the hypotenuse. So Cos (phi) = h/c which can be solved for h, h = c ^. cos (phi). Also recall the sine of an angle is the ratio of the side opposite to the hypotenuse, or sin (phi) = b/c which give b = c ^. sin(phi). You now have expressions for both h and b in terms of the known values. Substitute these into the equation for the area of a triangle and solve.

Area = 1/2 bh = (1/2)(c²)[cos(phi) sin(phi)] = 291 cm²