Question

find the area of a triangle using the sine function

Answer 1

Hey Baseballer again. The two legs defining angle BAC are equal, so the line perpendicular to the 3rd leg divides the 54 deg equally. This means that you have created a triangle within the triangle with a perpendicular angle, a diagonal of length 31 and a top angle of 27 deg (see image).

For such a triangle you can use the sine function: sin(27 deg) = X/31, or when solved for X: X = 31 * sin(27 deg). Solve for X and you'll find X = 14.1 units. As X is only half the base B of the trianle, you need to double 14.1, getting 28.2 units for B.

The height of the triangle follows from 31^2 = H^2 + 14.1^2, with H = 27.6 units.

Now the easy part: AREA = (B*H)/2, so 28.2 * 27.6 / 2 = 389,3 units, so almost 390 units.

Answer 2

For some reasons parts of the drawing disappeared. I just added X to make sure the drawing and the text still make sense....hope this works.

Answer 3

I think they want you to drop a perpendicular from B to the base AC, forming a right angle

notice that sin(54º)= opp/hyp = h/31
solve for h to get
\[ h= 31\sin(54º) \]

the area of the triangle is 1/2 base * height, so you get
\[ \text{Area }= \frac{1}{2} 31 \cdot 31 \cdot \sin(54º) \]
you can simplify using a calculator