Need help with trigonometry question please

Question

Two sides and an angle are given below. Determine whether the given information results in one​ triangle, two​ triangles, or no triangle at all. Solve any​ triangle(s) that results

b=8 c=10 B=20 degrees

Answer 1

Drawing this out might help.

If c is 10 units long and b is 8 and angle B is 20°, then we can draw the triangle ABC in the blue position. But, we can also draw the triangle in the red position as well. Notice that side b is still the same length and angle B is still 20° but it's side A that changes and the problem doesn't specify the length of A nor does it say anything about angle A or angle C.

So 2 triangles are possible.

To solve the triangles, use the Law of Sines.

sin(20)/8 = sin(C)/10 so

sin(C) = 10sin(20)/8 = 0.42753

C = sin^-1(0.42753) = 25.3106°

So angle C is 25.3° - this is obviously the red triangle.

For this triangle, angle A must be 180 - 20 - 25.3 = 134.7° and side a can be calculated from the Law of Sines by sin(20)/8 = sin(134.7)/a or a = 16.629

The other possible angle for angle C is 180° - 25.3106° = 154.7° - obviously the blue triangle (notice that the sin(154.7) is also 0.42753).

You can repeat the process to find the other angle and sides for this blue triangle.

Answer 2

Draw a picture......

1st, draw two lines with a 20 degree angle between them (Angle B).

Name one of the lines "c' and make a mark about 1" from point B. Label this with the length 10.

Label the other line "a"--we don't know it's length yet, so just make it about 2x the length of "c".

Looking at this diagram. picture where line "b" can be. It is shorter than "c", and will thus intersect with line "a" at one of 2 points: At one point, angle A will be acute and angle C will be obtuse. At the other point, C will be acute, and A obtuse.

Draw the 2 possibilities to scale---to do this, mark off 5 equal intervals on line c. Then the distance of 4 intervals will be 8.

So...the answer is "two triangles'. Use the law of sines to get all the numbers.