Solve triangle BCD if angle B = 45, angle D = 60, BC = √3 cm.

Solving a triangle means finding all its unknown elements.

We calculate the degree measure of the angle C using the theorem on the sum of the angles of a triangle.

The angles of a triangle add up to 180 °.

angle C = 180 ° – (60 ° + 45 °) = 75 °.

We find the unknown sides by the sine theorem.

ВС / sin 60 ° = DC / sin 45 °, we express CD:

CD = BC * sin 45 ° / sin 60 ° = √3 * (√2 / 2) * 2 / √3 = √2.

Similarly, we look for BD:

ВС / sin 60 ° = BD / sin 75 °.

sin 75 ° = sin (45 ° + 30 °) = sin 45 ° * cos 30 ° + cos 45 ° * sin 30 ° = (√2 * √3 / 4 + √2 / 4) = √2 (√3 + 1) / 4.

BD = BC * sin 75 ° / sin 60 ° = √3 * (√2 (√3 + 1) / 4) * 2 / √3 = √2 (√3 + 1) / 2.

Answer: angle C = 75 °, BD = √2 (√3 + 1) / 2, CD = √2.