# In triangle ABC, side b = 3 cm, side c = 4 cm, angle A = 135 degrees, find the remaining elements

**In triangle ABC, side b = 3 cm, side c = 4 cm, angle A = 135 degrees, find the remaining elements of the triangle using the theorem of sines and cosines.**

To find the side a, we use the cosine theorem. In the form of a formula, this theorem is written as follows:

a ^ 2 = b ^ 2 + c ^ 2 – 2 * b * c * cosA

a ^ 2 = 3 ^ 2 + 4 ^ 2-2 * 3 * 4 * cos135 = 9 + 16-24 * (- cos45) = 25 + 24 * (√ (2)) / 2 = 25 + 12 * √ ( 2)

a = √ (25 + 12 * √ (2))

Now we use the sine theorem to find the angles B and C.

a / sinA = b / sinB

(√ (25 + 12 * √ (2))) / sin135 = 3 / sinB

(√ (25 + 12 * √ (2))) * 2 / (√2) = 3 / sinB

sinB = 3 * (√ 2) / (√ (25 + 12 * √ 2)) * 2 = 3 / ((√ (25 + 12 * √ 2)) * (√2)) = 3 / (√ (50 + 24 * √ 2))

Hence, the angle B = arcsin (3 / (√ (50 + 24 * √2)))

Since the sum of the angles in a triangle is 180 degrees, then angle C = 180 – angle A – angle B = 180 – 135 – arcsin (3 / (√ (50 + 24 * √2))) = 45 – arcsin (3 / ( √ (50 + 24 * √ 2)))

Answer: a = √ (25 + 12 * √ (2)), angle B = arcsin (3 / (√ (50 + 24 * √ 2))), angle C = 45 – arcsin (3 / (√ (50+ 24 * √2)))