In triangle ABC AC = 4√2 cm, angle С = 45º, angle А = 30º. Find the side of the sun.

| February 12, 2021 | education

The sine theorem states that all sides of a triangle are proportional to opposite angles.

Thus, we get the following relationship: AC / sin B = BC / sin A = AB / sin C.

First, we calculate the angle B.

By the formula for the sum of all the angles of the triangle, we have: A + B + C = 180.

B = 180 – A – C.

B = 180 – 300 – 450.

B = 105.

Now we find the side of the BC by the formula AC / sin B = BC / sin A.

BC = AC * sin A / sin B.

ВС = 4√2 cm * sin 30 / sin 105.

sin 300 = ½.

sin 105 can be represented as the sine of the sum of two angles.

sin (60 + 45) = sin 60 * cos 45 + sin 45 * cos 60 = √3 / 2 * √2 / 2 + √2 / 2 * 1/2 = √6 / 4 + √2 / 4.

Now let’s substitute the obtained values.

BC = 4√2 cm * 1/2 / (√6 / 4 + √2 / 4).

BC = 2√2 cm * (√6 / 4 + √2 / 4).

Let’s expand the brackets and simplify.

BC = 2√2 * √6 / 4 + 2√2 * √2 / 4.

BC = √12 / 2 + √4 cm / 2.

BC = 2√3 / 2 + 2 cm / 2.

BC = √3 + 1.

Answer: BC = (√3 + 1) see.



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