In triangle ABC, side AB=210cm ,side BC=180cm and angle B =54° . Find the length of side AC?

Answer 1

To find the length of side AC, we can use the law of sines. The law of sines states that in any triangle, the ratio of the length of a side to the sine of its opposite angle is constant.

Let's call the length of side AC "x".

Using the law of sines, we have:

sin(A) / AC = sin(B) / AB

Since angle A is opposite side BC, and we know the lengths of sides BC and AB, we can substitute in the values:

sin(A) / x = sin(54°) / 210

Now, we can solve for x by cross-multiplying:

x = 210 * sin(A) / sin(54°)

To find the value of sin(A), we can use the fact that the sum of the angles in a triangle is equal to 180°. Thus, angle C is equal to 180° - 54° - A.

sin(C) = sin(180° - 54° - A) = sin(126° - A)

Now, we can rewrite the equation in terms of sin(C):

x = 210 * sin(A) / sin(54°) = 210 * sin(A) / sin(126° - A)

Next, we can use the fact that sin(180° - θ) = sin(θ):

x = 210 * sin(A) / sin(54°) = 210 * sin(A) / sin(A)

This simplifies to:

x = 210 cm

Therefore, the length of side AC is 210 cm.