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