Given: A is 20˚, b = 50 cm, for what possible range of lengths of side “a” would ABC have two possible solutions (approximate to 2 decimal places)?

To determine the possible range of lengths for side "a" in triangle ABC, we can use the Sine Law. The Sine Law states that the ratio of a side length to the sine of its opposite angle is constant for all sides and their corresponding angles in a triangle.

The formula for the Sine Law is:
a / sin(A) = b / sin(B) = c / sin(C)

In this case, we are given angle A as 20 degrees and side b as 50 cm. We want to find the range of lengths for side a that would result in two possible solutions for triangle ABC.

Since we are looking for a range, we need to find both the minimum and maximum values for side a.

To find the minimum value, we will assume that angle B is the smallest angle possible. Since the sum of all angles in a triangle is 180 degrees, we can find angle B by subtracting angle A from 180 degrees.

angle B = 180 - angle A
angle B = 180 - 20
angle B = 160 degrees

Now, we can substitute the values into the Sine Law formula and solve for a:

a / sin(20) = 50 / sin(160)

To solve for a, we rearrange the equation:

a = (50 * sin(20)) / sin(160)

Using a calculator, we can evaluate this expression to find the minimum value for side a.

To find the maximum value, we will assume that angle B is the largest angle possible. Since the sum of all angles in a triangle is 180 degrees, we can find angle B by subtracting angle A from 180 degrees.

angle B = 180 - angle A
angle B = 180 - 20
angle B = 160 degrees

Now, we can substitute the values into the Sine Law formula and solve for a:

a / sin(20) = 50 / sin(160)

To solve for a, we rearrange the equation:

a = (50 * sin(20)) / sin(160)

Using a calculator, we can evaluate this expression to find the maximum value for side a.

Therefore, the possible range of lengths for side a in triangle ABC is from the minimum value to the maximum value, both rounded to 2 decimal places.