Solve ∆ABC given angle A = 58ᵒ, b = 10.0 cm, and c = 14.0 cm. Round to one decimal place.
To solve ∆ABC, we need to use the Law of Sines which states that the ratio of the length of a side of a triangle to the sine of its opposite angle is constant.
First, we need to find angle B using the Law of Sines:
sin(B)/10.0 = sin(58°)/14.0
Cross-multiplying, we get:
sin(B) = (10.0 * sin(58°))/14.0
sin(B) ≈ 0.5842
Taking the inverse sine, we find that angle B is approximately 35.3°.
To find angle C, we know that the sum of all angles in a triangle is always 180°:
180° = 58° + 35.3° + angle C
angle C ≈ 86.7°
Now that we have all three angles, we can find the length of side a using the Law of Sines once again:
sin(58°)/10.0 = sin(86.7°)/a
Cross-multiplying, we get:
a = (10.0 * sin(86.7°))/sin(58°)
a ≈ 11.5 cm
Therefore, the lengths of the sides of ∆ABC are approximately:
a ≈ 11.5 cm
b = 10.0 cm
c = 14.0 cm