Find the area of each triangle ABC.


A = 59.80°, b = 15.00 cm, C = 53.10°

Verified step by step guidance

1. Identify the given elements of the triangle: angle A = 59.80°, side b = 15.00 cm, and angle C = 53.10°.
2. Calculate the third angle B using the triangle angle sum property: \(B = 180^\circ - A - C\).
3. Use the Law of Sines to find side a, which is opposite angle A: \(\frac{a}{\sin A} = \frac{b}{\sin B}\), so \(a = b \times \frac{\sin A}{\sin B}\).
4. Calculate the height (altitude) of the triangle relative to base b using the formula \(h = a \times \sin C\).
5. Find the area of the triangle using the formula \(\text{Area} = \frac{1}{2} \times b \times h\).

Key Concepts

Law of Sines

The Law of Sines relates the sides and angles of a triangle, stating that the ratio of a side length to the sine of its opposite angle is constant. It is useful for finding unknown sides or angles when given two angles and one side, as in this problem.

Triangle Angle Sum Property

The sum of the interior angles in any triangle is always 180°. Knowing two angles allows you to find the third angle, which is essential for applying trigonometric formulas correctly.

Area Formula Using Two Sides and Included Angle

The area of a triangle can be calculated using the formula (1/2)ab sin(C), where a and b are two sides and C is the included angle between them. This formula is especially useful when two sides and the included angle are known or can be found.