How To Calculate the Area of ​​a Triangle, Given the Lengths of Two Sides and the Measure of the Angle Between Them? Trigonometric Formula

• Side's length - Angle's measure - Side's length

• This method of calculating the area of ​​a triangle involves knowing the lengths of two sides and the measure of the angle between them.
• In a triangle △ABC with side lengths BC = a, AC = b, and AB = c, the lengths of a and b are known, as well as the measure of angle ∠C.

• We start with the classic formula of triangle's area:

• Area = ¹/₂ × a × h_a
• Where:
• a = the side of the triangle opposite angle ∠A
• h_a = the height AD from the vertex A to the side with length "a" (BC), where D belongs to the line containing the segment BC.
• 

  Height h_a as a function of the side b and the sine of angle ∠C

• == But we don't know the length of the height h_a. What next? ==
• Using notions of trigonometry, in the right triangle △ADC (∡D = 90°), formed to the right of the height h_a, we have:
• sin(∠C) = ^h_a / _b => h_a = b × sin(∠C).
• Substitute the height h_a into the classic formula of the triangle's area and get:

• The trigonometric formula:

• Area = ¹/₂ × a × b × sin(∠C)

Let's calculate the area of ​​a triangle with the side lengths a = 10, b = 9, and the angle measure of m(∠C) = 40 degrees.