# Video: Trigonometric Formulas for Area of Triangles

Which of the following is a formula that can be used to find the area of a triangle? [A] 1/2 𝑎𝑏 Cos 𝐶 [B] 1/2 𝑎𝑏 Sin 𝐶 [C] 1/3 𝑎𝑏 Sin 𝐶 [D] 1/4 𝑎𝑏 Cos 𝐶 [E] 1/4 𝑎𝑏 Sin 𝐶

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### Video Transcript

Which of the following is a formula that can be used to find the area of a triangle? A) one-half 𝑎𝑏 cos 𝐶, B) one-half 𝑎𝑏 sin 𝐶, C) one-third 𝑎𝑏 sin 𝐶, D) one-fourth 𝑎𝑏 cos 𝐶, or E) one-fourth 𝑎𝑏 sin 𝐶.

If we sketch a triangle and label it 𝐴, 𝐵, and 𝐶, the side length opposite vertex 𝐴 is usually labelled with a lower case 𝑎. The side length opposite vertex 𝐵 is labelled with a lower case 𝑏. And we label lower case 𝑐 the side length opposite vertex 𝐶. We have to remember that a triangle is half of a rectangle. And so, it’s unlikely that options C through E would be the answer.

We noticed that options A and B are dealing with the angle at vertex 𝐶, that’s this angle, and the lengths 𝑎 and 𝑏. At this point, we recognize that we have two sides and an included angle. And we know that the height of this triangle will be equal to 𝑏 times the sin of 𝐶. To use trigonometry to solve for the area of a triangle, we take one-half times 𝑎 times 𝑏 times sin of 𝐶, which is option B here.