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 π‘Žπ‘ Cos 𝐢 [B] 1/2 π‘Žπ‘ Sin 𝐢 [C] 1/3 π‘Žπ‘ Sin 𝐢 [D] 1/4 π‘Žπ‘ Cos 𝐢 [E] 1/4 π‘Žπ‘ Sin 𝐢

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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 π‘Žπ‘ cos 𝐢, B) one-half π‘Žπ‘ sin 𝐢, C) one-third π‘Žπ‘ sin 𝐢, D) one-fourth π‘Žπ‘ cos 𝐢, or E) one-fourth π‘Žπ‘ sin 𝐢.

If we sketch a triangle and label it 𝐴, 𝐡, and 𝐢, the side length opposite vertex 𝐴 is usually labelled with a lower case π‘Ž. The side length opposite vertex 𝐡 is labelled with a lower case 𝑏. And we label lower case 𝑐 the side length opposite vertex 𝐢. 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 𝐢, that’s this angle, and the lengths π‘Ž 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 𝐢. To use trigonometry to solve for the area of a triangle, we take one-half times π‘Ž times 𝑏 times sin of 𝐢, which is option B here.

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