Question Video: Using the Law of Cosines to Calculate an Unknown Length in a Triangle Mathematics

𝐴𝐵𝐶 is a triangle, where 𝐵𝐶 = 25 cm, 𝐴𝐶 = 13 cm, and 𝑚∠𝐶 = 142°. Find the length 𝐴𝐵 giving the answer to three decimal places.

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Video Transcript

𝐴𝐵𝐶 is a triangle, where 𝐵𝐶 is equal to 25 centimeters, 𝐴𝐶 is equal to 13 centimeters, and the measure of angle 𝐶 is 142 degrees. Find the length 𝐴𝐵 giving the answer to three decimal places.

We will begin by sketching the triangle 𝐴𝐵𝐶, where the length 𝐵𝐶 equals 25 centimeters, the length 𝐴𝐶 is equal to 13 centimeters, and the measure of angle 𝐶 is equal to 142 degrees. We need to find the length 𝐴𝐵, and we will let this be 𝑥 centimeters. As we know the lengths of two sides of our triangle as well as the included angle, we will use the law of cosines, otherwise known as the cosine rule. This states that 𝑐 squared is equal to 𝑎 squared plus 𝑏 squared minus two 𝑎𝑏 multiplied by the cos of angle 𝐶.

Substituting in the values from the diagram, we have 𝑥 squared is equal to 25 squared plus 13 squared minus two multiplied by 25 multiplied by 13 multiplied by the cos of 142 degrees. Typing the right-hand side of our equation into our calculator gives us 1306.2069 and so on. We can then square root both sides of this equation such that 𝑥 is equal to 36.1414 and so on. Rounding this to three decimal places, 𝑥 is equal to 36.141. The length 𝐴𝐵 to three decimal places is 36.141 centimeters.

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