Question Video: Using the Law of Cosines to Calculate the Measure of an Unknown Angle in a Triangle

The side lengths of a triangle are π‘Ž = 5 cm, 𝑏 = 7 cm, and 𝑐 = 10 cm. Amelia calculated, to one decimal place, the measure of the corresponding angles as 𝐴 = 27.7Β°, 𝐡 = 40.5Β°, and 𝐢 = 111.8Β°. Was she correct?

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Video Transcript

The side lengths of a triangle are π‘Ž is equal to five centimeters, 𝑏 is equal to seven centimeters, and 𝑐 is equal to 10 centimeters. Amelia calculated, to one decimal place, the measure of the corresponding angles as 𝐴 is equal to 27.7 degrees, 𝐡 is equal to 40.5 degrees, and 𝐢 is equal to 111.8 degrees. Was she correct?

We will begin by sketching the triangle. The side lengths of the triangle are five, seven, and 10 centimeters, where side length π‘Ž is opposite angle 𝐴, side length 𝑏 is opposite angle 𝐡, and side length 𝑐 is opposite angle 𝐢. In order to calculate the measure of any of the angles inside the triangle, we can use the law of cosines, otherwise known as the cosine rule. This states that the cos of angle 𝐢 is equal to π‘Ž squared plus 𝑏 squared minus 𝑐 squared all divided by two π‘Žπ‘. This will hold for all three angles such that the cos of angle 𝐴 is equal to 𝑏 squared plus 𝑐 squared minus π‘Ž squared all over two 𝑏𝑐. Likewise, the cos of angle 𝐡 is equal to π‘Ž squared plus 𝑐 squared minus 𝑏 squared all over two π‘Žπ‘.

Let’s begin by calculating angle 𝐴. The cos of angle 𝐴 is equal to seven squared plus 10 squared minus five squared all divided by two multiplied by seven multiplied by 10. The right-hand side simplifies to 31 over 35 so that the cos of angle 𝐴 is equal to 31 over 35. We can repeat this process for angle 𝐡 and angle 𝐢. The cos of angle 𝐡 is equal to five squared plus 10 squared minus seven squared all divided by two multiplied by five multiplied by 10. The cos of angle 𝐡 is therefore equal to 19 over 25. The cos of angle 𝐢 is equal to five squared plus seven squared minus 10 squared all divided by two multiplied by five multiplied by seven. This simplifies to negative 13 over 35. cos of angle 𝐢 equals negative 13 over 35.

We can now calculate all three angles by taking the inverse cosine of both sides of our three equations. The inverse cos of 31 over 35 is equal to 27.7. Therefore, angle 𝐴 is equal to 27.7 degrees. The inverse cos of 19 over 25 is 40.5. Therefore, 𝐡 is equal to 40.5 degrees. Angle 𝐢 is equal to 111.8 degrees as the inverse cos of negative 13 over 35 is equal to 111.8.

All three of these angles match Amelia’s calculations, and they sum to 180 degrees. We can therefore conclude that the correct answer is yes, the corresponding angles are 𝐴 is equal to 27.7 degrees, 𝐡 is equal to 40.5 degrees, and 𝐢 is equal to 111.8 degrees.

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