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In ∆ABC, a = 7, b = 5, c = 8. What is the measure of ∠BAC?

The measure of angle ∠BAC = ________ °

In ∆ABC, a = 7, b = 5, c = 8. What is the measure of ∠BAC? The measure of angle ∠BAC-example-1

1 Answer

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To find the measure of ∠BAC, we can use the law of cosines, which states that:

c^2 = a^2 + b^2 - 2ab cos(C)

Where c is the length of the side opposite angle C. In this case, we have:

c = 8

a = 7

b = 5

Substituting these values, we get:

8^2 = 7^2 + 5^2 - 2(7)(5) cos(C)

Simplifying:

64 = 49 + 25 - 70 cos(C)

70 cos(C) = 10

cos(C) = 10/70

cos(C) = 1/7

Using a calculator, we can find that the arccosine of 1/7 is approximately 82.46 degrees. Therefore, the measure of ∠BAC is approximately:

∠BAC = 180 - ∠ABC - ∠ACB

∠BAC = 180 - 90 - 82.46

∠BAC = 7.54 degrees (rounded to two decimal places)

So the measure of angle ∠BAC is approximately 7.54 degrees.

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