Question

a homeowner makes a corner bookshelf for his photos and knick - knacks in an inside corner of his den. the lengths of the sides of the corner bookshelf are 20 inches, 24 inches, and 29 inches. what are the respective angles, in degrees? round to the nearest hundredth.

Explanation:

Step1: Use the Law of Cosines

The Law of Cosines formula is \(c^{2}=a^{2}+b^{2}-2ab\cos C\), where \(a\), \(b\), \(c\) are the sides of the triangle and \(C\) is the angle opposite to side \(c\).
Let \(a = 20\), \(b=24\), \(c = 29\).
First, find the angle opposite to the side \(c\):

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\(C=\cos^{-1}(0.140625)\approx81.91^{\circ}\)

Step2: Find another angle

Let's find the angle opposite to the side \(a\). Let \(a = 20\), \(b = 29\), \(c=24\)

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\(A=\cos^{-1}(0.7319)\approx42.90^{\circ}\)

Step3: Find the third angle

Since the sum of angles in a triangle is \(180^{\circ}\), let the third angle be \(B\)
\(B=180^{\circ}-A - C\)
\(B=180-(42.90 + 81.91)=55.19^{\circ}\)

Answer:

The angles are approximately \(42.90^{\circ}\), \(55.19^{\circ}\), and \(81.91^{\circ}\)