Question

1. a. determine side a .

1. in a scalene triangle the measurement of an angle is 37 degrees 0 minutes the side opposite measures a. the measurements of the other sides are 3.000 inches and 2.800 inches.

Explanation:

Step1: Identify the Law to Use

We have a triangle with two sides (\(b = 3.000\) in, \(c = 2.800\) in) and the included angle (\(A = 37^\circ 0' = 37^\circ\)). To find the third side \(a\), we use the Law of Cosines: \(a^2=b^2 + c^2-2bc\cos A\).

Step2: Convert Angle to Decimal (if needed)

The angle is \(37^\circ\), so \(\cos(37^\circ)\approx0.7986\).

Step3: Substitute Values into the Formula

Substitute \(b = 3.000\), \(c = 2.800\), and \(\cos A\approx0.7986\) into the formula:

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Step4: Take the Square Root

Take the square root of \(a^2\) to find \(a\): \(a=\sqrt{0.06112}\approx0.247\) (Wait, this seems off. Wait, maybe I mixed up the sides. Wait, the Law of Cosines: if angle \(A\) is between sides \(b\) and \(c\), then \(a\) is opposite angle \(A\). Wait, maybe I made a mistake in the values. Wait, let's recalculate:

Wait, \(b = 3.000\), \(c = 2.800\), \(\cos(37^\circ)\approx0.7986\)

\(2bc\cos A=2\times3\times2.8\times0.7986 = 6\times2.8\times0.7986=16.8\times0.7986 = 13.41648\)

Then \(a^2=3^2 + 2.8^2-13.41648=9 + 7.84-13.41648=16.84 - 13.41648 = 3.42352\)

Then \(a=\sqrt{3.42352}\approx1.850\) inches. (Ah, I see, I made a mistake in the multiplication earlier. Let's do it correctly: \(2\times3\times2.8 = 16.8\), not \(2\times3\times2.8 = 16.8\), then \(16.8\times0.7986\approx13.416\). Then \(9 + 7.84=16.84\), \(16.84 - 13.416 = 3.424\), square root of \(3.424\) is approximately \(1.850\) inches.)

Wait, let's check the Law of Cosines again. The Law of Cosines is \(a^2 = b^2 + c^2-2bc\cos A\), where \(A\) is the angle between sides \(b\) and \(c\). So if the two sides are \(3\) and \(2.8\), and the included angle is \(37^\circ\), then:

\(b = 3\), \(c = 2.8\), \(A = 37^\circ\)

\(a^2=3^2 + 2.8^2-2\times3\times2.8\times\cos(37^\circ)\)

\(\cos(37^\circ)\approx0.7986\)

\(3^2 = 9\), \(2.8^2=7.84\), \(2\times3\times2.8 = 16.8\), \(16.8\times0.7986\approx13.416\)

So \(a^2=9 + 7.84-13.416 = 3.424\)

\(a=\sqrt{3.424}\approx1.85\) inches.

Answer:

\(\approx1.85\) inches (or more precisely, after recalculating, let's use a calculator for \(\cos(37^\circ)\) more accurately. \(\cos(37^\circ)\approx0.79863551\))

\(a^2 = 3^2+2.8^2 - 2\times3\times2.8\times0.79863551\)

\(=9 + 7.84-16.8\times0.79863551\)

\(=16.84 - 13.41707657\)

\(=3.42292343\)

\(a=\sqrt{3.42292343}\approx1.850\) inches. So the length of side \(a\) is approximately \(1.85\) inches.