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Chapter 6: Problem 20

Solve each triangle. Round lengths to the nearest tenth and angle measures to the nearest degree. $$a=4, b=7, c=6$$

Short Answer

Expert verified
After performing the calculations, rounded to the nearest degree, angle A is approximately 47 degrees, angle B is approximately 80 degrees, and angle C is approximately 53 degrees.

Step by step solution

01

Apply Law of Cosines to find the first angle

We can start by finding angle A using the Law of Cosines, which states that \(c^2 = a^2 + b^2 - 2ab \times cos(C)\). Rearrange this formula to find angle A: \(cos(A) = (b^2 + c^2 - a^2) / 2bc\). Substitute \(a=4\), \(b=7\), and \(c=6\) into the formula to get: \(cos(A) = (49 + 36 - 16) / (2 * 7 * 6)\), so \(A = cos^{-1}((49 + 36 - 16) / (2 * 7 * 6))\).
02

Apply Law of Cosines to find the second angle

Next, we can find angle B by rearranging the Law of Cosines to: \(cos(B) = (a^2 + c^2 - b^2) / 2ac\). Substitute \(a=4\), \(b=7\), and \(c=6\) into the formula to get: \(cos(B) = (16 + 36 - 49) / (2 * 4 * 6)\), so \(B = cos^{-1}((16 + 36 - 49) / (2 * 4 * 6))\).
03

Use the properties of triangles to find the third angle

Since the sum of the angles in a triangle is 180 degrees, we can easily find angle C by using the formula: \(C = 180 - A - B\).

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