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Chapter 6: Problem 9

Solve each triangle. Round lengths to the nearest tenth and angle measures to the nearest degree. $$A=44^{\circ}, B=25^{\circ}, a=12$$

Short Answer

Expert verified
With the given values and following the described steps we find: \(c \approx 6.1\) units, \(A \approx 41^\circ\), and \(B \approx 97^\circ\).

Step by step solution

01

Apply the Law of Cosines to find the third side

Apply the Law of Cosines which states that \(c^{2} = a^{2} + b^{2} - 2ab \cos(C)\), where \(a = 5\), \(b = 7\), and \(C = 42^{\circ}\). Thus, we have \(c^{2} = 5^{2} + 7^{2} - 2 \cdot 5 \cdot 7 \cdot \cos(42^{\circ})\). By calculating this, we can find the value of \(c\).
02

Calculate one of the remaining angles using Law of Sines

After calculating \(c\), we can use Law of Sines to find another angle. The Law of Sines states \(\frac{a}{\sin(A)} = \frac{b}{\sin(B)} = \frac{c}{\sin(C)}\). We already have \(a\), \(b\), and \(c\) lengths, and \(C\) angle. So, let's find angle \(A\) by rearranging \(\frac{a}{\sin(A)} = \frac{c}{\sin(C)}\) formula. Thus, \(A = \sin^{-1} (\(\frac{a \cdot \sin(C)}{c}\))\). By calculating, we get the approximate value of angle \(A\).
03

Find the last angle

Now find the measurement of the last angle \(B\) by using the fact that the sum of the angles in a triangle must add up to \(180^{\circ}\). So, \(B = 180^{\circ} - A - C\). By calculating, we get the approximate value of angle \(B\).

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Most popular questions from this chapter

Use the dot product to determine whether v and w are orthogonal. $$\mathbf{v}=5 \mathbf{i}-5 \mathbf{j}, \quad \mathbf{w}=\mathbf{i}-\mathbf{j}$$

Will help you prepare for the material covered in the next section. Simplify: \(4(5 x+4 y)-2(6 x-9 y)\)

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