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Chapter 5: Problem 29

Use the Law of Sines to solve (if possible) the triangle. If two solutions exist, find both. Round your answers to two decimal places. $$A=58^{\circ}, \quad a=11.4, \quad b=12.8$$

Short Answer

Expert verified
Therefore, the two possible solutions are \(B=73.67^{\circ}\), \(C=48.33^{\circ}\) and \(B=106.33^{\circ}\), \(C=15.67^{\circ}\)

Step by step solution

01

Employ the Law of Sines to find the unknown angle

Start by using the given information and inserting it into the law of sines formula: \[\frac{a}{\sin A} = \frac{b}{\sin B}\]This means that:\[\frac{11.4}{\sin 58^\circ} = \frac{12.8}{\sin B}\]Rearranging to isolate \(\sin B\):\[\sin B = \frac{12.8\sin 58^\circ}{11.4}\]Now calculate the value of \(\sin B\) by inputting the values of sin 58掳 which is approximately 0.8480480962. We get \(\sin B = 0.9542475901\)
02

Calculate the value of angle B

Calculate the value of angle B by taking the arcsin of \(\sin B\). \[B = \arcsin(\sin B) = \arcsin(0.9542475901)\]This gives a result of approximately \(73.67^\circ\). From this, we can deduce that angle B consists of two possible values which are \(B=73.67^\circ\) and \(B=180^\circ-73.67^\circ=106.33^\circ\)
03

Compute the remaining angle C for each scenario

In a triangle, the sum of the angles is \(180^\circ\). If we subtract the known angles from \(180^\circ\), we find the remaining angle.Therefore,for the first triangle: \[C = 180-58-73.67 = 48.33^{\circ}\]For the second triangle:\[C = 180-58-106.33 = 15.67^{\circ}\]Hence, we have two possible solutions for the triangle, which are \(B=73.67^{\circ}\) and \(C=48.33^{\circ}\), or \(B=106.33^{\circ}\) and \(C=15.67^{\circ}\)

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Most popular questions from this chapter

Use a graphing utility to approximate the solutions (to three decimal places) of the equation in the interval \([0,2 \pi)\). $$2 \sin x+\cos x=0$$

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