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Chapter 5: Problem 23

Use the Law of Sines to solve the triangle. Round your answers to two decimal places. $$A=110^{\circ} 15^{\prime}, \quad a=48, \quad b=16$$

Short Answer

Expert verified
After performing all the calculations using the steps outlined above, we find that \[ B = 20.98^\circ, C = 48.77^\circ, and c = 27.47 \]. These are the measurements needed to completely solve the triangle.

Step by step solution

01

Convert angle measure

First, convert the given angle A which is in degrees and minutes into decimal degrees. This can be done by dividing the minutes by 60 and adding the result to the degrees. \[A = 110^\circ 15' = 110^\circ + \frac{15}{60}^\circ = 110.25^\circ \].
02

Compute for angle B using Law of Sines

The next step is to use the Law of Sines to find the measure of angle B. According to the Law of Sines, \(\frac{a}{\sin A} = \frac{b}{\sin B}\). Rearrange the equation to solve for B: \[\sin B = \frac{b}{a} \sin A \]Substitute the known values of \(a = 48\), \(A = 110.25^\circ\), and \(b = 16\), we get \[\sin B = \frac{16}{48} \sin (110.25)\]. Calculate the value of \(\sin B\) and then use the arc sine function to find the measure of angle B.
03

Compute for angle C

Now we can find angle C by subtracting the sum of angles A and B from 180, as the sum of the interior angles in any triangle is 180 degrees. That gives \[C = 180 - A - B\].
04

Calculate the length of side c

Finally, we compute for the length of side c using the Law of Sines. We can rearrange the formula to solve for side c like this: \(c = a * \frac{\sin C}{\sin A}\). Substituting the known values will give the value for side c.

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