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Chapter 5: Problem 13

Use the Law of Sines to solve the triangle. Round your answers to two decimal places. $$A=35^{\circ}, \quad B=65^{\circ}, \quad c=10$$

Short Answer

Expert verified
The measures of the triangle are approximately: \(C = 80^{\circ}\), \(a = 7.07\) and \(b = 9.65\)

Step by step solution

01

Find the third angle

Since we know that the sum of all angles in a triangle equals 180 degrees, we can find the third angle of the triangle (angle C). We subtract the sum of the known angles from 180. Formally this is done like this: \(C = 180 - (A + B) = 180 - (35 + 65) = 80^{\circ}\)
02

Find the first unknown side

We can use the Law of Sines to find the first unknown side, 'a'. The law of sines states that the ratio of the length of a side of a triangle to the sine of its opposite angle is constant. This gives us the equation: \(\frac{a}{\sin(A)} = \frac{c}{\sin(C)}\). Substituting the known values gives: \(a = c \times \frac{\sin(A)}{\sin(C)} = 10 \times \frac{\sin(35^{\circ})}{\sin(80^{\circ})}\). Using a calculator to compute the sines and the multiplication gives \(a \approx 7.07\)
03

Find the second unknown side

We find the second unknown side, 'b', in the same way, using the equation: \(\frac{b}{\sin(B)} = \frac{c}{\sin(C)}\). Substituting in the known values, we have: \(b = c \times \frac{\sin(B)}{\sin(C)} =10 \times \frac{\sin(65^{\circ})}{\sin(80^{\circ})}\). Using a calculator we find \(b \approx 9.65\)

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Most popular questions from this chapter

Consider the equation \(2 \sin x-1=0\). Explain the similarities and differences between finding all solutions in the interval \(\left[0, \frac{\pi}{2}\right)\), finding all solutions in the interval \([0,2 \pi),\) and finding the general solution.

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