Solve the right triangle. Round your answer to the appropriate number of significant digits.
A) $a = 2,607 \mathrm { ft } , b = 1,468 \mathrm { ft } , \angle B = 29.39 ^ { \circ }$
B) $a = 2,607 \mathrm { ft } , b = 1,468 \mathrm { ft } , \angle B = 60.61 ^ { \circ }$
C) $a = 1,468 \mathrm { ft } , b = 2,607 \mathrm { ft } , \angle B = 60.61 ^ { \circ }$
D) $a = 1,468 \mathrm { ft } , b = 2,607 \mathrm { ft } , \angle B = 29.39 ^ { \circ }$
E) $a = 1,468 \mathrm { ft } , b = 2,607 \mathrm { ft } , \angle B = 119.4 ^ { \circ }$

Question

Solve the right triangle. Round your answer to the appropriate number of significant digits.  
A) $a = 2,607 \mathrm { ft } , b = 1,468 \mathrm { ft } , \angle B = 29.39 ^ { \circ }$ 
B) $a = 2,607 \mathrm { ft } , b = 1,468 \mathrm { ft } , \angle B = 60.61 ^ { \circ }$ 
C) $a = 1,468 \mathrm { ft } , b = 2,607 \mathrm { ft } , \angle B = 60.61 ^ { \circ }$ 
D) $a = 1,468 \mathrm { ft } , b = 2,607 \mathrm { ft } , \angle B = 29.39 ^ { \circ }$ 
E) $a = 1,468 \mathrm { ft } , b = 2,607 \mathrm { ft } , \angle B = 119.4 ^ { \circ }$

Quizplus · Accepted Answer

Using the law of sines, we have:
$$\frac{\sin A}{a}=\frac{\sin B}{b}\rightarrow a\sin B=b\sin A$$
Then, plug in the given values:
$$2607\sin B=1468\sin 60.61^{\circ}$$
Solve for $\sin B$:
$$\sin B \approx 0.6837$$
Using the Pythagorean theorem, we have:
$$c=\sqrt{a^2+b^2}=\sqrt{(1468\ \mathrm{ft})^2+(2607\ \mathrm{ft})^2}\approx 3004 \ \mathrm{ft}$$
Using the law of cosines, we have:
$$c^2=a^2+b^2-2ab\cos C\rightarrow \cos C=\frac{a^2+b^2-c^2}{2ab}$$
Then, plug in the given values:
$$\cos C=\frac{(1468\ \mathrm{ft})^2+(2607\ \mathrm{ft})^2-(3004\ \mathrm{ft})^2}{2(1468\ \mathrm{ft})(2607\ \mathrm{ft})}\approx 0.8104$$
Solve for $\angle C$:
$$\angle C\approx 36.6^{\circ}$$
Finally, use the fact that the sum of angles in a triangle is $180^{\circ}$ to find $\angle A$:
$$\angle A=180^{\circ}-\angle B-\angle C\approx 83.8^{\circ}$$
Therefore, the solution is $(a = 1,468 \mathrm { ft} , b =2,607 \mathrm { ft} , \angle B =60.61 ^ { \circ})$.

Solve the Right Triangle a=2,607ft,b=1,468ft,∠B=29.39∘a = 2,607 \mathrm { ft } , b = 1,468 \mathrm { ft } , \angle B = 29.39 ^ { \circ }a=2,607ft,b=1,468ft,∠B=29.39∘

Solve the Right Triangle $a = 2,607 \mathrm { ft } , b = 1,468 \mathrm { ft } , \angle B = 29.39 ^ { \circ }$