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Precalculus : Law of Cosines and Sines

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Example Question #31 : Use The Laws Of Cosines And Sines

Find the length of the missing side, $\displaystyle d.$

Possible Answers:

5.92 $\displaystyle 5.92$

8.10 $\displaystyle 8.10$

6.91 $\displaystyle 6.91$

6.37 $\displaystyle 6.37$

Correct answer:

6.37 $\displaystyle 6.37$

Explanation:

First, use the Law of Sines to find the measurement of angle $\displaystyle C.$

$\frac{5}{\sin 50}=\frac{3}{\sin C}$ $\displaystyle \frac{5}{\sin 50}=\frac{3}{\sin C}$

$5\sin C = 2.30$ $\displaystyle 5\sin C = 2.30$

$\sin C=0.46$ $\displaystyle \sin C=0.46$

$\displaystyle C= 27.39$

Recall that all the angles in a triangle need to add up to 180 $\displaystyle 180$ degrees.

$\displaystyle 50+27.39+B=180$

$\displaystyle B=102.61$

Now, use the Law of Sines again to find the length of $\displaystyle d$.

$\frac{d}{\sin 102.61}=\frac{5}{\sin 50}$ $\displaystyle \frac{d}{\sin 102.61}=\frac{5}{\sin 50}$

$d \sin 50= 4.88$ $\displaystyle d \sin 50= 4.88$

d=6.37 $\displaystyle d=6.37$

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Example Question #32 : Use The Laws Of Cosines And Sines

What is the largest possible angle, measured in degrees, in triangle ABC $\displaystyle ABC$ if $\angle A = 30^\circ$ $\displaystyle \angle A = 30^\circ$, AB = 10 $\displaystyle AB = 10$, and BC = 6 $\displaystyle BC = 6$?

Possible Answers:

$123.56^\circ$ $\displaystyle 123.56^\circ$

$144.56^\circ$ $\displaystyle 144.56^\circ$

$107.56^\circ$ $\displaystyle 107.56^\circ$

$156.56^\circ$ $\displaystyle 156.56^\circ$

$93.56^\circ$ $\displaystyle 93.56^\circ$

Correct answer:

$123.56^\circ$ $\displaystyle 123.56^\circ$

Explanation:

In the ambiguous SSA case, use Law of Sines to solve for the angle opposite the given side.

$\\* \frac {\sin A}{a} = \frac {\sin C}{c} \\* \\* \frac {\sin 30^\circ}{6} = \frac {\sin C}{10} \\* \\* C = \arcsin \left(\frac {10 \sin 30^\circ}{6}\right) \approx 56.44^\circ$ $\displaystyle \\* \frac {\sin A}{a} = \frac {\sin C}{c} \\* \\* \frac {\sin 30^\circ}{6} = \frac {\sin C}{10} \\* \\* C = \arcsin \left(\frac {10 \sin 30^\circ}{6}\right) \approx 56.44^\circ$

If $C = 56.44^\circ$ $\displaystyle C = 56.44^\circ$, then $B = 93.56^\circ$ $\displaystyle B = 93.56^\circ$.

However! Another possible value of C is $(180 - 56.44)^\circ = 123.56^\circ$ $\displaystyle (180 - 56.44)^\circ = 123.56^\circ$.

In this case the angles will be $30^\circ, 123.56^\circ, 26.44^\circ$ $\displaystyle 30^\circ, 123.56^\circ, 26.44^\circ$.

This is bigger than $93.56^\circ$ $\displaystyle 93.56^\circ$ and is consequently the answer.

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Precalculus : Law of Cosines and Sines

Study concepts, example questions & explanations for Precalculus

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Example Questions

Example Question #31 : Use The Laws Of Cosines And Sines

Example Question #32 : Use The Laws Of Cosines And Sines

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