Math / Trigonometry

QuestionSolve the triangle if $\angle A=113^{\circ}, a=31, b=18$ . If no answer exists, enter DNE for all answers. $\angle B$ is $\square$ degrees $\angle C$ is $\square$ degrees $c=$ $\square$ Assume $\angle \boldsymbol{A}$ is opposite side $\boldsymbol{a}, \angle \boldsymbol{B}$ is opposite side $\boldsymbol{b}$ , and $\angle \boldsymbol{C}$ is opposite side $\boldsymbol{c}$ . Round all answers to 2 decimal places as necessary. Question Help: Message instructor

Studdy Solution

STEP 1

1. The triangle is non-right and can be solved using the Law of Sines and the angle sum property.
2. $\angle A$ is opposite side $a$ , $\angle B$ is opposite side $b$ , and $\angle C$ is opposite side $c$ .
3. $\angle A = 113^\circ$ , $a = 31$ , and $b = 18$ .

STEP 2

1. Use the Law of Sines to find $\angle B$ .
2. Use the angle sum property to find $\angle C$ .
3. Use the Law of Sines to find side $c$ .

STEP 3

Use the Law of Sines to find $\angle B$ :
$\frac{a}{\sin A} = \frac{b}{\sin B}$
Substitute the known values:
$\frac{31}{\sin 113^\circ} = \frac{18}{\sin B}$
Solve for $\sin B$ :
$\sin B = \frac{18 \cdot \sin 113^\circ}{31}$
Calculate $\sin B$ :
$\sin B = \frac{18 \cdot 0.9272}{31} \approx 0.5389$
Find $\angle B$ :
$\angle B = \arcsin(0.5389) \approx 32.56^\circ$

STEP 4

Use the angle sum property to find $\angle C$ :
$\angle C = 180^\circ - \angle A - \angle B$
$\angle C = 180^\circ - 113^\circ - 32.56^\circ \approx 34.44^\circ$

STEP 5

Use the Law of Sines to find side $c$ :
$\frac{a}{\sin A} = \frac{c}{\sin C}$
Substitute the known values:
$\frac{31}{\sin 113^\circ} = \frac{c}{\sin 34.44^\circ}$
Solve for $c$ :
$c = \frac{31 \cdot \sin 34.44^\circ}{\sin 113^\circ}$
Calculate $c$ :
$c = \frac{31 \cdot 0.5658}{0.9272} \approx 18.92$
The solutions are:
$\angle B \approx 32.56^\circ$
$\angle C \approx 34.44^\circ$
$c \approx 18.92$

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