The law of sines states that 

\(\displaystyle \frac{sinA}{a}=\frac{sinB}{b}=\frac{sinC}{c}\).

In this triangle, we are looking for the side length c, and we are given angle A, angle B, and side b. The sum of the interior angles of a triangle is \(\displaystyle 180^{\circ}\); using subtraction we find that angle C = \(\displaystyle 57^{\circ}\).

We can now form a proportion that includes only one unknown, c:

\(\displaystyle \frac{sin52^{\circ}}{13}=\frac{sin57^{\circ}}{c}\)

Solving for c, we find that 

\(\displaystyle c=\frac{13sin57^{\circ}}{sin52^{\circ}}\approx13.84\).

First, find \(\displaystyle m\angle C\). The sum of the interior angles of a triangle is \(\displaystyle 180^\circ\), so \(\displaystyle m\angle C = 180^\circ - 78^\circ - 40^\circ\), or \(\displaystyle 62^\circ\).

Using this information, you can set up a proportion to find side b:

\(\displaystyle \frac{sinC}{c}=\frac{sinB}{b}\)

\(\displaystyle \frac{sin62^\circ}{14}=\frac{sin40^\circ}{b}\)

\(\displaystyle b=\frac{14sin40^\circ}{sin62^\circ}\)

\(\displaystyle b\approx10.2\)

To use the law of sines, first you must find the measure of \(\displaystyle \angle C\). Since the sum of the interior angles of a triangle is \(\displaystyle 180^\circ\), \(\displaystyle m\angle C=84^\circ\).

Law of sines:

\(\displaystyle \frac{\sin A}{a}=\frac{\sin C}{c}\)

\(\displaystyle \frac{\sin 75^\circ}{a}=\frac{\sin 84^\circ}{19.3}\)

\(\displaystyle a=\frac{19.3 \sin 75^\circ}{\sin 84^\circ}\)

\(\displaystyle a\approx 18.7\)

Because the angle with measure 117.28 degrees is opposite the side of length 6, and angle \(\displaystyle \theta\) is opposite the side of length 3, we can use the Law of Sines to solve for the measure of \(\displaystyle \theta\).

\(\displaystyle \frac{\sin(117.28^{\circ})}{6}=\frac{\sin(\theta)}{3}\)

\(\displaystyle \theta=\arcsin(\frac{3\sin(117.28^\circ)}{6})\)

\(\displaystyle \theta=26.38^\circ\)

The law of sines states that, given a triangle with sides a, b, and c and angles A, B, and C opposite to the corresponding sides,

\(\displaystyle \frac{a}{\sin A}=\frac{b}{\sin B}=\frac{c}{\sin C}\)

Since the sides and angles given are directly opposite, we can use the law of sines.

\(\displaystyle \frac{a}{\sin 55^{\circ}}=\frac{10}{\sin 60^{\circ}}\)

Solving for a, we get

\(\displaystyle a=\frac{10\sin 55^{\circ}}{\sin 60}\)

Evaluating the expression, we find that \(\displaystyle a=9.46\)

The law of sines states that, given a triangle, the following relationship is always true:

\(\displaystyle \frac{a}{\sin A}=\frac{b}{\sin B}=\frac{c}{\sin C}\)

where a, b, and c are sides and A, B, and C are the angles opposite to the sides.

This problem does not give us the length of the side opposite to the angle we want to find, so we have to find it indirectly.

We start by finding the measure of the unmarked angle, which we'll represent as B:

\(\displaystyle \frac{10}{\sin B}=\frac{12}{\sin 65^{\circ}}\)

Solving for \(\displaystyle \sin B\), we get

\(\displaystyle \frac{10\sin 65^{\circ}}{12}=\sin B\)

\(\displaystyle B=\arcsin(0.755)=49^\circ\)

Now that we have the measures of two angles, we can find the measure of the third by the \(\displaystyle 180^{\circ}\) theorem:

\(\displaystyle 65^\circ+49^\circ+A=180^\circ\)

\(\displaystyle A=180^{\circ}-65^\circ-49^\circ=66^\circ\)

Recall the Law of Sines:

\(\displaystyle \frac{a}{\sin A}=\frac{b}{\sin B}=\frac{c}{\sin C}\)

Start by finding the value of angle \(\displaystyle A\):

\(\displaystyle A+56+39=180\)

\(\displaystyle A=85\)

Plug in the given values into the Law of Sines:

\(\displaystyle \frac{4}{\sin 39}=\frac{a}{\sin 85}\)

Rearrange the equation to solve for \(\displaystyle a\):

\(\displaystyle a=(\sin 85)\frac{4}{\sin 39}=6.33\)

Make sure to round to two places after the decimal.

Recall the Law of Sines:

\(\displaystyle \frac{a}{\sin A}=\frac{b}{\sin B}=\frac{c}{\sin C}\)

Start by finding the value of angle \(\displaystyle C\):

\(\displaystyle C+36+76=180\)

\(\displaystyle C=68\)

Plug in the given values into the Law of Sines:

\(\displaystyle \frac{3}{\sin 76}=\frac{c}{\sin 68}\)

Rearrange the equation to solve for \(\displaystyle c\):

\(\displaystyle c=(\sin 68)\frac{3}{\sin 76}=2.87\)

Make sure to round to two places after the decimal.

Recall the Law of Sines:

\(\displaystyle \frac{a}{\sin A}=\frac{b}{\sin B}=\frac{c}{\sin C}\)

Start by finding the value of angle \(\displaystyle C\):

\(\displaystyle C+65+42=180\)

\(\displaystyle C=73\)

Plug in the given values into the Law of Sines:

\(\displaystyle \frac{8}{\sin 65}=\frac{c}{\sin 73}\)

Rearrange the equation to solve for \(\displaystyle c\):

\(\displaystyle c=(\sin 73)\frac{8}{\sin 65}=8.44\)

Make sure to round to two places after the decimal.

Recall the Law of Sines:

\(\displaystyle \frac{a}{\sin A}=\frac{b}{\sin B}=\frac{c}{\sin C}\)

Start by finding the value of angle \(\displaystyle C\):

\(\displaystyle C+71+28=180\)

\(\displaystyle C=81\)

Plug in the given values into the Law of Sines:

\(\displaystyle \frac{11}{\sin 71}=\frac{c}{\sin 81}\)

Rearrange the equation to solve for \(\displaystyle c\):

\(\displaystyle c=(\sin 81)\frac{11}{\sin 71}=11.49\)

Make sure to round to two places after the decimal.