10.1 Non-right Triangles: Law of Sines

Given $\alpha = 80° , a = 120 ,$ and $b = 121 ,$ find the missing side and angles. If there is more than one possible solution, show both.

Given $\alpha = 80° , a = 100 , b = 10 ,$ find the missing side and angles. If there is more than one possible solution, show both. Round your answers to the nearest tenth.

Determine the number of triangles possible given $a = 31 ,$ $b = 26 ,$ $\beta = 48° .$

Find the area of a triangle with sides $a = 90 , b = 52 ,$ and angle $\gamma = 102° .$ Round the area to the nearest integer.

Find the area of the triangle given $\beta = 42° ,$ $a = 7.2 \text{ft} ,$ $c = 3.4 \text{ft} .$ Round the area to the nearest tenth.

Describe the altitude of a triangle.

When can you use the Law of Sines to find a missing angle?

What type of triangle results in an ambiguous case?

Assume $\alpha$ is opposite side $a , \beta$ is opposite side $b ,$ and $\gamma$ is opposite side $c .$ Solve each triangle, if possible. Round each answer to the nearest tenth.

$\alpha = 35° , \gamma = 73° , c = 20$

Assume $\alpha$ is opposite side $a , \beta$ is opposite side $b ,$ and $\gamma$ is opposite side $c .$ Solve each triangle, if possible. Round each answer to the nearest tenth.

$a = 4 ,$ $\alpha = 60° ,$ $\beta = 100°$

Use the Law of Sines to solve for the missing side for each oblique triangle. Round each answer to the nearest hundredth. Assume that angle $A$ is opposite side $a ,$ angle $B$ is opposite side $b ,$ and angle $C$ is opposite side $c .$

Find side $b$ when $A = 37° ,$ $B = 49° ,$ $c = 5.$

Use the Law of Sines to solve for the missing side for each oblique triangle. Round each answer to the nearest hundredth. Assume that angle $A$ is opposite side $a ,$ angle $B$ is opposite side $b ,$ and angle $C$ is opposite side $c .$

Find side $c$ when $B = 37° , C = 21° ,$ $b = 23.$

Assume $\alpha$ is opposite side $a , \beta$ is opposite side $b ,$ and $\gamma$ is opposite side $c .$ Determine whether there is no triangle, one triangle, or two triangles. Then solve each triangle, if possible. Round each answer to the nearest tenth.

$\gamma = 113° , b = 10 , c = 32$

Assume $\alpha$ is opposite side $a , \beta$ is opposite side $b ,$ and $\gamma$ is opposite side $c .$ Determine whether there is no triangle, one triangle, or two triangles. Then solve each triangle, if possible. Round each answer to the nearest tenth.

$a = 12 ,$ $c = 17 ,$ $\alpha = 35°$

Assume $\alpha$ is opposite side $a , \beta$ is opposite side $b ,$ and $\gamma$ is opposite side $c .$ Determine whether there is no triangle, one triangle, or two triangles. Then solve each triangle, if possible. Round each answer to the nearest tenth.

$a = 7 ,$ $c = 9 ,$ $\alpha = 43°$

Assume $\alpha$ is opposite side $a , \beta$ is opposite side $b ,$ and $\gamma$ is opposite side $c .$ Determine whether there is no triangle, one triangle, or two triangles. Then solve each triangle, if possible. Round each answer to the nearest tenth.

$b = 13 , c = 5 , \gamma = 10°$

Assume $\alpha$ is opposite side $a , \beta$ is opposite side $b ,$ and $\gamma$ is opposite side $c .$ Determine whether there is no triangle, one triangle, or two triangles. Then solve each triangle, if possible. Round each answer to the nearest tenth.

$\beta = 119° , b = 8.2 , a = 11.3$

Use the Law of Sines to solve, if possible, the missing side or angle for each triangle or triangles in the ambiguous case. Round each answer to the nearest tenth.

Find angle $A$ when $a = 13 , b = 6 , B = 20°.$

Find the area of the triangle with the given measurements. Round each answer to the nearest tenth.

$a = 5 , c = 6 , \beta = 35°$

Find the area of the triangle with the given measurements. Round each answer to the nearest tenth.

$a = 32 , b = 24 , \gamma = 75°$

In order to estimate the height of a building, two students stand at a certain distance from the building at street level. From this point, they find the angle of elevation from the street to the top of the building to be 35°. They then move 250 feet closer to the building and find the angle of elevation to be 53°. Assuming that the street is level, estimate the height of the building to the neare…

A man and a woman standing $3 \frac{1}{2}$ miles apart spot a hot air balloon at the same time. If the angle of elevation from the man to the balloon is 27°, and the angle of elevation from the woman to the balloon is 41°, find the altitude of the balloon to the nearest foot.

A street light is mounted on a pole. A 6-foot-tall man is standing on the street a short distance from the pole, casting a shadow. The angle of elevation from the tip of the man’s shadow to the top of his head of 28°. A 6-foot-tall woman is standing on the same street on the opposite side of the pole from the man. The angle of elevation from the tip of her shadow to the top of her head is 28°. If…

Two streets meet at an 80° angle. At the corner, a park is being built in the shape of a triangle. Find the area of the park if, along one road, the park measures 180 feet, and along the other road, the park measures 215 feet.

The Bermuda triangle is a region of the Atlantic Ocean that connects Bermuda, Florida, and Puerto Rico. Find the area of the Bermuda triangle if the distance from Florida to Bermuda is 1030 miles, the distance from Puerto Rico to Bermuda is 980 miles, and the angle created by the two distances is 62°.