10.2 Non-right Triangles: Law of Cosines

Find the missing side and angles of the given triangle: $\alpha = 30° ,$ $b = 12 ,$ $c = 24.$

Given $a = 5 , b = 7 ,$ and $c = 10 ,$ find the missing angles.

Use Heron’s formula to find the area of a triangle with sides of lengths $a = 29.7 \text{ft} , b = 42.3 \text{ft} ,$ and $c = 38.4 \text{ft} .$

Find the area of a triangle given $a = 4.38 \text{ft} , b = 3.79 \text{ft},$ and $c = 5.22 \text{ft} .$

If you are looking for a missing side of a triangle, what do you need to know when using the Law of Cosines?

Explain what $s$ represents in Heron’s formula.

When must you use the Law of Cosines instead of the Pythagorean Theorem?

Assume $\alpha$ is opposite side $a , \beta$ is opposite side $b ,$ and $\gamma$ is opposite side $c .$ If possible, solve each triangle for the unknown side. Round to the nearest tenth.

$\alpha = 120° , b = 6 , c = 7$

Assume $\alpha$ is opposite side $a , \beta$ is opposite side $b ,$ and $\gamma$ is opposite side $c .$ If possible, solve each triangle for the unknown side. Round to the nearest tenth.

$\gamma = 115° , a = 18 , b = 23$

Assume $\alpha$ is opposite side $a , \beta$ is opposite side $b ,$ and $\gamma$ is opposite side $c .$ If possible, solve each triangle for the unknown side. Round 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 $c .$ If possible, solve each triangle for the unknown side. Round to the nearest tenth.

$\alpha = 43.1° , a = 184.2 , b = 242.8$

Assume $\alpha$ is opposite side $a , \beta$ is opposite side $b ,$ and $\gamma$ is opposite side $c .$ If possible, solve each triangle for the unknown side. Round to the nearest tenth.

$\beta = 50° , a = 105 , b = 45 {}$

Use the Law of Cosines to solve for the missing angle of the oblique triangle. Round to the nearest tenth.

$a = 14 , b = 13 , c = 20 ;$ find angle $C .$

Use the Law of Cosines to solve for the missing angle of the oblique triangle. Round to the nearest tenth.

$a = 13 , b = 22 , c = 28 ;$ find angle $A .$

Solve the triangle. Round to the nearest tenth.

$A = 35° , b = 8 , c = 11$

Solve the triangle. Round to the nearest tenth.

$C = 121° , a = 21 , b = 37$

Solve the triangle. Round to the nearest tenth.

$a = 3.1 , b = 3.5 , c = 5$

Use Heron’s formula to find the area of the triangle. Round to the nearest hundredth.

Find the area of a triangle with sides of length 18 in, 21 in, and 32 in. Round to the nearest tenth.

Use Heron’s formula to find the area of the triangle. Round to the nearest hundredth.

$a = \frac{1}{2} \text{m} , b = \frac{1}{3} \text{m} , c = \frac{1}{4} \text{m}$

Use Heron’s formula to find the area of the triangle. Round to the nearest hundredth.

$a = 1.6 \textrm{ }\text{yd} , b = 2.6 \textrm{ }\text{yd} , c = 4.1 \textrm{ }\text{yd}$

The sides of a parallelogram are 11 feet and 17 feet. The longer diagonal is 22 feet. Find the length of the shorter diagonal.

A pilot flies in a straight path for 1 hour 30 min. She then makes a course correction, heading 10° to the right of her original course, and flies 2 hours in the new direction. If she maintains a constant speed of 680 miles per hour, how far is she from her starting position?

Two airplanes take off in different directions. One travels 300 mph due west and the other travels 25° north of west at 420 mph. After 90 minutes, how far apart are they, assuming they are flying at the same altitude?

The four sequential sides of a quadrilateral have lengths 4.5 cm, 7.9 cm, 9.4 cm, and 12.9 cm. The angle between the two smallest sides is 117°. What is the area of this quadrilateral?

Find the area of a triangular piece of land that measures 30 feet on one side and 42 feet on another; the included angle measures 132°. Round to the nearest whole square foot.