Calculating Side Lengths with Sines & Cosines in Triangles

A triangle has two sides measuring 8 cm and 10 cm, with an included angle of 45 degrees. Calculate the length of the third side using the law of cosines.

• c ≈ 5.00 cm

• c ≈ 9.50 cm

• c ≈ 10.50 cm

• c ≈ 7.07 cm

In triangle ABC, angle A measures 30 degrees, angle B measures 60 degrees, and side a (opposite angle A) is 12 m. Use the law of sines to find the length of side b (opposite angle B).

• 12 m

• 12√3 m

• 15 m

• 10 m

A ladder leans against a wall, forming a 75-degree angle with the ground. If the foot of the ladder is 4 m from the wall, how long is the ladder? Use the law of sines to find the length.

• 15.59 m

• 18.40 m

• 12.75 m

• 10.25 m

A surveyor measures a triangular plot of land. One angle is 50 degrees, and the sides opposite to this angle are 20 m and 30 m. Calculate the third side using the law of cosines.

• 15.0 m

• 35.0 m

• 25.0 m

• 30.0 m

In triangle XYZ, angle X is 45 degrees, side x is 10 m, and side y is 15 m. Use the law of sines to find angle Y.

• 75 degrees

• 30 degrees

• 90 degrees

• 60.3 degrees

A boat sails from point A to point B, then to point C, forming a triangle. If angle A is 70 degrees, side AB is 15 km, and side AC is 10 km, find the length of side BC using the law of cosines.

• 16.2 km

• 14.9 km

• 18.0 km

• 12.5 km

A triangle has sides of lengths 7 cm and 9 cm, with an angle of 60 degrees between them. Calculate the length of the third side using the law of cosines.

• 7.50 cm

• 8.19 cm

• 10.00 cm

• 6.50 cm

In triangle DEF, angle D is 40 degrees, side d is 25 m, and side e is 30 m. Use the law of sines to find angle E.

• 60 degrees

• 50.5 degrees

• 70 degrees

• 30 degrees

A kite is flying at an angle of elevation of 30 degrees from a point on the ground. If the string is 50 m long, how high is the kite from the ground? Use trigonometric ratios to find the height.

• 35 meters

• 40 meters

• 15 meters

• 25 meters

A triangular park has sides measuring 50 m, 70 m, and an angle of 60 degrees between them. Calculate the area of the park using the law of sines and the formula for area.

• 800 m²

• 980 m²

• 1500 m²

• 1217.5 m²