# The Law of Cosines - practice problems

The law of cosines is a mathematical formula used in trigonometry that relates the sides of a triangle to the cosine of one of its angles. Specifically, it states that in any triangle with sides a, b, and c and angles A, B, and C opposite to those sides, the following equation holds:c

^{2}= a

^{2}+ b

^{2}- 2ab * cos(C)

where c is the length of the side opposite angle C, a is the length of the side opposite angle A, and b is the length of the side opposite angle B. The formula is also known as "cosine formula" or "cosine rule".

The law of cosines can be used to find the length of a side of a triangle when the lengths of the other two sides and the angle opposite the unknown side are known. It can also be used to find an angle of a triangle when the lengths of all three sides are known.

It is particularly useful in solving triangles that are not right triangles, where the Pythagorean theorem can not be applied.

The law of cosines can also be useful in solving problems involving distance and navigation, like finding the distance between two points on the surface of the earth, or finding the distance between two celestial bodies. It is also used in physics and engineering, such as in calculating the force required to bend a beam of a certain length and material properties.

Direction: Solve each problem carefully and show your solution in each item.

#### Number of problems found: 64

- Laws

From which law directly follows the validity of Pythagoras' theorem in the right triangle? ... - Scalene triangle

Solve the triangle: A = 50°, b = 13, c = 6 - Greatest angle

Calculate the greatest triangle angle with sides 197, 208, 299. - Triangle 2668

The triangle ABC has side lengths a = 14 cm, b = 20 cm, c = 7.5 cm. Find the sizes of the angles and the area of this triangle. - Largest angle of the triangle

Calculate the largest angle of the triangle whose sides have the sizes: 2a, 3/2a, 3a - Triangle ABC

Triangle ABC has side lengths m-1, m-2, and m-3. What has to be m to be a triangle a) rectangular b) acute-angled? - Diagonals

Calculate the length of the rhombus's diagonals if its side is long 5 and one of its internal angles is 80°. - Diagonals in diamond

In the rhombus is given a = 160 cm, alpha = 60 degrees. Calculate the length of the diagonals. - Side c

In △ABC a=6, b=6 and ∠C=110°. Calculate the length of the side c. - The angle of view

Determine the angle of view at which the observer sees a rod 16 m long when it is 18 m from one end and 27 m from the other. - Medians of isosceles triangle

The isosceles triangle has a base ABC |AB| = 16 cm and a 10 cm long arm. What is the length of the medians? - A rhombus

A rhombus has sides of the length of 10 cm, and the angle between two adjacent sides is 76 degrees. Find the length of the longer diagonal of the rhombus. - Find the area

Find the area of the triangle with the given measurements. Round the solution to the nearest hundredth if necessary. A = 50°, b = 30 ft, c = 14 ft - Heron backlaw

Calculate the missing side in a triangle with sides 25 and 13 and area 152. - Calculate 2

Calculate the largest angle of the triangle whose sides are 5.2cm, 3.6cm, and 2.1cm - Triangle and its heights

Calculate the length of the sides of the triangle ABC if v_{a}=5 cm, v_{b}=7 cm and side b are 5 cm shorter than side a. - Big tower

From the tower, which is 15 m high, and 30 m from the river, the river's width appeared at an angle of 15°. How wide is the river in this place? - Cosine

Cosine and sine theorem: Calculate all missing values from triangle ABC. c = 2.9 cm; β = 28°; γ = 14° α =? °; a =? cm; b =? cm - ABCD

AC= 40cm , angle DAB=38 , angle DCB=58 , angle DBC=90 , DB is perpendicular on AC , find BD and AD - Angles by cosine law

Calculate the size of the angles of the triangle ABC if it is given by: a = 3 cm; b = 5 cm; c = 7 cm (use the sine and cosine theorem).

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