# The Law of Cosines - practice problems - page 2 of 4

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: 68

- Four sides of trapezoid

In the trapezoid ABCD is |AB| = 73.6 mm; |BC| = 57 mm; |CD| = 60 mm; |AD| = 58.6 mm. Calculate the size of its interior angles. - Triangle 75

Triangle ABC has angle C bisected and intersected AB at D. Angle A measures 20 degrees, and angle B measures 40 degrees. The question is to determine AB-AC if length AD=1. - Two forces 3

Two forces with magnitudes 8 Newtons and 15 Newtons act at a point. If the resultant force is 17 Newtons, find the angle between the forces. - Triangle from median

Calculate the perimeter, area, and magnitudes of the triangle ABC's remaining angles: a = 8.4; β = 105° 35 '; and median ta = 12.5. - Parallelogram 65954

In the parallelogram ABCD AB = 8, BC = 5, BD = 7. Calculate the magnitude of the angle α = ∠DAB (in degrees). - The farmer

The farmer sees the back fence of the land, which is 50 m long at a viewing angle of 30 degrees. It is 92 m away from one end of the fence. How far is it from the other end of the fence? - Diagonals in diamond

In the rhombus is given a = 160 cm, alpha = 60 degrees. Calculate the length of the diagonals. - Observer 64354

At what angle of view does an object 70 m long appear to the observer, 50 m away from one end, and 80 m from the other end? - A triangle 7

A triangle lot has the dimensions a=15m, b=10m, and c=20m. What is the measure of the angle between the sides of b and c? - SAS calculation

Given the triangle ABC, if side b is 31 ft., side c is 22 ft., and angle A is 47°, find side a. Please round to one decimal. - The pond

We can see the pond at an angle of 65°37'. Its endpoints are 155 m and 177 m away from the observer. What is the width of the pond? - Three 235

Three houses form a triangular shape. House A is 50 feet from house C and house B is 60 feet from house C. The measure is angle ABC is 80 degrees. Draw a picture and find the distance between A and B. - Children playground

The playground has a trapezoid shape, and the parallel sides have a length of 36 m and 21 m. The remaining two sides are 14 m long and 16 m long. Find the size of the inner trapezoid angles. - Two chords

From the point on the circle with a diameter of 8 cm, two identical chords are led, which form an angle of 60°. Calculate the length of these chords. - Viewing angle

The observer sees a straight fence 60 m long at a viewing angle of 30°. It is 102 m away from one end of the enclosure. How far is the observer from the other end of the enclosure? - Calculate triangle

In the triangle, ABC, calculate the sizes of all heights, angles, perimeters, and area if given a=40cm, b=57cm, and c=59cm. - Piece of a wire

A piece of wire is bent into the shape of a triangle. Two sides have lengths of 24 inches and 21 inches. The angle between these two sides is 55°. What is the length of the third side to the nearest hundredth of an inch? A: The length of the third side is - Triangle's centroid

In the triangle ABC the given lengths of its medians tc = 9, ta = 6. Let T be the intersection of the medians (triangle's centroid), and the point S is the center of the side BC. The magnitude of the CTS angle is 60°. Calculate the length of the BC side t - Parallelogram 65334

In a parallelogram, the sum of the lengths of the sides a+b = 234. The angle subtended by the sides a and b is 60°. The diagonal size against the given angle of 60° is u=162. Calculate the sides of the parallelogram, its perimeter, and its area. - Circumscribed 81759

In triangle ABC, we know a = 4 cm, b = 6 cm, γ = 60°. Calculate the area and radius of the inscribed and circumscribed circle.

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