# 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: 78

- 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). - Isosceles 83157

Using the cosine theorem, prove that in an isosceles triangle ABC with base AB, c=2a cos α. - Isosceles 7929

ABCD isosceles trapezoid. A = 6cm, e = 7cm and delta angle = 105 °. Calculate the remaining pages.

- 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. - Sides ratio and angles

In triangle ABC, you know the ratio of side lengths a:b:c=3:4:6. Calculate the angle sizes of triangle ABC. - 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? - Largest angle of the triangle

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

- 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? - 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?

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