Exercises: Cosine rule

In a triangle $ABC$, the sides $b=5$ in, $c=8$ in and the included angle $A=60^\circ$ are known. Calculate the length of the side $a$.

In a triangle $ABC$, the lengths of the sides are $a=3$ in, $b=5$ in and $c=7$ in. Calculate the size of the angle $C$.

In a triangle $ABC$, the sides measure $a=10$ in, $b=12$ in and the included angle is $C=45^\circ$. Set up the Law of Cosines to calculate the exact value of the side $c$.

Determine the measure of the angle $B$ in a triangle whose sides are $a=7$ in, $b=8$ in and $c=9$ in.

A parallelogram has two adjacent sides measuring 4 in and 6 in. If the included angle between them is $120^\circ$, calculate the length of the longer diagonal.

Calculate the distance between two inaccessible points $A$ and $B$, knowing that from an observation point $C$ the distances $CA = 15$ ft and $CB = 20$ ft have been measured, and the visual angle $ACB$ is $40^\circ$.

In an isosceles triangle $ABC$, the sides $a$ and $b$ measure 6 in each, and the unequal angle $C=30^\circ$. Calculate the exact value of the side $c$, leaving the result in radical form.

Determine the largest angle of a triangle whose sides measure 4 in, 5 in and 6 in.

Two planes leave the same airport at the same time. Plane A flies at $800$ mph on a bearing of $040^\circ$ and plane B flies at $650$ mph on a bearing of $115^\circ$. What straight-line distance separates them after 1.5 hours of flight?

To measure the distance across a lake between points $A$ and $B$, a surveyor stands at point $C$. They measure the distances $CA = 125,5$ ft and $CB = 180,2$ ft, and the angle $ACB = 75^\circ$. Calculate the distance $AB$.

Two forces of 45 N and 60 N act on the same material point, forming an angle of $50^\circ$ between them. Calculate the magnitude of the resultant force.

The diagonals of a rhomboid measure 12 in and 16 in, and intersect forming an angle of $65^\circ$. Calculate the length of the four sides of the rhomboid.

A triangular roof truss has sides of 5 ft, 7 ft and 9 ft. Calculate the angle opposite the longest side.

On a soccer field, a player is 25 ft from one goalpost and 28 ft from the other post. Knowing that the goal has a regulation width of 7.32 ft, at what angle does the player see the goal from end to end?

Two satellites orbit the Earth. From a tracking station, the straight-line distances to the satellites are 400 miles and 600 miles, and the angle between their lines of sight is $35^\circ$. How far apart are the satellites?

In a circle of 10 in radius, a chord subtends a central angle of $100^\circ$. Calculate the length of the chord using the Law of Cosines in the triangle formed by the radii and the chord.

In a triangle, the lengths of the sides are $x$, $x+2$ and $x+4$. If the largest angle measures $120^\circ$, set up the corresponding equation, determine the value of $x$ and the exact lengths of the three sides.

Prove algebraically that in any triangle $ABC$ the following relationship holds:

$a^2 + b^2 + c^2 = 2(ab \cos C + bc \cos A + ca \cos B)$

In a triangle $ABC$, the sides measure $a=8$ in, $b=10$ in and $c=12$ in. Calculate the exact length of the median $m_a$ drawn from the vertex $A$ to the midpoint of the side $a$.

An inscribable (cyclic) quadrilateral $ABCD$ has sides of lengths $AB=3$ in, $BC=4$ in, $CD=5$ in and $DA=6$ in. Taking into account the properties of opposite angles in a cyclic quadrilateral, set up a system using the Law of Cosines in the triangles $ABC$ and $ADC$ to calculate the exact length of the diagonal $AC$.

If the algebraic relation $a^2 = b^2 + c^2 + \sqrt{2}bc$ holds in a triangle $ABC$, determine the exact measure of the angle $A$ without using a calculator.

A triangle has a fixed base of $c = 10$ in and a constant perimeter of $p = 26$ in. If the angle $C$ opposite the base measures $60^\circ$, set up a system of equations and find the exact lengths of the other two sides $a$ and $b$.

Two circles of radii 5 in and 8 in intersect at two points. The distance between their centers is 10 in. Calculate the length of the common chord to both circles. (Hint: Use the Law of Cosines to find the central angle in the triangle formed by the radii and the line of centers, and from there obtain the height of the triangle).

Calculate the acute angle formed by two main diagonals of a cube. To do this, express the length of the diagonal in terms of the edge $a$ and apply the Law of Cosines in the triangle formed by the halves of two diagonals and an edge of the cube.