Exercises: Cosine rule

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

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

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

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

A parallelogram has two adjacent sides measuring 4 cm and 6 cm. 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 $B$, given that from an observation point $C$ the distances measured are $CA = 15$ m and $CB = 20$ m, and the visual angle $ACB$ is $40^\circ$.

In an isosceles triangle $ABC$, sides $a$ and $b$ measure 6 cm each, and the non-equal angle is $C=30^\circ$. Calculate the exact value of side $c$, leaving the answer in surd form.

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

Two aeroplanes depart from the same airport at the same time. Aeroplane A flies at $800$ km/h on a bearing of $040^\circ$ and aeroplane B flies at $650$ km/h 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 $C$. He measures the distances $CA = 125.5$ m and $CB = 180.2$ m, and the angle $ACB = 75^\circ$. Calculate the distance $AB$.

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

The diagonals of a parallelogram measure 12 cm and 16 cm, and intersect forming an angle of $65^\circ$. Calculate the lengths of the four sides of the parallelogram.

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

On a soccer field, a player is 25 m from one goalpost and 28 m from the other goalpost. Given that the goal has a regulation width of 7.32 m, what angle does the goal subtend at the player's position?

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

In a circle of radius 10 cm, a chord subtends a central angle of $100^\circ$. Calculate the length of the chord using the cosine rule 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 relation holds:

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

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

A cyclic quadrilateral $ABCD$ has side lengths of $AB=3$ cm, $BC=4$ cm, $CD=5$ cm, and $DA=6$ cm. Taking into account the properties of opposite angles in a cyclic quadrilateral, set up a system of equations using the cosine rule in triangles $ABC$ and $ADC$ to calculate the exact length of diagonal $AC$.

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

A triangle has a fixed base $c = 10$ cm and a constant perimetre $p = 26$ cm. If 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 cm and 8 cm intersect at two points. The distance between their centers is 10 cm. Calculate the length of the common chord of both circles. (Hint: Use the cosine rule to find the central angle in the triangle formed by the radii and the line joining the centers, and from there calculate the altitude of the triangle).

Calculate the acute angle formed by two space diagonals of a cube. To do this, express the length of the diagonal in terms of edge $a$ and apply the cosine rule in the triangle formed by the halves of two diagonals and one edge of the cube.