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.

Aoife needs to calculate the distance between two beacons, $B$ and $C$, knowing the distances from her observation position $A$ ($b = AC = 6\text{ m}$ and $c = AB = 10\text{ m}$) and the angle formed by the lines of sight, $\widehat{A} = 120^\circ$. On her exam, she writes the following steps:

1. a) Without redoing the numerical operations at first, reason geometrically why Aoife's result cannot be correct by comparing the figure with a right-angled triangle with legs of $6\text{ m}$ and $10\text{ m}$.
2. b) Pinpoint the specific mathematical mistake made by Aoife, explain which property of trigonometric ratios she did not take into account, and calculate the correct value of side $a$ rounded to two decimal places.

Two straight sections of an officially approved trail in the region of Cazorla measure $b = 7\text{ km}$ and $c = 8\text{ km}$, and branch off forming an angle $\widehat{A} = 58^\circ$. The park committee wants to build a direct, straight path connecting the two free ends (side $a$).

1. a) Before carrying out any calculation with a calculator: considering that $58^\circ$ is very close to $60^\circ$ and that the sides are almost equal ($7\text{ km}$ and $8\text{ km}$), make a reasoned estimate of approximately how long side $a$ will be. Between which two extreme values must it necessarily lie according to the triangle inequality?
2. b) Apply the law of cosines with $\cos(58^\circ) \approx 0.530$ to calculate the exact length of $a$, rounding to two decimal places.
3. c) Compare your previous estimate with the obtained result and justify whether the value is physically plausible.

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 football pitch, 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 with known sides $a = 4\text{ cm}$, $b = 6\text{ cm}$, and $c = 9\text{ cm}$, the goal is to find the measure of its three angles by combining the law of cosines with the law of sines to save calculation time. Two classmates propose different strategies for the first step:

• Seán proposes: "Let's first calculate the smallest angle, $\widehat{A}$, using the law of cosines and, then, solve for the largest angle, $\widehat{C}$, using the law of sines."

• Aoife counters: "It is much safer to first calculate the largest angle, $\widehat{C}$, using the law of cosines and, from it, find $\widehat{A}$ using the law of sines."

1. a) Mathematically justify which of the two methods is preferable and explain what mathematical risk or ambiguity the other entails when using the inverse function $\operatorname{arcsen}$ on the interval $[0^\circ, 180^\circ]$.
2. b) Following the correct strategy, determine with reasoning whether the triangle is acute-angled or obtuse-angled by examining only the sign of $\cos \widehat{C}$, and calculate the value of the three angles of the triangle rounded to the nearest degree.

During fire surveillance operations by the Irish Forest Service in the Wicklow Mountains, a control tower located at observation point $P$ spots two columns of smoke at points inaccessible to each other, $F_1$ and $F_2$.

Using the station's laser rangefinder and goniometer, the following measurements are obtained:

• Distance to the first column: $d(P, F_1) = 3.4\text{ km}$.

• Distance to the second column: $d(P, F_2) = 5.2\text{ km}$.

• Horizontal angle formed by both lines of sight: $\widehat{F_1 P F_2} = 115^\circ$.

1. a) Model the layout of the three points using a triangle, identify the known elements, and calculate the straight-line distance separating both smoke sources (round to two decimal places).
2. b) The helitack crew has a $7\text{ km}$ emergency rapid-deployment hose. Justify whether it would be feasible with this equipment to connect both sources directly without having to pass through observation point $P$.

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 centres 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 centres, 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.

A triangular steel truss is designed for the roof of an industrial warehouse. Two of the structural bars have fixed lengths of $b = 5\text{ m}$ and $c = 8\text{ m}$.

1. a) Write an inequality in terms of the length of the third side, $a$, that guarantees that the angle $\widehat{A}$ enclosed between the two given bars is strictly obtuse ($90^\circ < \widehat{A} < 180^\circ$). Determine the interval of possible real values for the length $a$.
2. b) Now assume that the specifications are modified by fixing bar $a = 7\text{ m}$ and angle $\widehat{A} = 60^\circ$, keeping $c = 8\text{ m}$, but leaving the length of bar $b$ as an unknown to be dimensioned. Set up the quadratic equation that $b$ must satisfy by applying the law of cosines.
3. c) Solve the obtained equation, check whether the solutions found are geometrically viable using the triangle inequality, and explain why this case yields two different and valid triangular configurations with the same initial data.

During docking manoeuvres at the outer port of Ferrol, two tugboats pull simultaneously on a merchant ship, applying coplanar forces of fixed magnitudes $|\vec{F}_1| = 50\text{ kN}$ and $|\vec{F}_2| = 70\text{ kN}$. The tow cables form an angle $\alpha = 40^\circ$ between each other.

1. a) Reasonably estimate between which two extreme bounds the magnitude of the resultant force $\vec{R} = \vec{F}_1 + \vec{F}_2$ must lie, considering the limiting situations where the cables pull in the same direction and sense, or in opposite senses.
2. b) When representing the vector addition using the parallelogram rule, a triangle is obtained whose adjacent sides are $|\vec{F}_1|$ and $|\vec{F}_2|$, and whose third side represents the resultant vector $\vec{R}$. Justify geometrically why the angle opposite to $\vec{R}$ in that triangle measures $180^\circ - \alpha = 140^\circ$, and not $40^\circ$.
3. c) Apply the law of cosines to determine the exact magnitude of the resultant force $\vec{R}$ (in $\text{kN}$, rounded to one decimal place) and verify the accuracy of your estimate.
4. d) Due to structural safety requirements of the hull, the total resultant force cannot exceed $105\text{ kN}$. Set up the corresponding inequality and deduce the minimum angle $\alpha$ that the tugboat cables must form in order not to exceed this limit.