The 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 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 $c$.

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.

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

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

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.

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.

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.

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 $C$.

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

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

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.

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 $40^\circ$.

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 $AB$.

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.

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.

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.