Solving non-right-angled triangles

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

The three sides of a triangle measure $a = 3$ m, $b = 5$ m and $c = 7$ m. Determine the size of the largest angle in this triangle.

In a triangle, the following data are known: $a = 12$ m, $b = 8$ m and angle $A = 120^\circ$. Calculate the size of angle $B$.

Seán has to calculate the length of side $a$ of a triangle for which he knows $b = 5\text{ cm}$, $c = 8\text{ cm}$, and the included angle $A = 120^\circ$. In his notebook, he writes the following resolution:

1) Point out the calculation error Seán made when working with the signs.

2) Without having to redo the calculations yet, explain why a result of $a = 7\text{ cm}$ is geometrically impossible in this triangle (relate the magnitude of angle $A$ to the relative measures of the sides).

3) Correct the procedure and calculate the correct value of side $a$ rounded to two decimal places.

In the design of an articulated robotic arm, the first link measures $40$ cm and the second measures $35$ cm. If the angle formed by the first link with the horizontal base is $50^\circ$ and the end of the second link touches the horizontal base exactly, calculate the two possible distances from the origin of the arm to the contact point on the base.

In a triangle $ABC$, the angles $A$ and $B$ measure $30^\circ$ and $105^\circ$ respectively. If the side opposite angle $A$ has a length of $a = 8$ cm, find the length of side $c$ (opposite angle $C$).

In an isosceles triangle, the non-equal side (the base) measures $12$ m and the equal angles measure $75^\circ$ each. Determine the length of the equal sides using the sine rule.

In a triangle $ABC$, side $a$ measures $10$ cm and the opposite angle $A$ measures $45^\circ$. Calculate the radius $R$ of the circumscribed circle of the triangle.

Find the area of a triangle $ABC$ for which the sides $b = 14$ cm, $c = 20$ cm and the angle between them, $A = 60^\circ$, are known.

The adjacent sides of a parallelogram measure $6$ cm and $10$ cm, forming an obtuse angle of $120^\circ$. Calculate the length of the longer diagonal of the parallelogram.

In a triangle, we know sides $a = 6\text{ cm}$ and $b = 6\text{ cm}$, and the included angle between them $C = 62^\circ$.

1) Estimate: before performing any numerical calculations with formulas, reason whether the third side $c$ will be slightly greater than, less than, or equal to $6\text{ cm}$. To justify this, imagine what the figure would look like if the angle were exactly $60^\circ$ (equilateral triangle) and analyze what happens when opening the angle two degrees more.

2) Check: calculate the exact length of side $c$ using the law of cosines (round to two decimal places).

3) Assess whether the obtained result confirms your qualitative estimate and calculate the absolute error you would have made if you had directly approximated $c \approx 6\text{ cm}$.

In a triangle $ABC$ with sides $a=13$ cm, $b=14$ cm and $c=15$ cm, the circumscribed circle of radius $R$ and the inscribed circle of radius $r$ are drawn. Calculate the area of the triangle using Heron's formula and, from it, obtain the exact values of $R$ and $r$.

Two ships leave a port simultaneously along straight routes that form an angle of $75^\circ$ with each other. The first ship sails at a constant speed of $15$ knots and the second at $20$ knots. Calculate the distance in nautical miles separating the two ships after $2$ hours of navigation.

To measure the height $h$ of a wind turbine, we stand at a point $A$ on the ground and measure the angle of elevation to the top point, obtaining $40^\circ$. We move $30$ metres in a straight line towards the base of the wind turbine to reach a point $B$, from which the new angle of elevation is $55^\circ$. Calculate the height of the wind turbine.

A surveillance drone is at a point $C$ observing a straight stretch of road bounded by two cities $A$ and $B$ that are $5$ km apart. The angles of depression from the drone to the cities are such that in the triangle formed, the angle at $A$ is $35^\circ$ and the angle at $B$ is $62^\circ$. Calculate the altitude at which the drone is flying above the road.

A piece of farmland has the shape of a quadrilateral $ABCD$. The lengths of its sides are $AB = 50$ m, $BC = 40$ m, $CD = 30$ m and $DA = 60$ m. Knowing that the angle at vertex $B$ is $70^\circ$, calculate the total area of the land by dividing it into two triangles.

A regular heptagon (a polygon with $7$ equal sides) is inscribed in a circle of radius $10$ cm. Using the cosine rule and/or sine rule in the central triangles, calculate the perimeter and area of the heptagon.

The three sides of a triangle are known: $a = 4\text{ cm}$, $b = 7\text{ cm}$, and $c = 10\text{ cm}$. To solve it completely, you must find its three angles. Two classmates propose different paths to calculate the second angle after having obtained the first:

• Aoife: "First, I calculate the smallest angle, $A$, with the law of cosines. Then, to make the calculations faster and simpler, I calculate the largest angle, $C$, using the law of sines ($\frac{a}{\sin A} = \frac{c}{\sin C}$), and finally $B = 180^\circ - A - C$".

• Seán: "It is dangerous to calculate $C$ using the law of sines. It is much safer to calculate $C$ directly with the law of cosines, or use the law of sines to calculate $B$, but never for $C$".

1) Mathematically justify Seán's objection: what limitation does the $\arcsin$ function on the calculator have when looking for angles that could be obtuse?

2) Check the flaw in Aoife's method: calculate angle $A$ using the law of cosines and show what incorrect value for $C$ the law of sines would yield if applied directly.

3) Apply the safe path proposed by Seán to calculate the three angles of the triangle accurately.

Prove geometrically or using trigonometric rules (sine and cosine rules) that in any triangle $ABC$ the projection formula holds: $a = b \cos C + c \cos B$. Next, use this identity together with the sine rule to prove the sine compound angle formula: $\sin(B+C) = \sin B \cos C + \cos B \sin C$.

Given the triangle $ABC$ with sides $a = 7$ cm, $b = 5$ cm and $c = 8$ cm, the internal angle bisector of angle $C$ is drawn, intersecting side $AB$ at point $D$. Applying the cosine rule to find angle $C$ and using the properties of areas or the sine rule, calculate the exact length of the angle bisector segment $CD$ (denoted as $x$).

You can do it on paper and send a photo with your answer.

A regular square-based pyramid has a base side length of $100$ m. Knowing that the dihedral angle formed by two adjacent triangular faces is $105^\circ$, set up the geometry of the appropriate cross-section and calculate the total height of the pyramid.