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

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 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.

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 $30$ meters 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.

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.

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.

Two tugboats pull a barge using two cables that form an angle of $45^\circ$ with each other. If the tensions exerted by the cables are $4500$ N and $6000$ N, find the magnitude of the resultant force and the angle this resultant forms with the cable exerting the force of $6000$ N.

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 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.

A solar panel $2$ meters long is to be installed on a pitched roof that has an inclination of $15^\circ$ to the horizontal. To maximize its performance, the panel must form an angle of $35^\circ$ with respect to the horizontal. Calculate the length $x$ of the vertical support that must be placed at the upper end of the panel to connect it to the roof.

The sides of a triangle are given by the expressions $x$, $x+2$ and $x+4$ centimeters. Knowing that the largest angle of the triangle measures $120^\circ$, set up the corresponding equation, find the exact value of $x$ and calculate the perimeter of the triangle.

In a triangle $ABC$, it is known that side $a = 10$ cm, angle $B = 30^\circ$ and that the total area of the triangle is $20\sqrt{3}$ cm$^2$. Set up a system of equations to find the exact length of sides $b$ and $c$, and find the size of angles $A$ and $C$.

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: $\sin(B+C) = \sin B \cos C + \cos B \sin C$.

The hands of a wall clock measure $12$ cm (the minute hand) and $8$ cm (the hour hand). Calculate the exact distance in centimeters separating the tips of both hands when the clock shows exactly 10:10. Take into account the proportional displacement of the hour hand.

Two aeroplanes fly at the same altitude. Aeroplane 1 is at point $A$ and flies North at a constant speed of $800$ km/h. Aeroplane 2 is at point $B$, located exactly $200$ km North-East ($45^\circ$ from North) of point $A$. If aeroplane 2 flies West at a constant speed $v$ and both collide at point $C$, determine the value of speed $v$ and the time it takes for them to collide.

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 $r$.

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.