"Solving Non-Right Triangles: Law of Sines and Law of Cosines"

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

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

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

In an isosceles triangle, the unequal side (the base) measures $12$ ft and the equal angles measure $75^\circ$ each. Determine the length of the equal sides using the Law of Sines.

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

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

Find the area of a triangle $ABC$ of which we know the sides $b = 14$ in, $c = 20$ in and the angle they form, $A = 60^\circ$.

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

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

In a triangle $ABC$ with sides $a=13$ in., $b=14$ in., and $c=15$ in., 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 on straight routes that form an angle of $75^\circ$ between them. The first ship sails at a constant speed of $15$ knots and the second at $20$ knots. Calculate the distance in nautical miles that will separate the two ships after $2$ hours of sailing.

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

Two airplanes fly at the same altitude. Airplane 1 is at point $A$ and flies toward the north at a constant speed of $800$ mph. Airplane 2 is at point $B$, located exactly $200$ miles to the northeast ($45^\circ$ with respect to the north) of point $A$. If airplane 2 flies toward the west at a constant speed $v$ and both collide at a point $C$, determine the value of the speed $v$ and the time they take to collide.

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 highest point, obtaining $40^\circ$. We advance $30$ feet in a straight line towards the base of the wind turbine until reaching a point $B$, from where the new angle of elevation is $55^\circ$. Calculate the height of the wind turbine.

A surveillance drone is located at a point $C$ observing a straight stretch of road delimited by two cities $A$ and $B$ that are $5$ miles 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 with respect to the road.

A solar panel $2$ feet long is to be installed on a flat roof that has an inclination of $15^\circ$ with respect 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 join it to the roof.

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

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

A regular heptagon (polygon with $7$ equal sides) is inscribed in a circle of radius $10$ in. Using the Law of Cosines and/or Law of Sines in the central triangles, calculate the perimeter and the area of this heptagon.

Two tugboats pull a barge using two cables that form an angle of $45^\circ$ between them. 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.

The sides of a triangle are given by the expressions $x$, $x+2$ and $x+4$ inches. 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.

Prove geometrically or by using trigonometric laws (Law of Sines and Law of Cosines) that in any triangle $ABC$ the projection relation holds: $a = b \cos C + c \cos B$. Then, use this identity together with the Law of Sines to prove the formula for the sine of the sum of two angles: $\sin(B+C) = \sin B \cos C + \cos B \sin C$.

Given triangle $ABC$ with sides $a = 7$ in., $b = 5$ in., and $c = 8$ in. The interior angle bisector of angle $C$ is drawn, which intersects side $AB$ at point $D$. Applying the Law of Cosines to find the angle $C$ and using the properties of areas or the Law of Sines, calculate the exact length of the bisector segment $CD$ (denoted as $x$).

A regular square-based pyramid has a base edge of $100$ ft. 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.