Law of Sines

In a triangle $ABC$, the angles $A = 45^\circ$ and $B = 60^\circ$ are known, as well as the length of side $a = 12$ cm. Calculate the length of side $b$ using the sine rule.

In any triangle, side $c$ measures $24$ cm and the opposite angle $C$ measures $30^\circ$. Calculate the radius $R$ of the circumcircle of this triangle.

A triangle $PQR$ has angles $P = 50^\circ$ and $Q = 75^\circ$. If side $r = 20$ m, calculate the length of side $p$.

Given a triangle $ABC$ in which sides $a = 15$ cm and $b = 10$ cm and the obtuse angle $A = 120^\circ$ are known, determine the value of angle $B$.

Determine whether it is possible to construct a triangle $ABC$ with the following data: $a = 8$ cm, $b = 14$ cm, and $A = 45^\circ$. Justify your answer mathematically using the sine rule.

In triangle $ABC$, sides $a = 10$ cm, $b = 14$ cm, and angle $A = 30^\circ$ are known. Calculate the two possible values for angle $B$ and the corresponding lengths of side $c$ for each case.

In the construction of a triangle $ABC$, side $a = 10$ cm and angle $A = 45^\circ$ are fixed. Side $b$ has a variable length $x$ cm. Determine, giving reasons, the strict interval of real values of $x$ for which there exist exactly two distinct triangles satisfying these conditions simultaneously.

Two coastguard stations, $A$ and $B$, are separated by a distance of $50$ km along a straight coastline. A ship $S$ emits a distress signal. Station $A$ detects the signal forming an angle of $40^\circ$ with the coastline, while station $B$ detects it with an angle of $65^\circ$. Calculate the straight-line distance from each station to the ship.

A cable car ascends in a straight line from a base station $A$ to the summit $C$ of a mountain. From a point $D$, situated $200$ m behind $A$ in the same horizontal plane, the angle of elevation to $C$ is $22^\circ$. From station $A$, the angle of elevation to the summit is $35^\circ$. Calculate the length of the cable car wire $AC$ by solving triangle $ADC$.

An aircraft flies in a straight line at a constant altitude. A stationary observer on the ground initially spots it with an angle of elevation of $30^\circ$. Exactly $10$ seconds later, the aircraft, flying directly towards the observer's vertical line, is spotted with an angle of elevation of $45^\circ$. If the constant speed of the aircraft is $600$ km/h, calculate its altitude by solving the triangle formed by the observer and the two positions of the aircraft using the sine rule.

Calculate the perimeter of a regular heptagon (a 7-sided polygon) inscribed in a circle of radius $R = 10$ cm. Use the law of sines on the isosceles triangles formed from the center of the polygon.

A triangular plot of land has two angles measuring $60^\circ$ and $75^\circ$. The side between these two angles is bounded by a straight wall measuring $120$ m. Calculate the total area of the plot.

The slope of a mountain has a constant inclination of $15^\circ$ relative to the horizontal plane. Two points $A$ and $B$ are on the slope, with $B$ situated $50$ m further up than $A$ along the slope. A vertical telecommunications pole is installed at point $B$. From point $A$, the angle of elevation to the top of the pole is $42^\circ$ measured from the horizontal. Calculate the exact height of the pole.

In a triangle $ABC$, the proportion $\frac{a}{\sin A} = 24$ holds. If the sum of the angles $A + B = 120^\circ$ and side $a = 12\sqrt{3}$ cm, determine the exact length of side $c$.

In a triangle $ABC$, angle $A$ is exactly twice angle $B$ ($A = 2B$). Prove, using the sine rule and double-angle trigonometric identities, that the lengths of the sides satisfy the relation $a = 2b \cos B$.

Prove that the radius $R$ of the circumcircle of a triangle $ABC$ is directly related to the area of triangle $S$ by the formula:

$R = \frac{a \times b \times c}{4S}$

(Hint: Express the area of the triangle using the sine of one of its angles and combine it with the sine rule).

In a triangle $ABC$, the lengths of sides $a$, $b$, and $c$ are in arithmetic progression, satisfying the relation $2b = a + c$. Prove that the sines of the opposite angles maintain an analogous relation: $\sin A + \sin C = 2 \sin B$.