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.

In an isosceles triangle, the non-equal angle measures $40^\circ$ and the equal sides measure $15$ cm. Calculate the length of the remaining side by directly applying the sine rule.

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.

An ancient tower has tilted and now forms an angle of $85^\circ$ with the horizontal ground. A surveyor stands $40$ m from the base of the tower, directly in the direction in which it is leaning. From that point, he measures the angle of elevation to the top of the tower, obtaining $50^\circ$. Calculate the total length of the tower from its base to the top.

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 $ADC$.

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 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: $\sin A + \sin C = 2 \sin B$.

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, starting from the sine rule, that in any triangle $ABC$ the following trigonometric identity relating its sides and angles holds:

$a \sin(B - C) + b \sin(C - A) + c \sin(A - B) = 0$