Exercises: Sine rule

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.

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 $B$.

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 triangle $ABC$, it is known that the length of side $a$ is twice that of side $b$ ($a = 2b$). If $\sin A = 0.8$, find the exact value of $\sin B$.

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

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 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.

The sides of a triangle are proportional to the sines of their opposite angles. If the ratio between two sides is $\frac{a}{b} = \sqrt{2}$ and the angle $B = 30^\circ$, find the measure of angle $A$, knowing in advance that it is an acute angle.

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.

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.

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.

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 $c$.

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 $ADC$.

Two observers $A$ and $B$ are separated by $1200$ km on the Earth's surface (a distance we will consider straight for this local calculation). Both observe a satellite $S$ passing through the same vertical plane joining them, with its ground projection lying between $A$ and $B$. The angle of elevation measured from $A$ is $82^\circ$ and from $B$ is $85^\circ$. Calculate the exact distance from observer $A$ to the satellite.

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

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

In a triangle $ABC$, the interior angle bisector of angle $A$ intersects the opposite side $BC$ at point $D$. Apply the sine rule to triangles $ABD$ and $ACD$ to formally prove the angle bisector theorem, that is, that the following proportion holds:

$\frac{BD}{DC} = \frac{c}{b}$

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 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.

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 $ABC$, 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 $a = 2b \cos B$.

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

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.