Exercises: Sine rule

In a triangle $ABC$, the angles $A = 45^\circ$ and $B = 60^\circ$, and the length of the side $a = 12$ in are known. Calculate the length of the side $b$ using the law of sines.

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

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

In the triangle $ABC$, it is known that the length of side $a$ is double 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 the side $r = 20$ ft, calculate the length of the side $p$.

Determine if it is possible to construct a triangle $ABC$ with the following data: $a = 8$ in, $b = 14$ in and $A = 45^\circ$. Justify your answer mathematically using the law of sines.

In an isosceles triangle, the unequal angle measures $40^\circ$ and the equal sides measure $15$ in. Calculate the length of the unequal side by directly applying the law of sines.

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 the angle $A$, knowing beforehand that it is an acute angle.

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

Two Coast Guard stations, $A$ and $B$, are separated by a distance of $50$ miles 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 old tower has leaned and now forms an angle of $85^\circ$ with the horizontal ground. A surveyor stands $40$ ft from the base of the tower, exactly in the direction it is leaning. From that point, they measure 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 (7-sided polygon) that is inscribed in a circle of radius $R = 10$ in. Use the law of sines in the isosceles triangles formed from the center of the polygon.

A triangular plot of land has two angles that measure $60^\circ$ and $75^\circ$. The side between these two angles is bounded by a straight wall measuring $120$ ft. 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 the side $a = 12\sqrt{3}$ in, determine the exact length of the side $c$.

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

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

Prove, starting from the Law of Sines, 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 Law of Sines in 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 circumscribed circle of a triangle $ABC$ is directly related to the area of the triangle $S$ by the formula:

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

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

In the construction of a triangle $ABC$, the side $a = 10$ in and the angle $A = 45^\circ$ are fixed. The side $b$ has a variable length $x$ in. Determine, with reasoning, the strict interval of real values of $x$ for which there exist exactly two distinct triangles that satisfy these conditions simultaneously.

A mountain slope has a constant inclination of $15^\circ$ with respect to the horizontal plane. Two points $A$ and $B$ are on the slope, with $B$ located $50$ ft higher than $A$ along the slope. A vertical telecommunications pole has been installed at point $B$. From point $A$, the angle of elevation to the top end of the pole is $42^\circ$ measured from the horizontal. Calculate the exact height of the pole.

In a triangle $ABC$, the angle $A$ is exactly double the angle $B$ ($A = 2B$). Prove, using the Law of Sines and the double angle trigonometric identities, that the lengths of the sides satisfy the relationship $a = 2b \cos B$.

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

A plane flies in a straight line at a constant altitude. A static observer on the ground initially spots it with an angle of elevation of $30^\circ$. Exactly $10$ seconds later, the plane, which is flying directly towards the vertical of the observer, is spotted with an angle of elevation of $45^\circ$. If the constant speed of the plane is $600$ mph, calculate the altitude at which it is flying by solving the triangle formed by the observer and the two positions of the plane using the Law of Sines.