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 a Trigonometry exam, Marcos must solve a triangle where $a = 14\text{ cm}$, $b = 9\text{ cm}$, and $A = 35^\circ$ are known. To find angle $B$, Marcos writes:

And concludes: "Since the sine function is positive in both the first and second quadrants, there are two possible solutions: $B_1 = \arcsin(0,3687) \approx 21,63^\circ$ and $B_2 = 180^\circ - 21,63^\circ = 158,37^\circ$. Therefore, there are two distinct triangles that satisfy the given data". Find the conceptual error Marcos makes, explain using two different geometric arguments why the solution $B_2$ is impossible in this triangle, and calculate the correct measure of the remaining angles $B$ and $C$.

In a triangle $ABC$, the angles $A = 70^\circ$ and $B = 30^\circ$ are known, as well as the length of side $a = 20\text{ m}$.

1. a) Before carrying out any numerical calculation, reasonably estimate whether side $b$ will measure more or less than half of side $a$. Base your estimate on comparing the exact value of $\sin 30^\circ$ with the behavior of the sine function as the angle approaches $90^\circ$.
2. b) Apply the law of sines to calculate the exact length of $b$, rounding the result to two decimal places.
3. c) Check whether the value obtained confirms your initial estimate and reflect: why in a triangle are the sides not proportional to the angles, but rather to their sines?

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 perimetre of a regular heptagon (a 7-sided polygon) inscribed in a circle of radius $R = 10$ cm. Use the sine rule on the isosceles triangles formed from the centre 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.

The forest fire surveillance service of Castilla-La Mancha has two observation posts, Post $A$ and Post $B$, located on a plain and separated by a straight-line distance of $6\text{ km}$. Both posts spot the same smoke column at a point $C$.

From Post $A$, the line of sight to the fire forms an angle of $42^\circ$ with the straight segment $AB$. From Post $B$, the line of sight to the fire forms an angle of $68^\circ$ with that same segment $AB$.

1. a) Model the situation using a triangle and calculate, using the law of sines, the distance from the fire to each of the surveillance posts (round the results to two decimal places).
2. b) If a helicopter based adjacent to Post $B$ is dispatched and flies in a straight line toward the fire at an average speed of $150\text{ km/h}$, calculate how many minutes it will take to reach the site of the fire.

To construct a triangular piece $ABC$, an operator fixes side $b = 10\text{ cm}$ and angle $A = 30^\circ$, but can vary at will the length of the opposite side $a$.

1. a) From the law of sines, write the expression that allows calculating $\sin B$ as a function of the length $a$.
2. b) Reasonably determine for which exact values or intervals of the length $a$ each of the following cases occurs:

i) It is not possible to construct any triangle with those measurements.

ii) Exactly one triangle can be constructed, and it is a right triangle.

iii) Two distinct, non-congruent triangles can be constructed.

iv) A single triangle can be constructed, and it is not a right triangle.

1. c) Geometrically interpret what the critical value of $a$ found in case ii) represents in relation to the altitude drawn from vertex $C$.

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:

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

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

It is desired to calculate the perimeter of a triangular flowerbed $ABC$ inscribed in a circular roundabout of radius $R = 8\text{ m}$. The interior angles $A = 45^\circ$ and $B = 60^\circ$ are known.

To solve the problem, two strategies are proposed:

• Strategy 1: Draw the altitude from vertex $C$ to divide the triangle into two right triangles, calculate the altitudes and projections using elementary trigonometry of special angles, and solve for the sides using systems of equations.

• Strategy 2: Calculate the third angle $C$ and apply the extended law of sines ($\frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C} = 2R$).

1. a) Reasonably compare both methods, assessing their operational efficiency (number of calculation steps) and the risk of propagation of rounding errors. Justify which of the two is preferable.
2. b) Apply the most advantageous strategy to find the exact value of the perimeter in terms of radicals (use that $\sin 75^\circ = \sin(45^\circ + 30^\circ) = \frac{\sqrt{6} + \sqrt{2}}{4}$) and obtain its approximate numerical value in meters to two decimal places.

A telecommunications company installs a rigid triangular structure $ABC$ whose three vertices must rest on a circular guide of radius $R = 10\text{ m}$. Due to technical constraints of the available bars, side $a$ is fixed to measure $14\text{ m}$ and side $b$ is fixed to measure $18\text{ m}$.

1. a) Use the extended law of sines to obtain all possible angular values for vertices $A$ and $B$ (rounded to two decimal places of a degree).
2. b) Thoroughly analyze the four possible combinations of angle pairs $(A, B)$ and determine how many geometrically viable and non-congruent triangles can be constructed. For each valid triangle, calculate the measure of angle $C$ and the length of the remaining side $c$.
3. c) Technical regulations require that the structure be strictly acute-angled in order to distribute wind stresses evenly. Justify whether any of the viable triangles satisfies this requirement and, if so, calculate the surface area bounded by this structure using the area formula that relates two sides and the sine of the included angle.