The Law of Sines
You're standing on the shore of a lake, and there is a boat dock on the far side. You can't walk across and you can't stretch a tape measure over the water. How far away is the dock? SOH-CAH-TOA won't help, because this triangle has no right angle. You need a tool that works for any triangle, and the Law of Sines is the first one.
The Law of Sines
In any triangle $ABC$, each side is proportional to the sine of the opposite angle. Side $a$ is opposite angle $A$, side $b$ is opposite $B$, and side $c$ is opposite $C$.
To use the law you need a matching pair, a side and its opposite angle, plus one more piece of information. That means you know two angles and a side (AAS or ASA), or two sides and an angle opposite one of them (SSA).
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…
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.
A triangle $PQR$ has angles $P = 50^\circ$ and $Q = 75^\circ$. If the side $r = 20$ ft, calculate the length of the side $p$.
A triangle $PQR$ has angles $P = 50^\circ$ and $Q = 75^\circ$. If the side $r = 20$ ft, calculate the length of the side $p$.
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…
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.
Sides, sines, and the circumscribed circle
The common ratio in the Law of Sines is not just a number. It is the diameter of the circle through the three vertices, the circumscribed circle.
Since $\sin$ is never more than $1$, no side can be longer than $2R$, the diameter. That makes sense, because a chord is never longer than a diameter.
The law also gives you ratios. From $\frac{a}{b} = \frac{\sin A}{\sin B}$, the ratio of two sides equals the ratio of the sines of their opposite angles.
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 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 $\sin B$.
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 $\sin B$.
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…
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 $c$.
Finding an angle: the ambiguous case
When you know two sides and an angle opposite one of them (SSA), you solve for another angle by isolating its sine. Two different angles between $0^\circ$ and $180^\circ$ can have the same sine, because $\sin(180^\circ - x) = \sin x$. So you may get two valid triangles.
Drag the slider $k$. The line $y = k$ crosses the sine curve twice, at two angles from $0$ to $\pi$ radians ($0^\circ$ to $180^\circ$). Those two angles add up to $180^\circ$, and they are the two candidates for the angle you want.
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 $B$.
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 $B$.
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 $B = 30^\circ$,…
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…
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.
How many triangles?
You can decide how many triangles exist before solving, by comparing the side opposite the given angle with the height. Suppose you know angle $A$, the side $b$ next to it, and the side $a$ opposite it. Swing side $a$ like a compass from $C$, and compare it with the height $h = b\sin A$ from $C$ to the base line.
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…
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 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…
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.
Triangulation: distances from two angles
Surveyors and navigators avoid measuring hard-to-reach distances directly. They measure one baseline and two angles, then use the Law of Sines on the triangle they form.
Back to the start: that is how you find the distance to a dock you can't reach. You only walk along the shore.
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…
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.
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…
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.
Angles of elevation
An angle of elevation is always measured from the horizontal. When you stand at two points on a level line and look up at the same target, the two triangles share information. The key fact is that the interior angle of the big triangle at the closer point is $180^\circ$ minus its angle of elevation.
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…
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.
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,…
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 $ADC$.
Slopes and moving targets
Two other settings use the same idea. On a hillside, the ground is not horizontal, but angles of elevation are still measured from the horizontal. Subtract or add the slope angle to get the angles inside your triangle.
A moving object works like the water tower: two sightings of the same object give two angles and a baseline, and the baseline is the distance the object traveled.
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…
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.
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 $10$…
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.
Area and regular polygons
The Law of Sines also gives areas. If you know two sides and the angle between them, the area is half their product times the sine of that angle.
For a regular polygon inscribed in a circle, draw segments from the center to each vertex. You get congruent isosceles triangles whose legs equal the radius $R$.
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…
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.…
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.
Proofs with the Law of Sines
In a proof you can't plug in numbers, so you work with the extended form. Since $\frac{a}{\sin A} = 2R$, every side can be written as
$$a = 2R\sin A, \qquad b = 2R\sin B, \qquad c = 2R\sin C.$$Substitute these into the statement and the sides turn into sines, which you can handle with trigonometric identities. Three facts are useful:
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$A + B + C = 180^\circ$, so $\sin C = \sin(180^\circ - (A + B)) = \sin(A + B)$.
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$\sin(180^\circ - x) = \sin x$.
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The identity $\sin(x+y)\sin(x-y) = \sin^2 x - \sin^2 y$.
Prove, starting from the Law of Sines, that in any triangle $ABC$ the following trigonometric identity relating its sides and angles holds:…
Prove, starting from the Law of Sines, that in any triangle $ABC$ the following trigonometric identity relating its sides and angles holds:
$a \sin(B - C) + b \sin(C - A) + c \sin(A - B) = 0$
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:…
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 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…
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: $\sin A + \sin C = 2 \sin B$.
Proofs with auxiliary segments and double angles
Some proofs add a segment to the triangle. When you draw a segment from a vertex to the opposite side, you get two smaller triangles and can apply the Law of Sines in both. The two triangles share a side and the angles where they meet on the line are supplementary, so their sines are equal. This is what makes the ratios match up.
Other proofs link an angle and its double. Then the double angle identity $\sin 2B = 2\sin B\cos B$ helps.
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…
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:
$\frac{BD}{DC} = \frac{c}{b}$
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…
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 $a = 2b \cos B$.