The Law of Sines
Two lifeguards sit in towers $A$ and $B$, 400 feet apart along a Lake Michigan beach. Both spot a swimmer $S$ in trouble. The guard in tower $A$ measures $52^\circ$ between the beach line and the swimmer, and the guard in tower $B$ measures $71^\circ$. Nobody can stretch a tape measure across the water, and the triangle has no right angle. How far is the swimmer from each tower? You need a tool that works in any triangle. That tool is the Law of Sines.
The Law of Sines
In a right triangle you use $\sin\theta = \frac{\text{opposite}}{\text{hypotenuse}}$. In a general triangle, drop an altitude to make two right triangles. That is all you need.
The altitude $h$ is a leg of two right triangles. In the left one, $h = b\sin A$. In the right one, $h = a\sin B$. Set them equal: $b\sin A = a\sin B$, so $\dfrac{a}{\sin A} = \dfrac{b}{\sin B}$. The same argument works for the third side.
The Law of Sines needs a complete pair: one side together with its opposite angle. Then one more piece of information lets you find the rest.
| What you know | What to use |
|---|---|
| AAS or ASA (two angles and any side) | Law of Sines. One triangle. |
| SSA (two sides and an angle opposite one of them) | Law of Sines. Be careful: 0, 1 or 2 triangles (next sections). |
| SAS or SSS | Law of Cosines (its own page). |
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 sine…
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 sine rule.
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 sine rule.
The circumscribed circle and ratios of sines
The common value of the three fractions has a geometric meaning. It is the diameter of the circle through the three vertices.
Two quick consequences:
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If two angles are known, the third is $180^\circ$ minus their sum. One complete pair then gives $2R$, and every other side follows.
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The ratio of two sides equals the ratio of the sines of their opposite angles: $\dfrac{a}{b} = \dfrac{\sin A}{\sin B}$.
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 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: two possible answers
When the unknown is an angle (you know two sides and an angle opposite one of them), you solve for a sine: $\sin B = \dfrac{b\sin A}{a}$. But two different angles share the same sine, because
$$\sin(180^\circ - x) = \sin x.$$Your calculator's $\sin^{-1}$ only gives the acute one. The other candidate is $180^\circ - x$.
Think of side $a$ as a bar hinged at $C$ that swings down to the ground. It can touch the ground in two places, $B_1$ and $B_2$. Both triangles have the same $A$, $a$ and $b$.
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 $B$.
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? The SSA case
You can tell how many triangles exist before solving. Given angle $A$, side $b$ (adjacent to $A$) and side $a$ (opposite $A$), compare $a$ with the altitude $h = b\sin A$.
The shortest way from $C$ to the ground is $h$. So side $a$ has to be at least that long.
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 sine rule.
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.
Two observers: finding distances
Surveyors and sailors measure two angles from the ends of a baseline they can measure, then use the Law of Sines to find distances they cannot.
The hook is closed: the swimmer is about 451 ft from tower $A$ and 376 ft from tower $B$. Tower $B$ has the shorter run, which makes sense because it sees the swimmer at the larger angle.
Observers can also be standing on the ground looking up at something between them, like a satellite. Draw the picture first: if the object is between the two observers, the angles of elevation are the two interior base angles of the triangle.
Two coastguard 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 coastguard 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 from the same side
Sometimes both measurements are taken from the same side of the object, for example walking toward a hill or watching a plane approach. An angle of elevation is measured up from the horizontal. The triangle's own angles come from two facts:
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The interior angle at the nearer point is $180^\circ - (\text{its elevation})$.
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The angle at the object is the difference of the two elevations (exterior angle theorem).
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 $ADC$.
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 toward 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.
Leaning objects and sloped ground
When the ground is not level or the object is not vertical, the only change is how you find the interior angles. Draw the picture, mark every angle you know from the horizontal or the vertical, and then ask what the angle inside the triangle is.
If the pole leaned away from you at $80^\circ$ measured on its far side, the angle inside the triangle would be $180^\circ - 80^\circ = 100^\circ$.
Now sloped ground. A vertical pole stands at $B$ on a hillside. Be careful with the angle at $B$: the pole is vertical, but the slope tilts.
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 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.
Area and regular polygons
Once you know two sides and the angle between them, the area comes from the altitude $h = b\sin C$:
For a regular polygon with $n$ sides inscribed in a circle of radius $R$, join the center to two neighboring vertices. You get an isosceles triangle with two sides $R$ and apex angle $\dfrac{360^\circ}{n}$.
The same ideas give neat formulas. For example, using $a = 2R\sin A$ and $b = 2R\sin B$ in $\text{Area} = \tfrac12 ab\sin C$:
Calculate the perimeter of a regular heptagon (7-sided polygon) that is inscribed in a circle of radius $R = 10$ in. Use the sine rule in the isosceles triangles formed…
Calculate the perimeter of a regular heptagon (7-sided polygon) that is inscribed in a circle of radius $R = 10$ in. Use the sine rule 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.
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 sine rule).
Proving identities with the Law of Sines
Many proofs have the same first move. Let $k$ be the common value $\dfrac{a}{\sin A} = \dfrac{b}{\sin B} = \dfrac{c}{\sin C}$ (so $k = 2R$). Then
$$a = k\sin A, \qquad b = k\sin B, \qquad c = k\sin C.$$Substitute these, cancel $k$, and finish with trigonometric identities such as $\sin(x \pm y) = \sin x\cos y \pm \cos x\sin y$, $\sin 2x = 2\sin x\cos x$, and $\sin(180^\circ - x) = \sin x$.
Prove, starting from the sine rule, that in any triangle $ABC$ the following trigonometric identity relating its sides and angles holds:…
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 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: $\sin A + \sin C = 2 \sin B$.
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 $a = 2b \cos B$.
Using the Law of Sines in two triangles
A segment from a vertex to the opposite side splits the triangle into two smaller triangles that share that segment. Apply the Law of Sines in both, and use a key fact: the two angles at the point $D$ on the side are a linear pair, so they are supplementary and have equal sines.
For the angle bisector theorem you do the same steps with the bisector, where $\angle BAD = \angle CAD$, so their sines are equal as well.
In a triangle $ABC$, the interior angle bisector of angle $A$ intersects the opposite side $BC$ at point $D$. Apply the sine rule 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 sine rule in triangles $ABD$ and $ACD$ to formally prove the angle bisector theorem, that is, that the following proportion holds:
$\frac{BD}{DC} = \frac{c}{b}$