Picture a surveyor on the Dublin quays who wants the width of the Liffey without crossing it. Triangles in surveying, navigation or engineering almost never have a right angle, so you need a tool that works with any triangle. The sine rule is the first one, and by the end of this page you will use it to find an inaccessible distance, just like that surveyor.
The rule and the circumscribed circle
In any triangle $ABC$, each side is proportional to the sine of its opposite angle, and this ratio is always the diameter of the circle that passes through the three vertices (the circumcircle). The sine rule is in the Formulae and Tables booklet.
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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…
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.
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 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.
When the unknown is an angle (side-side-angle), solving for the sine can give two valid solutions: an acute angle and its supplementary angle, because $\sin(180^\circ-x)=\sin x$.
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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 $B$.
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 $B$.
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…
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 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…
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.
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 $a$. a)…
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 $a$.
a) From the law of sines, write the expression that allows calculating $\sin B$ as a function of the length $a$.
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.
c) Geometrically interpret what the critical value of $a$ found in case ii) represents in relation to the altitude drawn from vertex $C$.
In surveying and navigation (for example, finding the distance to a ship off the Cliffs of Moher), the usual approach is to measure two angles from two points (never an inaccessible distance directly) and apply the sine rule to the resulting triangle.
Back to the start: the surveyor on the Liffey measures a baseline on one bank and two angles to a point on the far bank, then uses the sine rule exactly as above.
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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…
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…
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.
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…
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 $ADC$.
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…
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.
The sine rule also solves regular polygons (by dividing them into isosceles triangles from the centre) and area problems when you know one side and the two adjacent angles.
Proofs using the sine rule
Substituting $a = 2R\sin A$, $b = 2R\sin B$, $c = 2R\sin C$ turns problems about sides into problems about sines. The area formula $\tfrac12 ab\sin C$ then gives $S = \dfrac{abc}{4R}$.
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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…
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.…
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 interior angle bisector of angle $A$ intersects the opposite side $BC$ at point $D$. Apply the sine rule to 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 to triangles $ABD$ and $ACD$ to formally prove the angle bisector theorem, that is, that the following proportion holds:
If a proof involves $A = 2B$, use $\sin 2B = 2\sin B\cos B$ and $\cos 2B = \cos^2 B - \sin^2 B$ (Formulae and Tables).
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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…
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 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…
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.