The sine rule
Until now you have only known how to solve right-angled triangles. In surveying, navigation or engineering, triangles almost never have a right angle: you need a tool that works with any triangle. The sine rule is the first one.
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.
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.
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…
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.
Finding an angle: the ambiguous case
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 $\sin(180^\circ-x)=\sin x$.
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.
Triangulation: distances from two angles
In surveying and navigation, 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.
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 $ADC$.
Area and combined problems
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.
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.
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.