Solving Oblique Triangles
Two boats leave the port of Samaná on different headings, and each travels at its own speed. How far apart are they after a while? There is no right angle anywhere: you need the law of sines and the law of cosines working together. At the end of the page we solve it.
Which data you have, which law to use
Solving an oblique triangle means finding all three of its sides and all three of its angles. With three pieces of data (including at least one side) you can almost always do it, but the route depends on which data you have. In the side-side-angle case (the angle opposite one of the given sides) there can be zero, one, or two solutions (the ambiguous case); below you will see how to find both.
In a triangle $ABC$, the sides $a = 5$ cm and $b = 8$ cm are known, as well as the included angle $C = 60^\circ$ between them. Calculate the length of side $c$.
In a triangle $ABC$, the sides $a = 5$ cm and $b = 8$ cm are known, as well as the included angle $C = 60^\circ$ between them. Calculate the length of side $c$.
The three sides of a triangle measure $a = 3$ m, $b = 5$ m and $c = 7$ m. Determine the size of the largest angle in this triangle.
The three sides of a triangle measure $a = 3$ m, $b = 5$ m and $c = 7$ m. Determine the size of the largest angle in this triangle.
In a triangle, the following data are known: $a = 12$ m, $b = 8$ m and angle $A = 120^\circ$. Calculate the size of angle $B$.
In a triangle, the following data are known: $a = 12$ m, $b = 8$ m and angle $A = 120^\circ$. Calculate the size of angle $B$.
Solving the whole triangle
The two laws are combined in whatever order is convenient. By convention, each side gets the lowercase letter of its opposite vertex.
The ambiguous case: two solutions
When you know two sides and an angle opposite one of them, the sine of the other angle can correspond to two different angles: an acute one and its supplement ($180^\circ-\theta$), because $\sin\theta=\sin(180^\circ-\theta)$. If both give a valid triangle (angle sum less than $180^\circ$), there are two solutions.
In a triangle $ABC$, the angles $A$ and $B$ measure $30^\circ$ and $105^\circ$ respectively. If the side opposite angle $A$ has a length of $a = 8$ cm, find the length…
In a triangle $ABC$, the angles $A$ and $B$ measure $30^\circ$ and $105^\circ$ respectively. If the side opposite angle $A$ has a length of $a = 8$ cm, find the length of side $c$ (opposite angle $C$).
In an isosceles triangle, the non-equal side (the base) measures $12$ m and the equal angles measure $75^\circ$ each. Determine the length of the equal sides using the…
In an isosceles triangle, the non-equal side (the base) measures $12$ m and the equal angles measure $75^\circ$ each. Determine the length of the equal sides using the sine rule.
In a triangle $ABC$, side $a$ measures $10$ cm and the opposite angle $A$ measures $45^\circ$. Calculate the radius $R$ of the circumscribed circle of the triangle.
In a triangle $ABC$, side $a$ measures $10$ cm and the opposite angle $A$ measures $45^\circ$. Calculate the radius $R$ of the circumscribed circle of the triangle.
In the design of an articulated robotic arm, the first link measures $40$ cm and the second measures $35$ cm. If the angle formed by the first link with the horizontal…
In the design of an articulated robotic arm, the first link measures $40$ cm and the second measures $35$ cm. If the angle formed by the first link with the horizontal base is $50^\circ$ and the end of the second link touches the horizontal base exactly, calculate the two possible distances from the origin of the arm to the contact point on the base.
Area without needing the height
If you know two sides and the angle between them, there is a direct formula for the area.
A parallelogram is split by a diagonal into two equal triangles, and its diagonal comes from the law of cosines. With sides $6$ and $10$ and an angle of $120^\circ$, the longer diagonal $d$ is opposite the obtuse angle: $d^2=6^2+10^2-2\times6\times10\cos120^\circ=36+100+60=196$, so $d=14$ cm.
Find the area of a triangle $ABC$ for which the sides $b = 14$ cm, $c = 20$ cm and the angle between them, $A = 60^\circ$, are known.
Find the area of a triangle $ABC$ for which the sides $b = 14$ cm, $c = 20$ cm and the angle between them, $A = 60^\circ$, are known.
The adjacent sides of a parallelogram measure $6$ cm and $10$ cm, forming an obtuse angle of $120^\circ$. Calculate the length of the longer diagonal of the…
The adjacent sides of a parallelogram measure $6$ cm and $10$ cm, forming an obtuse angle of $120^\circ$. Calculate the length of the longer diagonal of the parallelogram.
Triangulation, navigation, and forces
Triangulation fixes the position of an inaccessible point by measuring angles and one distance from known points.
Two forces also form a triangle (the parallelogram rule). With tensions $F_1=4500$ N and $F_2=6000$ N that form $45^\circ$, the triangle has sides $F_1$ and $F_2$ with angle $180^\circ-45^\circ=135^\circ$ between them, and the resultant is the third side.
Two ships leave a port simultaneously along straight routes that form an angle of $75^\circ$ with each other. The first ship sails at a constant speed of $15$ knots and…
Two ships leave a port simultaneously along straight routes that form an angle of $75^\circ$ with each other. The first ship sails at a constant speed of $15$ knots and the second at $20$ knots. Calculate the distance in nautical miles separating the two ships after $2$ hours of navigation.
Two tugboats pull a barge using two cables that form an angle of $45^\circ$ with each other. If the tensions exerted by the cables are $4500$ N and $6000$ N, find the…
Two tugboats pull a barge using two cables that form an angle of $45^\circ$ with each other. If the tensions exerted by the cables are $4500$ N and $6000$ N, find the magnitude of the resultant force and the angle this resultant forms with the cable exerting the force of $6000$ N.
To measure the height $h$ of a wind turbine, we stand at a point $A$ on the ground and measure the angle of elevation to the top point, obtaining $40^\circ$. We move…
To measure the height $h$ of a wind turbine, we stand at a point $A$ on the ground and measure the angle of elevation to the top point, obtaining $40^\circ$. We move $30$ meters in a straight line towards the base of the wind turbine to reach a point $B$, from which the new angle of elevation is $55^\circ$. Calculate the height of the wind turbine.
A surveillance drone is at a point $C$ observing a straight stretch of road bounded by two cities $A$ and $B$ that are $5$ km apart. The angles of depression from the…
A surveillance drone is at a point $C$ observing a straight stretch of road bounded by two cities $A$ and $B$ that are $5$ km apart. The angles of depression from the drone to the cities are such that in the triangle formed, the angle at $A$ is $35^\circ$ and the angle at $B$ is $62^\circ$. Calculate the altitude at which the drone is flying above the road.
A solar panel $2$ meters long is to be installed on a pitched roof that has an inclination of $15^\circ$ to the horizontal. To maximize its performance, the panel must…
A solar panel $2$ meters long is to be installed on a pitched roof that has an inclination of $15^\circ$ to the horizontal. To maximize its performance, the panel must form an angle of $35^\circ$ with respect to the horizontal. Calculate the length $x$ of the vertical support that must be placed at the upper end of the panel to connect it to the roof.
Polygons: dividing into triangles
Any quadrilateral or polygon can be solved by dividing it into triangles with a diagonal. A regular polygon with $n$ sides inscribed in a circle of radius $r$ is divided into $n$ isosceles triangles with central angle $\frac{360^\circ}{n}$ and two sides equal to $r$. The side of the polygon comes from the law of cosines and the area is $n\times\frac12r^2\sin\frac{360^\circ}{n}$.
A piece of farmland has the shape of a quadrilateral $ABCD$. The lengths of its sides are $AB = 50$ m, $BC = 40$ m, $CD = 30$ m and $DA = 60$ m. Knowing that the angle…
A piece of farmland has the shape of a quadrilateral $ABCD$. The lengths of its sides are $AB = 50$ m, $BC = 40$ m, $CD = 30$ m and $DA = 60$ m. Knowing that the angle at vertex $B$ is $70^\circ$, calculate the total area of the land by dividing it into two triangles.
A regular heptagon (a polygon with $7$ equal sides) is inscribed in a circle of radius $10$ cm. Using the cosine rule and/or sine rule in the central triangles,…
A regular heptagon (a polygon with $7$ equal sides) is inscribed in a circle of radius $10$ cm. Using the cosine rule and/or sine rule in the central triangles, calculate the perimeter and area of the heptagon.
Advanced cases
Some problems ask you to set up an equation or combine formulas.
For the $13$, $14$, $15$ triangle: $s=21$, $S=\sqrt{21\cdot8\cdot7\cdot6}=84$, $R=\frac{13\cdot14\cdot15}{336}=\frac{65}{8}$, and $r=\frac{84}{21}=4$.
In motion problems (airplanes, ships), draw the triangle of positions, mark the angles relative to north, and apply the appropriate law. With the clock, remember that the minute hand moves $6^\circ$ per minute and the hour hand $0.5^\circ$ per minute.
Projection formula
Dihedral angle: the pyramid
The dihedral angle between two faces that share an edge is measured in a plane perpendicular to that edge. In a regular pyramid with a square base, the adjacent faces $VAB$ and $VBC$ share the edge $VB$. If you draw the perpendiculars to $VB$ from $A$ and from $C$, they land at the same point $P$, and angle $APC$ is the dihedral angle.
The sides of a triangle are given by the expressions $x$, $x+2$ and $x+4$ centimeters. Knowing that the largest angle of the triangle measures $120^\circ$, set up the…
The sides of a triangle are given by the expressions $x$, $x+2$ and $x+4$ centimeters. Knowing that the largest angle of the triangle measures $120^\circ$, set up the corresponding equation, find the exact value of $x$ and calculate the perimeter of the triangle.
In a triangle $ABC$, it is known that side $a = 10$ cm, angle $B = 30^\circ$ and that the total area of the triangle is $20\sqrt{3}$ cm$^2$. Set up a system of equations…
In a triangle $ABC$, it is known that side $a = 10$ cm, angle $B = 30^\circ$ and that the total area of the triangle is $20\sqrt{3}$ cm$^2$. Set up a system of equations to find the exact length of sides $b$ and $c$, and find the size of angles $A$ and $C$.
Prove geometrically or using trigonometric rules (sine and cosine rules) that in any triangle $ABC$ the projection formula holds: $a = b \cos C + c \cos B$. Next, use…
Prove geometrically or using trigonometric rules (sine and cosine rules) that in any triangle $ABC$ the projection formula holds: $a = b \cos C + c \cos B$. Next, use this identity together with the sine rule to prove the sine compound angle formula: $\sin(B+C) = \sin B \cos C + \cos B \sin C$.
The hands of a wall clock measure $12$ cm (the minute hand) and $8$ cm (the hour hand). Calculate the exact distance in centimeters separating the tips of both hands…
The hands of a wall clock measure $12$ cm (the minute hand) and $8$ cm (the hour hand). Calculate the exact distance in centimeters separating the tips of both hands when the clock shows exactly 10:10. Take into account the proportional displacement of the hour hand.
Two aeroplanes fly at the same altitude. Aeroplane 1 is at point $A$ and flies North at a constant speed of $800$ km/h. Aeroplane 2 is at point $B$, located exactly…
Two aeroplanes fly at the same altitude. Aeroplane 1 is at point $A$ and flies North at a constant speed of $800$ km/h. Aeroplane 2 is at point $B$, located exactly $200$ km North-East ($45^\circ$ from North) of point $A$. If aeroplane 2 flies West at a constant speed $v$ and both collide at point $C$, determine the value of speed $v$ and the time it takes for them to collide.
In a triangle $ABC$ with sides $a=13$ cm, $b=14$ cm and $c=15$ cm, the circumscribed circle of radius $R$ and the inscribed circle of radius $r$ are drawn. Calculate the…
In a triangle $ABC$ with sides $a=13$ cm, $b=14$ cm and $c=15$ cm, the circumscribed circle of radius $R$ and the inscribed circle of radius $r$ are drawn. Calculate the area of the triangle using Heron's formula and, from it, obtain the exact values of $R$ and $r$.
Given the triangle $ABC$ with sides $a = 7$ cm, $b = 5$ cm and $c = 8$ cm, the internal angle bisector of angle $C$ is drawn, intersecting side $AB$ at point $D$.…
Given the triangle $ABC$ with sides $a = 7$ cm, $b = 5$ cm and $c = 8$ cm, the internal angle bisector of angle $C$ is drawn, intersecting side $AB$ at point $D$. Applying the cosine rule to find angle $C$ and using the properties of areas or the sine rule, calculate the exact length of the angle bisector segment $CD$ (denoted as $x$).
You can do it on paper and send a photo with your answer.
A regular square-based pyramid has a base side length of $100$ m. Knowing that the dihedral angle formed by two adjacent triangular faces is $105^\circ$, set up the…
A regular square-based pyramid has a base side length of $100$ m. Knowing that the dihedral angle formed by two adjacent triangular faces is $105^\circ$, set up the geometry of the appropriate cross-section and calculate the total height of the pyramid.