Solving Oblique Triangles
Regina wants to know how far apart two cell phone towers on a hill in Oaxaca are, but she can't walk up to them: she can only measure one distance from where she stands and two angles with a theodolite. How does she get the distance without measuring it? With triangles that do not have a right angle and with two tools: the law of sines and the law of cosines.
What data you have, which theorem to use
An oblique triangle is one that has no right angle. "Solving" it means finding its three sides and its three angles. You almost always start with three pieces of data (at least one is a side), and depending on which ones they are, one theorem is more direct than the other.
The law of cosines is the Pythagorean theorem with a correction: if $C=90^\circ$, $\cos C=0$ and you get $c^2=a^2+b^2$.
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 theorems are combined in whatever order is convenient. By convention, side $a$ is opposite angle $A$, and likewise for the others.
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.
Area of a triangle without its height
If you know two sides and the angle between them, you don't need the height.
A parallelogram is split into two equal triangles by a diagonal; so its area is $ab\sin\theta$, and its diagonal is found with the law of cosines.
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 and applications
Triangulation fixes the position of an inaccessible point by measuring angles and distances from known points. The same method works for ships, heights of towers, forces, and structural repairs.
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.
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 toward 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.
Quadrilaterals and polygons: dividing into triangles
Any polygon can be solved with diagonals that divide it into triangles, applying sines and cosines to each one. If it is a regular polygon with $n$ sides, the central triangles have a central angle of $\dfrac{360^\circ}{n}$.
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.
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.
Forces, angles, and longer problems
Many problems mix theorems. Always make a drawing, mark the data, and look for the triangle where you have the combination from the first box.
For forces with an angle $\theta$ between them, the resultant is the diagonal of the parallelogram. Careful: the angle of the triangle that forms is the supplementary one, $180^\circ-\theta$, so $R^2=F_1^2+F_2^2+2F_1F_2\cos\theta$.
Angle between the hands of a clock
For clock problems, you need the angle between the hands. The minute hand turns $360^\circ$ in $60$ minutes, that is, $6^\circ$ per minute. The hour hand turns $360^\circ$ in $12$ hours: $30^\circ$ per hour and also $0.5^\circ$ per minute, because it keeps moving between one hour and the next. Measured from 12, the two angles are:
-
minute hand: $6^\circ\times\text{minutes}$;
-
hour hand: $30^\circ\times\text{hours}+0.5^\circ\times\text{minutes}$.
The angle between them is the difference (if it exceeds $180^\circ$, subtract it from $360^\circ$). That angle is the one included between the two hands, so the law of cosines is used.
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.
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.
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.
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.
The projection formula
If you draw the altitude from $A$ to side $a$, its foot $D$ divides $a$ into two parts. Each part is the projection of another side onto $a$.
In right triangle $ABD$, $BD=c\cos B$. In right triangle $ADC$, $DC=b\cos C$. Since $a=BD+DC$:
With the law of sines you get the formula for the sine of a sum:
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: $\sin(B+C) = \sin B \cos C + \cos B \sin C$.
Heron, radii, and challenge exercises
With the three sides, Heron's formula gives the area without angles: with the semiperimeter $s=\dfrac{a+b+c}{2}$,
$$S=\sqrt{s(s-a)(s-b)(s-c)}$$And combining formulas: $S=\dfrac{abc}{4R}$ (from $a=2R\sin A$ and $S=\frac12bc\sin A$) and $S=rs$ (the triangle is divided into three triangles of height $r$).
In problems with unknowns, set up the theorem as an equation. With the angle bisector, the area of the large triangle is the sum of the two parts: $\frac12ab\sin C=\frac12bx\sin\frac C2+\frac12ax\sin\frac C2$. In the pyramid, draw the cross section that passes through a lateral edge: the perpendiculars to that edge from the two neighboring vertices of the base form an isosceles triangle with the dihedral angle at the vertex, and its base is the diagonal of the base of the pyramid. In bearing problems, the $45^\circ$ angle to the Northeast, measured from the North direction, usually gives a triangle with angles of $45^\circ$ and $90^\circ$.
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 $C$.
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.
Back to the beginning: Regina measures a baseline and two angles, finds the third angle by subtracting from $180^\circ$, and gets the distance to the tower with the law of sines, without walking up to it.