You're standing on the shore of Lake Michigan and there's a buoy floating out on the water. You can't walk out to measure how far away it is. But you can measure angles from two spots on the shore. Is that enough to find the distance? By the end of this page you'll know it is.
Choosing the right law
An oblique triangle is a triangle with no right angle. To solve one means to find all three sides and all three angles. We label it the standard way: angle $A$ is opposite side $a$, angle $B$ is opposite side $b$, and angle $C$ is opposite side $c$.
You have two tools. Both come from the right-triangle ideas you already know.
The Law of Sines needs a matching pair (an angle and its opposite side). The Law of Cosines needs either the angle between two known sides or all three sides. Look at what you're given and the choice is almost automatic.
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In a triangle $ABC$, the sides $a = 5$ in and $b = 8$ in are known, and the angle between them is $C = 60^\circ$. Calculate the length of the side $c$.
In a triangle $ABC$, the sides $a = 5$ in and $b = 8$ in are known, and the angle between them is $C = 60^\circ$. Calculate the length of the side $c$.
Solving a triangle usually takes two steps: one law to get a side or angle, then the other law (or the angle sum $A+B+C=180^\circ$) to finish.
An isosceles triangle works the same way. Its two base angles are equal, so one base angle gives you all three angles, and one side then gives you the rest by the Law of Sines.
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In a triangle $ABC$, the angles $A$ and $B$ measure $30^\circ$ and $105^\circ$ respectively. If the side opposite the angle $A$ has a length of $a = 8$ in, find the…
In a triangle $ABC$, the angles $A$ and $B$ measure $30^\circ$ and $105^\circ$ respectively. If the side opposite the angle $A$ has a length of $a = 8$ in, find the length of the side $c$ (opposite the angle $C$).
In an isosceles triangle, the unequal side (the base) measures $12$ ft and the equal angles measure $75^\circ$ each. Determine the length of the equal sides using the…
In an isosceles triangle, the unequal side (the base) measures $12$ ft and the equal angles measure $75^\circ$ each. Determine the length of the equal sides using the law of sines.
The adjacent sides of a parallelogram measure $6$ in and $10$ in, and form an obtuse angle of $120^\circ$. Calculate the length of the longer diagonal of the…
The adjacent sides of a parallelogram measure $6$ in and $10$ in, and form an obtuse angle of $120^\circ$. Calculate the length of the longer diagonal of the parallelogram.
Suppose you know two sides and an angle that is not between them (SSA). Picture side $b$ lying along the ground and angle $A$ opening upward. Side $a$ is like a swinging arm of fixed length. It can fall short of the ray, touch it, or cross it in two places.
To solve SSA, use the Law of Sines to get $\sin B=\dfrac{b\sin A}{a}$, then count the possibilities.
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In a triangle the following data are known: $a = 12$ ft, $b = 8$ ft and the angle $A = 120^\circ$. Calculate the measure of the angle $B$.
In a triangle the following data are known: $a = 12$ ft, $b = 8$ ft and the angle $A = 120^\circ$. Calculate the measure of the angle $B$.
In the design of an articulated robotic arm, the first link measures $40$ in and the second measures $35$ in. If the angle the first link forms with the horizontal base…
In the design of an articulated robotic arm, the first link measures $40$ in and the second measures $35$ in. If the angle the first link forms with the horizontal base is $50^\circ$ and the end of the second link exactly touches the horizontal base, calculate the two possible distances from the origin of the arm to the point of contact on the base.
If you know two sides and the angle between them, you don't need to find a height first.
When you know all three sides and no angle, use Heron's formula.
That second fact is the extended Law of Sines: $$\frac{a}{\sin A}=\frac{b}{\sin B}=\frac{c}{\sin C}=2R$$ So if you know one side and its opposite angle, you know the circumradius. For example, with $b=12$ and $B=60^\circ$ you get $2R=\dfrac{12}{\sin60^\circ}=8\sqrt3$, so $R=4\sqrt3\approx6.93$.
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Find the area of a triangle $ABC$ of which we know the sides $b = 14$ in, $c = 20$ in and the angle they form, $A = 60^\circ$.
Find the area of a triangle $ABC$ of which we know the sides $b = 14$ in, $c = 20$ in and the angle they form, $A = 60^\circ$.
In a triangle $ABC$, the side $a$ measures $10$ in and the opposite angle $A$ measures $45^\circ$. Calculate the radius $R$ of the circumscribed circle of this triangle.
In a triangle $ABC$, the side $a$ measures $10$ in and the opposite angle $A$ measures $45^\circ$. Calculate the radius $R$ of the circumscribed circle of this triangle.
In a triangle $ABC$ with sides $a=13$ in, $b=14$ in and $c=15$ in, the circumscribed circle of radius $R$ and the inscribed circle of radius $r$ are drawn. Calculate the…
In a triangle $ABC$ with sides $a=13$ in, $b=14$ in and $c=15$ in, 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 $r$.
Surveyors, pilots and ship captains find distances they can't measure directly by building a triangle from angles and one measured length. This is triangulation.
Directions are often given as bearings. A bearing of $\text{N}\,40^\circ\,\text{E}$ means you face north and turn $40^\circ$ toward east. The angle between two paths from the same point is the difference of their directions. Draw a north line at each point, and use the fact that north lines are parallel, so alternate interior angles are equal.
Now back to the buoy.
So one measured length and two angles are enough to locate the buoy. The same idea builds a whole map: measure one baseline carefully, then every new point is another triangle that shares a side with the last.
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Two ships leave a port simultaneously on straight routes that form an angle of $75^\circ$ between them. The first ship sails at a constant speed of $15$ knots and the…
Two ships leave a port simultaneously on straight routes that form an angle of $75^\circ$ between them. The first ship sails at a constant speed of $15$ knots and the second at $20$ knots. Calculate the distance in nautical miles that will separate the two ships after $2$ hours of sailing.
Two airplanes fly at the same altitude. Airplane 1 is at point $A$ and flies toward the North at a constant speed of $800$ mph. Airplane 2 is at point $B$, located…
Two airplanes fly at the same altitude. Airplane 1 is at point $A$ and flies toward the North at a constant speed of $800$ mph. Airplane 2 is at point $B$, located exactly $200$ miles to the Northeast ($45^\circ$ with respect to the North) of point $A$. If airplane 2 flies toward the West at a constant speed $v$ and both collide at a point $C$, determine the value of the speed $v$ and the time they take to collide.
To find the height of something you can't climb, measure the angle of elevation (measured up from the horizontal) from two spots in a straight line with its base. The two sightings, plus the distance between you, form a triangle you can solve.
A drone looking down at two cities uses an angle of depression, which is also measured from the horizontal. Alternate interior angles make it equal to the angle of elevation from the ground back up to the drone. Once the triangle is solved, the altitude is just $\text{side}\cdot\sin(\text{angle})$ in a right triangle.
The same method handles tilted objects. To find the angle between two lines that start at the same point, compare how each one tilts from the horizontal. For example, a panel tilted $35^\circ$ above a roof that is tilted $15^\circ$ makes a $20^\circ$ angle with the roof. A vertical segment makes $90^\circ$ with the horizontal, so it makes $90^\circ+15^\circ=105^\circ$ with the downhill part of the roof. Mark every angle on a drawing before you choose a law.
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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 highest point, obtaining $40^\circ$. We…
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 highest point, obtaining $40^\circ$. We advance $30$ feet in a straight line toward the base of the wind turbine until reaching a point $B$, from where the new angle of elevation is $55^\circ$. Calculate the height of the wind turbine.
A surveillance drone is located at a point $C$ observing a straight stretch of road delimited by two cities $A$ and $B$ that are $5$ miles apart. The angles of…
A surveillance drone is located at a point $C$ observing a straight stretch of road delimited by two cities $A$ and $B$ that are $5$ miles 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 with respect to the road.
A solar panel $2$ feet long is to be installed on a flat roof that has an inclination of $15^\circ$ with respect to the horizontal. To maximize its performance, the…
A solar panel $2$ feet long is to be installed on a flat roof that has an inclination of $15^\circ$ with respect 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 join it to the roof.
Sometimes the triangle isn't drawn, but it's there.
Forces. Two forces pulling on the same object add like vectors: complete the parallelogram, and the diagonal is the resultant. If the forces make an angle $\theta$ with each other, the triangle formed by the two forces and the resultant has an interior angle of $180^\circ-\theta$ between the force sides. Since $\cos(180^\circ-\theta)=-\cos\theta$, the Law of Cosines gives $$R^2=F_1^2+F_2^2+2F_1F_2\cos\theta$$ with a plus sign.
Clocks. The angle between the hands of a clock is the difference of their positions measured from 12. The minute hand moves $360^\circ\div60=6^\circ$ per minute. The hour hand moves $360^\circ\div12=30^\circ$ per hour, which is $0.5^\circ$ per minute, so it keeps creeping between the numbers.
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Two tugboats pull a barge using two cables that form an angle of $45^\circ$ between them. 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 $45^\circ$ between them. 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.
The hands of a wall clock measure $12$ in (the minute hand) and $8$ in (the hour hand). Calculate the exact distance in inches separating the tips of both hands when the…
The hands of a wall clock measure $12$ in (the minute hand) and $8$ in (the hour hand). Calculate the exact distance in inches separating the tips of both hands when the clock shows exactly 10:10. Take into account the proportional displacement of the hour hand.
Any polygon can be cut into triangles by drawing diagonals, and you can solve each triangle separately.
A regular polygon inscribed in a circle of radius $R$ splits into $n$ identical isosceles triangles with two sides $R$ and a central angle $\theta=\dfrac{360^\circ}{n}$.
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An agricultural plot of land has the shape of a quadrilateral $ABCD$. The measures of its sides are $AB = 50$ ft, $BC = 40$ ft, $CD = 30$ ft and $DA = 60$ ft. Knowing…
An agricultural plot of land has the shape of a quadrilateral $ABCD$. The measures of its sides are $AB = 50$ ft, $BC = 40$ ft, $CD = 30$ ft and $DA = 60$ ft. Knowing that the angle at the vertex $B$ is $70^\circ$, calculate the total area of the land by dividing it into two triangles.
A regular heptagon (polygon with $7$ equal sides) is inscribed in a circle of radius $10$ in. Using the law of cosines and/or the law of sines in the central triangles,…
A regular heptagon (polygon with $7$ equal sides) is inscribed in a circle of radius $10$ in. Using the law of cosines and/or the law of sines in the central triangles, calculate the perimeter and the area of this heptagon.
When a side is given as an expression, or when you know the area but not a side, set up an equation first and then solve.
When the area is given, the formula $S=\tfrac12ab\sin C$ turns it into an equation for a missing side.
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The sides of a triangle are given by the expressions $x$, $x+2$ and $x+4$ inches. 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 $x+4$ inches. 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 the side $a = 10$ in, the angle $B = 30^\circ$ and that the total area of the triangle is $20\sqrt{3}$ in$^2$. Set up a system to…
In a triangle $ABC$, it is known that the side $a = 10$ in, the angle $B = 30^\circ$ and that the total area of the triangle is $20\sqrt{3}$ in$^2$. Set up a system to find the exact value of the sides $b$ and $c$, and find out the measure of the angles $A$ and $C$.
An angle bisector. The bisector of angle $C$ splits triangle $ABC$ into two triangles with area $\tfrac12a\,x\sin\tfrac C2$ and $\tfrac12b\,x\sin\tfrac C2$, where $x$ is the bisector's length. Their sum is the whole area $\tfrac12ab\sin C$.
A solid. The dihedral angle between two faces is measured in a plane perpendicular to their common edge. For a square pyramid, adjacent lateral faces meet along a lateral edge. Draw the perpendiculars from the two neighboring base vertices to that edge. By symmetry they meet at one point $P$, and the angle at $P$ is the dihedral angle.
A proof. The Laws of Sines and Cosines also prove other identities. Some problems call the next one the projection relation: in any triangle, $a=b\cos C+c\cos B$. Geometrically, the altitude from $A$ splits side $a$ into pieces $c\cos B$ and $b\cos C$. Here is an algebraic proof.
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Given the triangle $ABC$ with sides $a = 7$ in, $b = 5$ in and $c = 8$ in. The interior angle bisector of angle $C$ is drawn, which intersects side $AB$ at point $D$.…
Given the triangle $ABC$ with sides $a = 7$ in, $b = 5$ in and $c = 8$ in. The interior angle bisector of angle $C$ is drawn, which intersects side $AB$ at point $D$. Applying the law of cosines to find the angle $C$ and using the properties of areas or the law of sines, calculate the exact length of the bisector segment $CD$ (denoted as $x$).
A regular square-based pyramid has a base edge of $100$ ft. Knowing that the dihedral angle formed by two adjacent triangular faces is $105^\circ$, set up the geometry…
A regular square-based pyramid has a base edge of $100$ ft. 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.
Prove geometrically or by using the trigonometric laws (Law of Sines and Law of Cosines) that in any triangle $ABC$ the projection relation holds:…
Prove geometrically or by using the trigonometric laws (Law of Sines and Law of Cosines) that in any triangle $ABC$ the projection relation holds: $a = b \cos C + c \cos B$. Then, use this identity together with the law of sines to prove the formula for the sine of the sum of two angles: $\sin(B+C) = \sin B \cos C + \cos B \sin C$.
Back to the buoy: two angles from the shore plus one measured baseline gave you a triangle with a known side and two known angles. The Law of Sines then gave the distance to the buoy.
Let's practice! 8 medium exercises in this section