How do surveyors measure the height of something they can't climb or reach, like a cell tower, a cliff, or the Gateway Arch? They don't use a giant tape measure. They stand on the ground, measure one distance and one angle, and let a right triangle do the rest.
Solving a right triangle
To solve a triangle means to find all of its sides and angles. In a right triangle you already know one angle ($90^\circ$), so you only need two more pieces of information, and at least one of them must be a side.
For an acute angle $\theta$ of a right triangle, the sides are named relative to $\theta$.
Which ratio should you use? Pick the one that contains the side you know and the side you want.
You know
You want
Use
angle and hypotenuse
opposite
$\sin$
angle and hypotenuse
adjacent
$\cos$
angle and adjacent
opposite
$\tan$
two sides
the angle
$\sin^{-1}$, $\cos^{-1}$ or $\tan^{-1}$
two sides
third side
Pythagorean Theorem
On the coordinate plane the legs are just horizontal and vertical distances. For $A(0,0)$, $B(5,0)$ and $C(5,12)$, the legs are $AB = 5$ and $BC = 12$, and the hypotenuse is $AC = 13$.
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A right triangle has legs measuring $5$ in and $12$ in. Calculate the length of the hypotenuse and the measure in degrees of its two acute angles.
A right triangle has legs measuring $5$ in and $12$ in. Calculate the length of the hypotenuse and the measure in degrees of its two acute angles.
The points $A(0,0)$, $B(8,0)$ and $C(8,6)$ form a right triangle on the Cartesian plane. Calculate the trigonometric ratios (sine, cosine and tangent) of the angle…
The points $A(0,0)$, $B(8,0)$ and $C(8,6)$ form a right triangle on the Cartesian plane. Calculate the trigonometric ratios (sine, cosine and tangent) of the angle located at the vertex $A$.
A ladder of length $4$ ft is leaning against a vertical wall. If the base of the ladder is $1.2$ ft from the wall, determine the angle the ladder makes with the…
A ladder of length $4$ ft is leaning against a vertical wall. If the base of the ladder is $1.2$ ft from the wall, determine the angle the ladder makes with the horizontal ground and the height its top end reaches on the wall.
The angle of elevation is the angle between the horizontal and your line of sight when you look up at something. It is the classic way to find a height you can't climb.
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An obelisk casts a shadow $15$ ft long on the horizontal ground at the moment the sun's rays form an angle of $42^\circ$ with the horizontal. Calculate the height of the…
An obelisk casts a shadow $15$ ft long on the horizontal ground at the moment the sun's rays form an angle of $42^\circ$ with the horizontal. Calculate the height of the obelisk.
A child is flying a kite held by a string of length $45$ ft, which is kept completely taut. If the string forms an angle of $55^\circ$ with the horizontal ground, at…
A child is flying a kite held by a string of length $45$ ft, which is kept completely taut. If the string forms an angle of $55^\circ$ with the horizontal ground, at what height is the kite? (Assume the child holds the string at ground level).
In a rectangular computer screen, the diagonal measures $60$ in and forms an angle of $29^\circ$ with the bottom base of the frame. Calculate the width and height of the…
In a rectangular computer screen, the diagonal measures $60$ in and forms an angle of $29^\circ$ with the bottom base of the frame. Calculate the width and height of the screen.
Looking down from a high point, the angle between the horizontal and your line of sight is the angle of depression. The horizontal line at the top is parallel to the ground, so by alternate interior angles the angle of depression equals the angle of elevation measured from the object up to you.
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From the lantern of a lighthouse, located $45$ ft above sea level, two ships are observed sailing in line with the base of the lighthouse. The angle of depression at…
From the lantern of a lighthouse, located $45$ ft above sea level, two ships are observed sailing in line with the base of the lighthouse. The angle of depression at which the closer ship is seen is $34^\circ$, and that of the further ship is $18^\circ$. Calculate the exact distance separating the two ships.
Sometimes you can't measure the distance to the base. Then take the angle of elevation from two points on a line to the base. Both right triangles share the same height $h$, which gives a system of two equations.
When the angles are $30^\circ$, $45^\circ$ or $60^\circ$, the answer can be written exactly with radicals. These values come from two special triangles.
Angle
$\sin$
$\cos$
$\tan$
$30^\circ$
$\dfrac{1}{2}$
$\dfrac{\sqrt{3}}{2}$
$\dfrac{\sqrt{3}}{3}$
$45^\circ$
$\dfrac{\sqrt{2}}{2}$
$\dfrac{\sqrt{2}}{2}$
$1$
$60^\circ$
$\dfrac{\sqrt{3}}{2}$
$\dfrac{1}{2}$
$\sqrt{3}$
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From a point on the ground, the nacelle of a wind turbine is observed with an angle of elevation of $32^\circ$. After advancing $25$ ft in a straight line toward the…
From a point on the ground, the nacelle of a wind turbine is observed with an angle of elevation of $32^\circ$. After advancing $25$ ft in a straight line toward the base of the tower, the new angle of elevation is $50^\circ$. Set up a system of trigonometric equations and calculate the height of the wind turbine.
Two observers, $A$ and $B$, are separated by a distance of $300$ ft on a horizontal plain. Both look toward a stationary drone located exactly in the vertical plane…
Two observers, $A$ and $B$, are separated by a distance of $300$ ft on a horizontal plain. Both look toward a stationary drone located exactly in the vertical plane passing through the straight line joining them, and situated spatially between them. Observer $A$ sees the drone with an angle of elevation of $60^\circ$ and observer $B$ sees it with an angle of $45^\circ$. Calculate the exact height at which the drone is flying, expressing the result with radicals.
A vertical pole casts a shadow on flat ground. At a certain time in the morning, when the sun's elevation is $\alpha = 30^\circ$, the shadow measures $L$ feet. A couple…
A vertical pole casts a shadow on flat ground. At a certain time in the morning, when the sun's elevation is $\alpha = 30^\circ$, the shadow measures $L$ feet. A couple of hours later, when the sun's elevation has increased to $\beta = 45^\circ$, it is found that the shadow has shortened by exactly $10$ feet. Set up a system of equations and calculate the exact height of the pole expressed with radicals.
Many shapes aren't right triangles, but they split into right triangles when you draw the right segment: the altitude of an isosceles triangle, the apothem of a regular polygon, or the half-diagonals of a rhombus.
Isosceles triangle. The altitude to the base is a perpendicular bisector: it cuts the base in half and bisects the apex angle. Each half is a right triangle.
Regular polygon with $n$ sides inscribed in a circle of radius $r$. Join the center to the vertices to get $n$ identical isosceles triangles with central angle $\dfrac{360^\circ}{n}$. Cut one in half with the apothem $a$.
Rhombus. The diagonals of a rhombus are perpendicular and bisect each other, so the half-diagonals are the legs of four congruent right triangles whose hypotenuse is a side of the rhombus.
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An isosceles triangle has two equal sides measuring $14$ in each, and the unequal angle (the one between them) measures $40^\circ$. By drawing the height corresponding…
An isosceles triangle has two equal sides measuring $14$ in each, and the unequal angle (the one between them) measures $40^\circ$. By drawing the height corresponding to the base to divide it into two right triangles, calculate the length of this base.
Calculate the area and the perimeter of a regular hexagon inscribed in a circle of radius $12$ in. (Remember to divide the polygon into triangles to use trigonometry).
Calculate the area and the perimeter of a regular hexagon inscribed in a circle of radius $12$ in. (Remember to divide the polygon into triangles to use trigonometry).
The diagonals of a rhombus measure $24$ in and $10$ in. Knowing that the diagonals of a rhombus intersect perpendicularly at their midpoint, calculate the measure of the…
The diagonals of a rhombus measure $24$ in and $10$ in. Knowing that the diagonals of a rhombus intersect perpendicularly at their midpoint, calculate the measure of the sides of the rhombus and the size of its four interior angles.
From a point $P$ outside a circle of radius $8$ in, the two tangent lines to it are drawn. If the distance from the point $P$ to the center of the circle is $17$ in,…
From a point $P$ outside a circle of radius $8$ in, the two tangent lines to it are drawn. If the distance from the point $P$ to the center of the circle is $17$ in, calculate the total angle formed between the two tangent lines.
Two intersecting circles have radii of $10$ in and $17$ in. The length of the common chord to both circles is $16$ in, and the centers lie on opposite sides of this…
Two intersecting circles have radii of $10$ in and $17$ in. The length of the common chord to both circles is $16$ in, and the centers lie on opposite sides of this chord. Calculate the distance separating the centers of the two circles and determine the angle formed between their respective tangent lines at one of the points of intersection.
A road with a $12\%$ grade rises $12$ ft for every $100$ ft of horizontal distance (some problems say "gradient"). So grade is a tangent written as a percent: $\text{grade} = \dfrac{\text{rise}}{\text{run}}\times 100\%$ and $\tan\alpha = \dfrac{\text{rise}}{\text{run}}$.
The run is the horizontal distance. The ramp length or road surface is the hypotenuse.
Projections. A segment of length $L$ tilted at an angle $\alpha$ from the horizontal has a horizontal "shadow" of $L\cos\alpha$ if the light falls straight down, and a vertical height of $L\sin\alpha$.
If the light falls perpendicular to a tilted roof, the "ground" for the shadow is the roof itself. A panel tilted at $\alpha$ to the roof then casts a shadow of length $L\cos\alpha$ on the roof.
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A straight wheelchair access ramp has a length of $6$ ft and overcomes a vertical drop of $0.45$ ft. Determine the angle of inclination of the ramp with respect to the…
A straight wheelchair access ramp has a length of $6$ ft and overcomes a vertical drop of $0.45$ ft. Determine the angle of inclination of the ramp with respect to the horizontal.
A traffic sign warns of a prolonged gradient of $15\%$. This indicates that for every $100$ ft advanced in the horizontal projection, there is a vertical ascent of $15$…
A traffic sign warns of a prolonged gradient of $15\%$. This indicates that for every $100$ ft advanced in the horizontal projection, there is a vertical ascent of $15$ ft. Calculate the angle of inclination of the road and the actual distance a vehicle will travel on the road surface if it advances $600$ ft in the horizontal projection.
A rectangular solar panel $2.5$ ft long is inclined at $35^\circ$ to the horizontal of a flat roof. Calculate the length of the shadow the panel casts on the roof when…
A rectangular solar panel $2.5$ ft long is inclined at $35^\circ$ to the horizontal of a flat roof. Calculate the length of the shadow the panel casts on the roof when the sun's rays strike completely perpendicular to the plane of the roof.
Harder problems combine several right triangles. These three ideas cover most of them.
1. Two tops, one horizontal distance. If two objects have different heights, the vertical side of your triangle is the difference of the heights, not either height alone. For a building with an antenna, find the height to the base and the height to the top separately, then subtract.
2. A shared distance. When you stand at the midpoint between two objects, the horizontal distance to each is the same, so you can find it from one object and reuse it for the other.
3. Bearings and moving objects. A ship moving at constant speed travels distance = rate × time. A knot is one nautical mile per hour. Directions such as "angle with respect to north" are measured from the north line, so draw a compass sketch with north pointing up.
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Two vertical industrial chimneys of heights $20$ ft and $35$ ft are built on flat ground. From the midpoint of the straight line segment joining their bases, the top of…
Two vertical industrial chimneys of heights $20$ ft and $35$ ft are built on flat ground. From the midpoint of the straight line segment joining their bases, the top of the first chimney is observed with an angle of elevation of $35^\circ$. With what angle of elevation will the top of the second chimney be observed from that same midpoint?
A telecommunications antenna is installed on the flat roof of a building. From a point on the ground, located $45$ ft from the base of the building, a surveyor observes…
A telecommunications antenna is installed on the flat roof of a building. From a point on the ground, located $45$ ft from the base of the building, a surveyor observes the base of the antenna with an angle of elevation of $38^\circ$ and the top of the antenna with an angle of elevation of $46^\circ$. Calculate the height of the building and the exact length of the antenna.
A coast guard detects that a ship $A$ is sailing in a straight line toward the East at a constant speed of $20$ knots, and a ship $B$ is sailing toward the North at $15$…
A coast guard detects that a ship $A$ is sailing in a straight line toward the East at a constant speed of $20$ knots, and a ship $B$ is sailing toward the North at $15$ knots. At a given instant ($t=0$), both ships are at exactly the same point, where their paths cross. At what angle with respect to the North direction will the captain of ship $A$ observe ship $B$ exactly $2$ hours after crossing?
The hardest problems have unknown sides, exact answers or a statement to prove. Use the same toolbox, with some algebra.
Sides given by an expression. Write the Pythagorean Theorem with the expressions, expand, and solve the quadratic equation. Reject any solution that gives a negative length. Some problems ask for the answer as an "irreducible fraction", which just means a fraction in simplest form. The smallest angle is opposite the shortest side.
Proofs with definitions. To prove an identity about a ramp or any right triangle, start from the definitions: $\sin\alpha = \dfrac{h}{L}$, $\cos\alpha = \dfrac{b}{L}$, where $h$ is the vertical drop, $b$ the horizontal base and $L$ the length of the ramp. Solve each for the length you need, then combine and simplify.
Rectangles inside triangles. Combine trig with similar triangles. In an isosceles triangle with base $B$ and height $H$, a rectangle with width $w$ and height $k$ resting on the base satisfies $\dfrac{w}{B} = \dfrac{H - k}{H}$, because the small triangle above the rectangle is similar to the whole triangle. If the diagonals of the $w \times k$ rectangle cross at an angle $\varphi$ facing the width, then half of that angle has opposite side $\dfrac{w}{2}$ and adjacent side $\dfrac{k}{2}$, so $\tan\dfrac{\varphi}{2} = \dfrac{w}{k}$.
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In a right triangle, the length of one leg is $x$ in, that of the other leg is $(x+7)$ in and the hypotenuse measures $(x+14)$ in. Set up and solve the corresponding…
In a right triangle, the length of one leg is $x$ in, that of the other leg is $(x+7)$ in and the hypotenuse measures $(x+14)$ in. Set up and solve the corresponding equation to determine the value of $x$. Then, calculate the exact value (as an irreducible fraction) of the sine and cosine of the smallest angle of the triangle.
A straight ramp is to be built to overcome a vertical drop of height $h$. If the angle of inclination of the ramp with respect to the horizontal is $\alpha$, prove using…
A straight ramp is to be built to overcome a vertical drop of height $h$. If the angle of inclination of the ramp with respect to the horizontal is $\alpha$, prove using the definitions of the trigonometric ratios that the sum of the length of the base of the ramp (its horizontal projection) plus the length of the ramp itself is equal to $h \cdot \frac{1 + \cos(\alpha)}{\sin(\alpha)}$.
In an isosceles triangle whose base measures $12$ in and whose height above this base is $18$ in, a rectangle is inscribed so that one of its sides rests on the base of…
In an isosceles triangle whose base measures $12$ in and whose height above this base is $18$ in, a rectangle is inscribed so that one of its sides rests on the base of the triangle and the other two vertices touch the equal sides. If the diagonals of this rectangle intersect forming an angle of $60^\circ$ (this being the angle subtending the base of the rectangle), set up the trigonometric and geometric system necessary to calculate the exact dimensions of this rectangle.
Back to the start: the Gateway Arch is $630$ ft tall. Standing $300$ ft from its base, you'd look up at an angle of $\tan^{-1}\dfrac{630}{300} = \tan^{-1}(2.1) \approx 64.5^\circ$. Run it the other way, and a $64.5^\circ$ angle plus the $300$ ft distance give you the height: $300\tan 64.5^\circ \approx 629$ ft, with no climbing.
Summary
Let's practice! 8 medium exercises in this section