Marcus wants to know how tall the flagpole in front of his high school is, and he is not about to climb it. He stands 30 ft from the base, aims his phone's clinometer app at the top of the pole, and reads $52^\circ$. Can that one angle and one distance tell him the height? By the end of this page you will see that they can.
Solving a right triangle
To solve a triangle means to find all of its sides and all of its angles. In a right triangle you already know one angle ($90^\circ$), so you only need two more pieces of information: two sides, or one side and one acute angle.
On the coordinate plane, a horizontal or vertical segment has length equal to the difference of coordinates. Take $A(0, 0)$, $B(12, 0)$, $C(12, 5)$. The legs are $AB=12$ and $BC=5$, so $AC=\sqrt{12^2+5^2}=13$. At vertex $A$: $\sin A=\dfrac{5}{13}$, $\cos A=\dfrac{12}{13}$, $\tan A=\dfrac{5}{12}$.
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A right-angled 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-angled 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 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 points $A(0,0)$, $B(8,0)$ and $C(8,6)$ form a right-angled triangle on the Cartesian plane. Calculate the trigonometric ratios (sine, cosine and tangent) of the…
The points $A(0,0)$, $B(8,0)$ and $C(8,6)$ form a right-angled triangle on the Cartesian plane. Calculate the trigonometric ratios (sine, cosine and tangent) of the angle located at the vertex $A$.
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).
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?
The horizontal dashed line and the ground are parallel, so the angle of depression equals the angle of elevation from the object back up to the observer (alternate interior angles). That means the height is the side opposite the angle at the object.
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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 cannot measure the distance to the base. The fix is to take two angle readings and write two equations that share the same height $h$.
When the angles are $30^\circ$, $45^\circ$ or $60^\circ$, you can keep exact values with radicals:
$\theta$
$\sin\theta$
$\cos\theta$
$\tan\theta$
$30^\circ$
$\dfrac12$
$\dfrac{\sqrt3}{2}$
$\dfrac{\sqrt3}{3}$
$45^\circ$
$\dfrac{\sqrt2}{2}$
$\dfrac{\sqrt2}{2}$
$1$
$60^\circ$
$\dfrac{\sqrt3}{2}$
$\dfrac12$
$\sqrt3$
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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 towards 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 towards 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.
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.
Two observers, $A$ and $B$, are separated by a distance of $300$ ft on a horizontal plain. Both look towards 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 towards 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.
Many figures hide right triangles. Cut the figure so that the right triangles appear.
Isosceles triangle: the altitude to the base splits it into two congruent right triangles. It halves the apex angle and halves the base.
Rectangle: a diagonal is the hypotenuse of a right triangle whose legs are the width and the height.
Rhombus: the diagonals are perpendicular and bisect each other, so four congruent right triangles appear. Each side is a hypotenuse, and the diagonal angles are halves of the rhombus angles.
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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-angled triangles, calculate the length of this base.
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.
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.
A regular polygon inscribed in a circle splits into $n$ congruent isosceles triangles. Each one has its apex at the center, with central angle $\dfrac{360^\circ}{n}$, and two sides equal to the radius $r$. Draw the altitude of one of them (the apothem $a$) to get right triangles.
Tangent lines. A tangent to a circle is perpendicular to the radius at the point of tangency. So from an outside point $P$, the segment to the center $O$ and the two tangent segments make right triangles, and $PO$ bisects the angle between the tangents.
Intersecting circles. The line through the two centers is the perpendicular bisector of the common chord. So half of the chord is a leg of two right triangles, one in each circle, with the radii as hypotenuses. The tangent lines at an intersection point are perpendicular to the radii, so the angle between the tangents equals the angle between the radii (or its supplement).
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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).
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.
Roads and ramps are described by a grade (a percent slope).
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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.
In harder problems, the unknown is not a number you can plug in but a letter you must solve for. Name the unknown, write one relation for each fact (Pythagorean Theorem, a trig ratio, or similar triangles), and solve.
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In a right-angled 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…
In a right-angled 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.
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.
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.
Moving objects. Distance $=$ rate $\times$ time. Ships on perpendicular courses make the two legs of a right triangle, and you can find directions with the inverse tangent. An angle "with respect to North" is measured from the North line toward the object.
Proofs. To prove an identity, do not use a calculator or numbers. Write each side length with a definition ($\sin$, $\cos$ or $\tan$), then manipulate the symbols.
For the sum $b+\ell$ you follow the same steps: just add instead of subtract.
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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 $h \cdot \frac{1 + \cos(\alpha)}{\sin(\alpha)}$.
A Coast Guard detects that a ship $A$ is sailing in a straight line towards the East at a constant speed of $20$ knots, and a ship $B$ is sailing towards the North at…
A Coast Guard detects that a ship $A$ is sailing in a straight line towards the East at a constant speed of $20$ knots, and a ship $B$ is sailing towards 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?
Back to Marcus: the 30 ft from the pole is the adjacent side, the height is the opposite side, so $h=30\tan52^\circ\approx 38.4$ ft (plus the height of his phone above the ground).
Summary
Let's practice! 8 medium exercises in this section