Exercises: Solving right-angled triangles

A right-angled triangle has legs measuring $5$ cm and $12$ cm. Calculate the length of the hypotenuse and the measure in degrees of its two acute angles.

A ladder $4$ m in length is resting against a vertical wall. If the foot of the ladder is $1,2$ m from the wall, determine the angle the ladder makes with the horizontal ground and the height reached by its top end on the wall.

An obelisk casts a shadow $15$ m long on the horizontal ground at the moment when the sun's rays form an angle of $42^\circ$ with the horizontal. Calculate the height of the obelisk.

A child flies a kite attached to a string of length $45$ m, which is held 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).

On a rectangular computer screen, the diagonal measures $60$ cm and forms an angle of $29^\circ$ with the bottom edge of the frame. Calculate the width and height of the screen.

A straight wheelchair ramp has a length of $6$ m and covers a vertical rise of $0,45$ m. Determine the angle of inclination of the ramp relative to the horizontal.

An isosceles triangle has two equal sides measuring $14$ cm each, and the non-equal angle (the included angle between them) measures $40^\circ$. By drawing the altitude to the base to divide it into two right-angled triangles, calculate the length of that base.

Points $A(0,0)$, $B(8,0)$, and $C(8,6)$ form a right-angled triangle in the Cartesian plane. Calculate the trigonometric ratios (sine, cosine, and tangent) of the angle at vertex $A$.

To calculate the distance between the base of a coastal lighthouse $24\text{ m}$ high above sea level and an anchored boat, Lucas measures an angle of depression of $20^\circ$ from the lighthouse lantern. In his notebook, he writes:

1. a) Without using a calculator, explain through elementary geometric reasoning (mentally comparing with the shape of a $45^\circ$ triangle) why a result of $8,74\text{ m}$ is clearly absurd for an angle of depression of only $20^\circ$.
2. b) Identify the conceptual error Lucas made by placing the angle of depression with respect to the vertical instead of the horizontal.
3. c) Correct the formulation and calculate the actual horizontal distance $x$ between the lighthouse and the boat.

In the renovation of a health centre in Andalucía, a straight access ramp is planned to bridge a vertical elevation gain of $1,2\text{ m}$. Due to urban space limitations on the pavement, the maximum available horizontal run is exactly $10\text{ m}$.

1. a) Before calculating: reason whether the angle of inclination of the ramp will be greater or smaller than $10^\circ$. To argue this, consider what angle of inclination would correspond to a $100\,\%$ slope (equal legs) and how the tangent function behaves for small acute angles.
2. b) Calculate the exact measure of the angle of inclination $\alpha$ of the ramp with the horizontal and the length a user will travel along it in a straight line.
3. c) Spanish accessibility regulations (Código Técnico de la Edificación) state that a ramp of this length must not exceed a slope of $6\,\%$ (that is, $\tan(\alpha) \le 0,06$). Check whether the designed ramp complies with the standard and indicate whether your initial estimate from part a) was consistent.

From a point on the ground, the nacelle of a wind turbine is observed with an angle of elevation of $32^\circ$. Moving $25$ m 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.

Calculate the area and perimetre of a regular hexagon inscribed in a circle of radius $12$ cm. (Remember to divide the polygon into triangles to use trigonometry).

From the lantern room of a lighthouse, located $45$ m above sea level, two ships are observed sailing in line with the base of the lighthouse. The angle of depression to the nearer ship is $34^\circ$, and to the farther ship is $18^\circ$. Calculate the exact distance separating the two ships.

The diagonals of a rhombus measure $24$ cm and $10$ cm. Given that the diagonals of a rhombus bisect each other at right angles, calculate the length of the sides of the rhombus and the size of its four interior angles.

A road sign warns of a steep gradient of $15\%$. This indicates that for every $100$ m travelled in the horizontal projection, there is a vertical rise of $15$ m. Calculate the angle of inclination of the road and the actual distance a vehicle will travel on the road surface if it advances $600$ m in the horizontal projection.

Two vertical industrial chimneys of heights $20$ m and $35$ m are built on flat ground. From the midpoint of the line segment joining their bases, the top of the first chimney is observed with an angle of elevation of $35^\circ$. What will be the angle of elevation of the top of the second chimney from the same midpoint?

From a point $P$ outside a circle of radius $8$ cm, two tangent lines to the circle are drawn. If the distance from point $P$ to the centre of the circle is $17$ cm, calculate the total angle between the two tangent lines.

A rectangular solar panel $2,5$ m long is inclined at $35^\circ$ to the horizontal on a flat roof. Calculate the length of the shadow cast by the panel on the roof when the sun's rays strike completely perpendicular to the plane of the roof.

On a section of the Pajares mountain pass (Asturias), a DGT road sign warns of a dangerous steep incline of «$12\,\%$», indicating a vertical rise of $12\text{ m}$ for every $100\text{ m}$ of forward movement in horizontal projection. A vehicle's odometer indicates that it has travelled $2,5\text{ km}$ along the tarmac continuously uphill on this section.

1. a) Model the situation using a right-angled triangle, precisely identifying which geometric element (leg or hypotenuse) corresponds to the distance recorded by the vehicle, and determine the road's angle of inclination $\alpha$ in sexagesimal degrees.
2. b) Calculate the altitude gained by the vehicle in metres and the actual horizontal distance it has travelled.
3. c) A driver mentally calculates the elevation gain by directly multiplying the distance along the tarmac by the percentage: $2500 \cdot 0,12 = 300\text{ m}$. Explain mathematically what conceptual confusion between trigonometric ratios they have made, calculate the error in metres of their approximation, and reason why this difference turns out to be so small for typical road slopes.

When solving a surveying problem to determine the height $h$ of the Mudéjar tower of San Martín (Teruel) by measuring angles from two points aligned with its base on level ground, a student wrote down on the blackboard only the final equation they arrived at after setting up a system and eliminating the horizontal distance variable:

1. a) Reconstruct the complete text of the original problem: describe the relative position of the two observation points with respect to the tower, specify the distance between them, and clarify the surveying meaning of the angles $28^\circ$ and $44^\circ$.
2. b) Justify geometrically why a subtraction of ratios appears in the parentheses rather than an addition: where should the two observers have been positioned for the terms to be added instead of subtracted?
3. c) Solve the equation and find the height $h$ of the Mudéjar tower to two decimal places.

A telecommunications antenna is installed on the flat roof of a building. From a point on the ground, situated $45$ m 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.

In a right-angled triangle, the length of one leg is $x$ cm, the other leg is $(x+7)$ cm and the hypotenuse is $(x+14)$ cm long. 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 in the triangle.

Two observers, $A$ and $B$, are separated by a distance of $300$ m on a horizontal plain. Both look towards a stationary drone that is located exactly in the vertical plane passing through the straight line joining them, positioned 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 altitude at which the drone is flying, expressing the result in surd form.

Two intersecting circles have radii of $10$ cm and $17$ cm. The length of the common chord of both circles is $16$ cm, and the centres lie on opposite sides of this chord. Calculate the distance between the centres of the two circles and determine the angle formed between their respective tangent lines at one of the points of intersection.

It is desired to construct a straight ramp to overcome a vertical drop of height $h$. If the angle of inclination of the ramp relative to the horizontal is $\alpha$, prove using the definitions of trigonometric ratios that the sum of the length of the ramp's base (its horizontal projection) plus the length of the ramp itself is equal to $h \cdot \frac{1 + \cos(\alpha)}{\sin(\alpha)}$.

A coastguard 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 moment ($t=0$), both ships are at exactly the same point, where their paths cross. At what angle relative to North will the captain of ship $A$ observe ship $B$ exactly $2$ hours after crossing?

In an isosceles triangle whose base is $12$ cm long and whose altitude above this base is $18$ cm, a rectangle is inscribed such that one of its sides lies 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 required to calculate the exact dimensions of this rectangle.

A vertical post casts a shadow on flat ground. At a point in the morning, when the sun's angle of elevation is $\alpha = 30^\circ$, the shadow measures $L$ metres. A couple of hours later, when the sun's angle of elevation has increased to $\beta = 45^\circ$, it is found that the shadow has shortened by exactly $10$ metres. Set up a system of equations and calculate the exact height of the post expressed in surd form.

On a vertical cliff facing the sea stands a watchtower $18\text{ m}$ high. From a coastal patrol boat stationary in the water, the angles of elevation to the base of the tower ($\alpha = 24^\circ$) and to its top ($\beta = 31^\circ$) are measured. We wish to calculate the height $H$ of the cliff above sea level.

Two officers propose different trigonometric methods:

• Method 1 (Angles with the horizontal): Use the unknown horizontal distance $d$ to the cliff via $\tan(24^\circ) = \frac{H}{d}$ and $\tan(31^\circ) = \frac{H+18}{d}$, isolating $d$ from the first and substituting into the second.

• Method 2 (Angles with the vertical / zenith angles): Work with the angles that the lines of sight make with respect to the vertical of the cliff ($\theta_1 = 90^\circ - 24^\circ = 66^\circ$ and $\theta_2 = 90^\circ - 31^\circ = 59^\circ$), expressing $d = H \cdot \tan(66^\circ)$ and $d = (H+18) \cdot \tan(59^\circ)$, and equating both expressions.

1. a) Solve the problem by applying both methods completely to find $H$.
2. b) Algebraically analyse both approaches: which of the two requires isolating an unknown in the denominator, and which leads directly to a linear equation in $H$ without algebraic fractions?
3. c) If in a field notebook the tangent values are rounded to only two decimal places in the intermediate steps, show numerically which of the two methods propagates a greater error in the final result and justify which one you would choose to programme the vessel's navigation software.

A technical team is planning a photovoltaic solar array on the flat horizontal roof of a logistics centre in Zaragoza. Each row of panels is $2,4\text{ m}$ long on its inclined plane and is mounted facing south with a fixed elevation angle of $36^\circ$ with respect to the roof.

1. a) Model the cross-section of a row using a right-angled triangle and calculate the vertical height $h$ of its upper edge and the length of its projected horizontal base $b$.
2. b) At the winter solstice at 12:00 (solar time), the sun's rays reach their minimum inclination of the year at that latitude, with a solar elevation angle of only $\gamma = 25^\circ$ above the horizon. Model the right-angled triangle describing the shadow cast by the row and deduce the minimum horizontal distance $D$ that must exist between the rear of one row and the foot of the next to prevent any shading on the panels.
3. c) If the roof measures exactly $45\text{ m}$ deep in the north-south direction, determine the maximum number of complete rows that can be installed while guaranteeing the complete absence of shadows at the winter solstice, and indicate how many metres of roof depth remain unused at the end.