Exercises: Solving right-angled triangles and applications

Solve the right-angled triangle in the figure, that is, calculate all its sides and all its angles.

Solve a right-angled triangle in which one side measures $7$ cm and its opposite angle is $40^\circ$.

The two shorter sides of a right-angled triangle measure $5$ cm and $9$ cm. Calculate its acute angles and the hypotenuse.

The hypotenuse of a right-angled triangle measures $15$ cm and one of its shorter sides measures $9$ cm. Find its acute angles and the other side.

From a point on the ground located $10$ m from the base of a tree, its top is observed with an angle of elevation of $30^\circ$.

How tall is the tree?

From a point on the ground $25$ m from the facade of a building, the roof is seen with an angle of elevation of $50^\circ$. What is the height of the building?

A chimney $30$ m high is observed from a point on the ground $45$ m from its base. What is the angle of elevation of its top?

Determine whether the following statement is true or false: "knowing the three angles of a right-angled triangle allows you to calculate its three sides". Justify your answer with an example.

Marcos wants to calculate how high the top end of a $3\text{ m}$ long ladder reaches when leaning against a vertical wall. Using a homemade protractor, he finds that the ladder forms an angle of $70^\circ$ with the horizontal ground.

To solve it, Marcos writes the following approach in his notebook:

1. a) Without doing any calculations with your calculator, explain why Marcos's result ($1,03\text{ m}$) makes no physical sense for a $3\text{ m}$ ladder tilted at $70^\circ$ (a very steep angle).
2. b) Explain what conceptual mistake Marcos made when choosing the trigonometric ratio and what that value of $1,03\text{ m}$ actually represents in reality.
3. c) Write the correct approach, calculate the actual height reached by the ladder, and round the result to two decimal places.

Following the technical recommendations of the IDAE (Institute for the Diversification and Saving of Energy) to optimize solar energy capture in homes in central Spain, a family installs a photovoltaic solar panel structure on their flat roof facing south with an inclination of $35^\circ$ relative to the horizontal. The length of the inclined panel (which acts as the hypotenuse of the structure) is $1,80\text{ m}$.

1. a) Describe which element of the right triangle (hypotenuse, opposite leg, or adjacent leg with respect to the $35^\circ$ angle) represents the maximum height of the panel above the roof and which represents the shadow or horizontal footprint it occupies on the ground.
2. b) Calculate at what height above the roof the upper edge of the photovoltaic panel sits.
3. c) Calculate the horizontal distance occupied by the base of the structure on the roof to know how much free space it leaves around it. Round both results to two decimal places of a meter.

A water park slide has a $18$ m ramp that forms an angle of $25^\circ$ with the ground, as shown in the figure.

1. a) How high is the starting platform? b) What horizontal distance does the slide occupy?

From the top of a $60$ m cliff, a buoy is sighted with an angle of depression of $25^\circ$, as shown in the figure.

1. a) How far from the foot of the cliff is the buoy? b) How far is it from the observer?

A hot air balloon is tethered to the ground by a taut rope of $80$ m that forms an angle of $60^\circ$ with the ground, as shown in the figure.

1. a) How high is the balloon? b) If a gust of wind tilts the rope to form $45^\circ$ with the ground, how many metres does the balloon descend?

To measure the width of a river, you stand directly opposite a tree on the other bank and walk $30$ m along your bank. From there, the tree is seen forming an angle of $55^\circ$ with the bank, as shown in the figure.

How wide is the river?

From a point on the ground $20$ m from a building, the base of the aerial on the roof is seen with an angle of elevation of $38^\circ$ and its top with $52^\circ$, as shown in the figure.

How tall is the aerial?

The ramp at a skatepark has an incline of $18^\circ$ and a length of $6$ m, as shown in the figure.

1. a) What vertical height does the ramp cover? b) What is its gradient expressed as a percentage?

An isosceles triangle has equal sides of $10$ cm and an unequal angle of $50^\circ$. a) Calculate the height perpendicular to the unequal side (hint: this height divides the triangle into two right-angled triangles). b) Calculate the area of the triangle.

A student claims: "from $30$ m away from a tower I see its top at an angle of $30^\circ$, so the tower is $30 \cdot \sen\ 30^\circ = 15\ \text{m}$ high". Identify the mistake, explain which trigonometric ratio should have been used, and calculate the correct height.

On a mountain in the Sierra de Guadarrama, an environmental ranger wants to find the height of a fire lookout tower. Standing on flat ground at a horizontal distance of $35\text{ m}$ from the base of the tower, he uses an inclinometer held at his eye level ($1,60\text{ m}$ above the ground) and measures an angle of elevation of $43^\circ$ to the top of the cabin.

1. a) Estimate: Since the measured angle ($43^\circ$) is very close to $45^\circ$, and knowing that in a right triangle with an acute angle of $45^\circ$ both legs have the same length, make a reasoned estimate of the total height of the tower without using a calculator. Do you expect the height above the ranger's eye level to be slightly greater or slightly less than $35\text{ m}$?
2. b) Calculate: Set up the appropriate trigonometric ratio and calculate precisely the total height of the tower above the ground (adding the observer's height). Round to two decimal places.
3. c) Check and evaluate: Compare the exact result with your previous estimate, calculate the difference between the two, and discuss whether the initial mental estimation was a practical and reasonable method for a first assessment in the field.

Spanish accessibility regulations in public spaces (Technical Building Code, DB-SUA) require gentle slopes so that wheelchair users can move safely. During the construction of a municipal sports center, a straight ramp has been designed to overcome a vertical drop of $0,75\text{ m}$, and the project specifies that the inclined surface (the ramp you travel along) must measure exactly $9,40\text{ m}$ in length.

1. a) Set up the trigonometric equation where the unknown is the angle of inclination $\alpha$ of the ramp with the horizontal and find its value in degrees (rounded to one decimal place).
2. b) Check whether the ramp complies with the technical recommendation that this angle should not exceed $5^\circ$.
3. c) Reverse construction: Design the data for an alternative access route yourself: write the statement of a problem where the starting data are the horizontal distance occupied by the ramp ($d$) and the angle of inclination set at $\alpha = 4^\circ$, such that the mathematical solution necessarily yields the exact same vertical drop of $0,75\text{ m}$. Calculate and show the exact numerical value of that horizontal distance $d$ that should appear in your problem (round to two decimal places).

From a certain point in a valley, the summit of a mountain is seen with an angle of elevation of $35^\circ$. Walking back $250$ m in a straight line, the summit is seen with an angle of $22^\circ$, as shown in the figure.

1. a) Set up a system of two equations using the tangent in each of the two right-angled triangles. b) Solve the system. c) What is the height of the mountain above the observer's eye level?

From the top of a $45$ m lighthouse, two ships are observed in line with the lighthouse: the nearer one with an angle of depression of $32^\circ$ and the farther one with $20^\circ$, as shown in the figure.

What distance separates the two ships?

A group of students wants to measure the width of a lake without crossing it. Opposite a buoy $B$ located on the far bank, they mark a point $P$ on their bank; from $P$ they walk $60$ m along the bank in a perpendicular direction to a point $Q$, and from $Q$ they measure the angle $\widehat{PQB} = 48^\circ$. a) Draw a diagram of the situation. b) Calculate the width of the lake. c) If a straight footbridge were built from $P$ to $B$ with an additional $5\ \%$ in length for the supports, what length would the footbridge be?

You can do it on paper and send a photo with your answer.

A mountain pass has a first section of $2{,}3$ km with a constant incline of $9^\circ$ and a second section of $1{,}8$ km with an incline of $6^\circ$. a) What vertical height is gained in each section? b) What is the total vertical gain of the mountain pass? c) What is the average gradient of the entire pass, expressed as a percentage? (Use the horizontal distances covered.)

Ana and Bruno are standing $50$ m apart on a flat field. Between them, in the same vertical plane, a kite is flying: Ana sees it with an angle of elevation of $55^\circ$ and Bruno with $65^\circ$, as shown in the figure.

1. a) At what height is the kite flying? b) At what horizontal distance from Ana is it?

From a point on the ground $30$ m from the base of a column, the top edge of the column is seen with an angle of elevation of $25^\circ$ and the top of the statue that crowns it with $40^\circ$. a) How tall is the column? b) How tall is the statue? c) With what angle of elevation would the whole structure be seen from a point $50$ m from the base?

A weather balloon rises vertically from a point $400$ m from an observer, as shown in the figure.

At a given moment, the observer sees it with an angle of elevation of $45^\circ$ and, one minute later, with $60^\circ$. a) What height was the balloon at in each observation? b) How much did it ascend in that minute? c) What was its average speed of ascent, in metres per second?

A zip wire connects two towers that are $22$ m and $14$ m tall, separated horizontally by $60$ m, as shown in the figure.

1. a) What is the difference in height between the two ends? b) What is the length of the cable, assuming it is taut and straight? c) At what angle of descent does the user slide?

A technical team needs to determine the height of a telecommunications tower located inside a fenced enclosure that cannot be entered to measure distances to its base. To solve the problem, two specialists discuss which field strategy to use from outside the fence:

• Strategy A (Double angle with inclinometer): Stand at a point $P$ on flat ground and measure an angle of elevation of $40^\circ$. Then, move back exactly $15\text{ m}$ in a straight horizontal line to a point $Q$ and measure a new angle of elevation of $28^\circ$. It requires setting up and solving a system of two equations using the tangent.

• Strategy B (Measurement with laser rangefinder): Stand at point $P$, measure the $40^\circ$ angle of elevation with the inclinometer, and aim at the top of the tower with a long-range laser rangefinder, which measures a direct line-of-sight distance (hypotenuse) of $33,87\text{ m}$.

1. a) Find the height of the tower using Strategy A: set up the system of equations using $\tg(40^\circ)$ and $\tg(28^\circ)$, eliminate the corresponding unknown by equating expressions, and calculate the height $h$ of the tower (round to two decimal places).
2. b) Find the height of the tower using Strategy B: calculate the height $h$ using a single trigonometric ratio from the rangefinder data.
3. c) Compare both methods and justify which of the two is more convenient in professional practice, analyzing the algebraic difficulty of the calculations, the number of measurements needed in the field, and the risk of carrying over rounding errors.

The Fuente Dé cable car, located in the Picos de Europa National Park (Cantabria), spans a large rock cliff in a single straight span with no intermediate towers. The official geographical data for the facility are:

• Altitude of the lower departure station: $1070\text{ m}$ above sea level.

• Altitude of the upper arrival station ("El Cable"): $1823\text{ m}$ above sea level.

• Length of the straight cable between both stations (distance traveled by the cabin): $1450\text{ m}$.

1. a) Model the situation using a right triangle: calculate the vertical drop covered by the cable car and identify what each magnitude ($1450\text{ m}$ and the calculated vertical drop) represents among the sides of the right triangle (hypotenuse, opposite leg, or adjacent leg with respect to the inclination with the horizontal ground).
2. b) Calculate the average angle of inclination that the cable forms with the horizontal (round to one decimal place of a degree).
3. c) Calculate the horizontal distance separating both stations (the straight-line projection onto a topographic map). Round to whole meters.
4. d) Suppose the cabin stops due to a safety shutdown when it has traveled exactly halfway along the cable ($725\text{ m}$ from the departure station). Reason, by applying properties of proportionality and triangle similarity (without using the calculator for trigonometric ratios), what altitude above sea level the cabin is at that moment. Then, numerically confirm your deduction by calculating that altitude using the sine of the angle obtained in part b).

Topic challenge. How tall is your environment really? You are going to calculate the height of an inaccessible object near your home (a building, tree, or streetlight) and analyze the behavior of its shadow using trigonometry.

1. Data collection: On a sunny day, measure the length of the chosen object's shadow with a tape measure. At that exact moment, note the time and look up the solar elevation ($\alpha$) in your town on AEMET or a weather website, or calculate that angle by measuring the shadow of a $1,0\text{ m}$ vertical reference object.

2. Mathematical solving: Model the situation using a right triangle. Apply the appropriate trigonometric ratio ($\tan \alpha = \frac{\text{height}}{\text{shadow}}$) and solve for the actual height of the object. Then, calculate how long its shadow will be when the Sun is at an elevation of $30^\circ$.

3. Digital report: Prepare a short one-page report (or a spreadsheet) containing the data table, the sketch of the solved triangle, and a practical recommendation: what is the minimum distance at which a planter or solar panel must be placed so that it is not shaded by the object during the middle hours of the day?

Assessment criteria:

• Rigor in obtaining real data and geometric formulation of the triangle.

• Correctness in the trigonometric calculation and algebraic solving with decimal precision.

• Quality of the presentation and practical justification of the final proposal.