Exercises: Applications of trigonometry

A $6\text{ m}$ ladder leans against the wall of a house and makes a $70^\circ$ angle with the ground. How high up the wall does the ladder reach? Use your calculator in degree mode and round to the nearest hundredth.

A jacaranda tree casts a $12\text{ m}$ shadow at a moment when the sun's rays make a $35^\circ$ angle with the ground. How tall is the tree? Round to the nearest hundredth.

The access ramp at a school in Puebla rises $0.6\text{ m}$ over a horizontal run of $7.2\text{ m}$. What angle does the ramp make with the ground? Round to the nearest hundredth of a degree.

For a school project, you need to calculate the length of the parallel of latitude $40^\circ$ N. Knowing that the Earth's radius is $R \approx 6\,371$ km, calculate:

1. a) The radius of the circle formed by this parallel (use the expression $r = R \times \cos(\text{latitud})$ and justify it from the right-angled triangle in the figure).
2. b) The total length (circumference) of this parallel.

From the top of a cliff $32\text{ m}$ above sea level in Acapulco, Valeria sees a boat at an angle of depression of $14^\circ$. A few minutes later, the boat has moved in a straight line toward the foot of the cliff, and she now sees it at an angle of depression of $20^\circ$.

1. a) What was the horizontal distance from the boat to the foot of the cliff at first?
2. b) And at the end?
3. c) How many meters did the boat travel? Round your answers to the nearest tenth.

Sofía stands $40\text{ m}$ from the base of the Angel of Independence in Mexico City and sees the top of the statue at an angle of elevation of $47^\circ$. Her eyes are $1.55\text{ m}$ above the ground. According to her measurement, what is the total height of the monument? Round to the nearest tenth.

Mateo lets out $80\text{ m}$ of string, pulled tight, and his kite ends up $52\text{ m}$ above his hand.

1. a) What angle does the string make with the horizontal? Round to the nearest hundredth of a degree.
2. b) What is the horizontal distance from Mateo to the kite? Round to the nearest hundredth.

Two cities, A and B, are on the same parallel at a latitude of $45^\circ \text{ N}$. City A has a longitude of $20^\circ \text{ E}$ and city B a longitude of $70^\circ \text{ E}$.

Given that the radius of the Earth is approximately $6371 \text{ km}$:

1. Calculate the radius of the $45^\circ \text{ N}$ parallel (Remember that the radius of a parallel at a latitude $\alpha$ is calculated as $r = R \cdot \cos \alpha$; for $45^\circ$ use the exact value $\cos 45^\circ = \frac{\sqrt{2}}{2}$).

2. Determine the distance in kilometers separating both cities along the arc of the parallel.

A 9th-grade class wants to estimate the height of the Pyramid of the Sun in Teotihuacan. From a point $P$ at ground level, they see the top at an angle of elevation of $30^\circ$. They walk $50\text{ m}$ in a straight line toward the pyramid, to a point $Q$, and from there they see the top at an angle of elevation of $45^\circ$. (All distances are horizontal, measured to the point directly below the top.)

1. a) Let $h$ be the height and $d$ the horizontal distance from $Q$. Write two equations using the tangent.
2. b) Solve the system and find the height of the pyramid. Round to the nearest tenth.
3. c) What was the horizontal distance from point $P$ to the top?

From the window of an apartment in Monterrey, $24\text{ m}$ above the ground, Diego sees the roof of the building across the street at an angle of elevation of $32^\circ$ and the base of that same building at an angle of depression of $50^\circ$.

1. a) What is the horizontal distance between the two buildings?
2. b) How tall is the building across the street?
3. c) Explain why the angle of depression from the window equals the angle of elevation at which the window would be seen from the base of the other building.

Round your answers to the nearest hundredth.