Geometry and Trigonometry · Trigonometry of general angles
Applications of Trigonometry
Sine, cosine, and tangent let you measure things you can't reach with a tape measure: the height of a kite, the length of a ramp, or the size of the Earth. First you will solve height and distance problems in right triangles. Then comes a large-scale application: the Earth is (almost) a sphere, and with the cosine you can calculate how big a parallel is —one of those horizontal lines on maps— and the distance between two cities on the same one.
Heights and distances with sine, cosine, and tangent
Almost every trigonometry problem in Secondary Year 3 follows the same plan: find a right triangle, mark the angle you know, and decide which ratio connects the given data with the unknown.
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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…
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.
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…
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.
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…
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.)
a) Let $h$ be the height and $d$ the horizontal distance from $Q$. Write two equations using the tangent.
b) Solve the system and find the height of the pyramid. Round to the nearest tenth.
c) What was the horizontal distance from point $P$ to the top?
A parallel is a circle made up of all the points on Earth that are at the same latitude (the angle relative to the plane of the equator). The equator, at latitude $0º$, has the same radius as the whole Earth; any other parallel is smaller, because it "narrows" as it gets closer to the poles.
Look at the right triangle formed by the center of the Earth, the center of the parallel, and a point on the parallel: the hypotenuse is $R$ (Earth's radius), and the leg adjacent to the latitude angle $\alpha$ is precisely $r$ (the radius of the parallel).
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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: a)…
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:
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).
b) The total length (circumference) of this parallel.
Once you know the radius $r$ of a parallel, calculating its length (its perimeter) uses the same formula as always for a circle: $2\pi r$. The difference from the equator is that here the radius is no longer $R$, but the $r$ reduced by the cosine of the latitude.
Let's practice! 4 medium exercises in this section
If two cities are on the same parallel but at different longitudes (the east-west angle), the distance between them along the parallel is an arc of that smaller circle, not of the whole globe. You first need the radius of the parallel, and then the usual arc-length formula with the angle that separates them.
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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…
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 $6371 \text{ km}$:
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 $\cos 45^\circ = \frac{\sqrt{2}}{2}$).
Determine the distance in kilometers separating both cities along the arc of the parallel.