Applications of Trigonometry (3D)
Trigonometry is a very old branch of mathematics that remains extremely useful today. If ever a distance or angle needs to be measured, and it cannot be measured directly, trigonometry might offer some helpful tools. Questions like,
- How far apart are those stars?
- How tall are the Egyptian pyramids?
- How much inclination does this road have over the next mile?
- What are the boundaries of this property?
- How tall is that mountain?
Here’s how: To measure something with trigonometry you only need to know a single distance and an angle. For example, you could easily measure the height of an electrical line pole even though you’d be unlikely to have the equipment to measure the height directly.
You can measure the distance from the pole, can measure the angle between the pole and the ground (should be very close to 90°), and can measure the angle from the ground to the pole’s top. While this would be a huge triangle, if two triangles, regardless of length size, share the same angles, they’re proportional. That’s how we go about using trigonometry to find distances or angles that cannot be measured directly.
In this case, suppose the angle was 28° and the distance was 40 feet. We’d have this triangle:
Here we know the horizontal distance, but need the vertical distance, so we’d use tangent.
The take-away is this. If a triangle can be constructed within a 3-dimensional shape, or over very long distances, then trigonometry can be used. The trick is seeing the triangles, recognizing the important information, and identifying the key formulas to be used.
Let’s see an example with several triangles and angles, that would likely require the use of sine, cosine, tangent, as well as the sine and cosine rules.
In the diagram below we have a building with a flagpole on top. The measured horizontal distance to the build’s base is 55 m, and the building is 15 m wide. The angle to the top of the building from the ground is 30°, and the angle between the top of the building and the flagpole is 12°. A guy-wire (tension wire) runs from the top of the flagpole to the top of the building (F to D).
- How tall is the building?
- How tall is the flagpole?
- What’s the distance EA? What about FA?
- What’s the angle between from the point of the guy wire (D) to the top of the flagpole (F)?
Q: How tall is the building?
We can answer this with tangent, because we know the angle and the horizontal distance.
Q: How tall is the flagpole?
This is a slightly trickier question. Let’s assume the question is not asking the distance of the flagpole above the ground, but instead, the height of the flagpole, regardless of where it is set.
The height of the pole above the ground will be the building’s height plus the pole’s length. We know the building’s height. We can find the height of both with the sine function, our angle this time being 42°. But, we must subtract 27.5 m from that to get the flagpole’s height.
This makes the building and flagpole 36.8021… m. Keep in mind that calculators round the irrational numbers. If you round and then perform future calculations with the rounding of the rounded answer (you rounded the calculators already approximated answer), those rounding errors get compounded. While this is only subtraction that comes next, it is a good habit to allow the calculator to “remember” and use the previous answer. It is a good idea to write the approximation on paper, but use the “ans” function on your calculator instead of your written number.
Since the height of the building was 27.5, and the height of the pole and building combined is 36.8, the pole is the difference of the two, 9.3 meters.
Q: What are the distances EA and FA?
These are both a hypotenuse of a right triangle. You know two sides so you could use the Pythagorean theorem. However, if your heights are off due to some mistake, then you’d be using an incorrect value. It is better, if possible, to only use what was provided as information to arrive at answers.
The distance EA is the hypotenuse of a 30 – 60 – 90 triangle, where the long side is 55 m. The long side is the hypotenuse times “the square root of three over two.” We can use that to set up a simple equation.
We must multiply by the reciprocal. Since we are using a calculator for this, it is very quick (no need to rationalize the denominator with a calculator).
EA = 63.5 m
Note: We will be rounding to three significant figures here. Use whatever rounding rule is conventional (as with time or money), or is provided in the instructions.
For the distance FA we have to use trigonometry. We have a right triangle with an angle of 42°, and an adjacent side of 55 m. We need the hypotenuse. This is cosine, or secant.
Note: It is a great practice to try these calculations twice, just to double-check your calculator work.
Q: The distance of the guy wire (from D to F)?
We have a right triangle here, and we know the height of each leg. The guy wire is the hypotenuse.
The distance of the wire is 17.6 m, rounded to three significant figures.
We ran through this example to show you that you have to be creative and use the tools at your disposal. It is tricky to see what information is most useful, and sometimes how to break apart a problem into pieces that make sense.
It is easy to get crossed up and confused, so having different approaches to a problem can be helpful! For reference, here is a table of all of the tools and requirements at your disposal, so far.