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Sine and Cosine Tutorial




Key concepts:

Trigonometry

Sine and cosine

Geometry



Look, we get it; trigonometry kind of sucks. As if regular geometry wasn't bad enough by itself, you are now being asked to remember all of these cosine and sine things (whatever they are). Let's get the basics straight:

Question 1:

"You point a laser range finder towards the tip of the CN tower at an angle of 70 degrees. It tells you that the tip is 588.8m away from where you are standing. Determine the height of the CN tower and how far away it is."

To solve this question, we need the following:

  1. A sketch of the situation we are trying to analyze

  2. Some equations that relate different quantities in the problem to each other.

  3. A way to double check if the answer makes sense.

Let us start with 1.

  1. A sketch of the situation we are trying to analyze

The sketch below shows what we know (the direct distance from you to the tower tip) and the angle you are holding the laser range finder at. It also shows what we want to find (the height of the tower and the distance to it).



So what on earth do we do now? To solve 2. we need some trigonometry:

2. Some equations that relate different quantities in the problem to each other.

Let us first think about how cosine and sine are even defined:



(Yes, I know this is a dumb way to remember it, but trust me; it actually works!)

In the problem we are trying to solve, we can plug in the numbers and isolate d:





So we should be around 200m away from the tower.

What about the height of the tower?




(Yes, I know this is an even dumber way to remember it, but sometimes dumb solutions are the smartest ones.)

Let us try to plug in the numbers and solve for the height:




Now for 3.

3. A way to double check if the answer makes sense.

The no-brainer solution is just to look up the height of the CN-tower. According to Wikipedia it is 553.3m tall. That is a good sign our answer is correct.



TIP: I know it seems silly to look up a wikipedia page like this, but double checking answers and thinking critically about your results is extremely important. I cannot tell you how many times I have gone through a lengthy calculation only to get the wrong result because I was too lazy to verify my answer - goodbye extra marks!


Question 2:

"You are standing 100m away from the base of the CN-Tower. You look up at a 79.75 degree angle and stare directly at its tip. Use this information to determine the height of the CN-Tower."

To solve this question, we need the following:

  1. A sketch of the situation we are trying to analyze

  2. Some equations that relate different quantities in the problem to each other.

  3. A way to double check if the answer makes sense.

Let us get started with 1.:

  1. A sketch of the situation we are trying to analyze



The first thing we notice is that the tip of the tower, its base and the place we are standing creates a triangle. What kind of triangle is this? It must be a right angle triangle!




Making a drawing always makes the problem a lot easier! Anyways, let us get started with 2.

2. Some equations that relate different quantities in the problem to each other.

Right now, we know 2 out of 3 angles in the triangle. One is 79.75 degrees, the other is 90 degrees, so the final one must be 180-90-79.75=10.25 degrees. We know all of the angles, but only one of the sides of the triangle. Luckily, we can use the Law of Sines:


In the present case, we could write down the following expression and solve for h: