What is Sine, Cosine and Tangent?

The Sine, Cosine and Tangent functions express the ratios of sides of a right triangle.

You can remember this by the acryonym SOHCAHTOA:

  • SOH: ##sin = “opposite”/”hypothenuse”##
  • CAH: ##cos = “adjacent”/”hypothenuse”##
  • TOA: ##tan = “opposite”/”adjacent”##

Remember that these are not properties of a triangle, but of an angle.

This triangle for example:

If we take the ##sin(30°)##, it is equal to ##”opposite”/”hypothenuse” = 1/2## The ##sin(60°)## however is not equal. It is ##”opposite”/”hypothenuse” = sqrt(3)/2##

So, the trigonometric functions are not properties of triangles, but of angles.

Why is this useful?

There are so many uses for the trig functions, but one of the easiest is to solve a triangle.

For example: what if I asked you what the length of the line |BC|

There are two things we know of this triangle. The angle ##|Â| = 40°## and the side ##|AC| = 6##. To find |BC|, we need to find a function that relates ##|AC|,|Â|## and ##|BC|##. The answer is ##tan(40°) = “opposite”/”adjacent” = |BC|/|AC| = |BC|/6##

So to find |BC|, we solve for |BC| in the last equation: ##tan(40°) = |BC|/6## ##tan(40°) * 6 = |BC|##

But now we face a problem: we don’t know the value of ##tan(40°)##. For this, there exists a simple device, called a calculator. Smply enter ##tan(40°)## in your calculator. I like to use Google where you should enter ##tan(40 ” deg”)##. This gives you ##0.839099631##.

=> ##|BC| = 6*0.839099631 = 5.03459779##

Now, as a challenge for you: find ##|AB|## (hypothenuse)

Hope this helped.

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