Understanding The Unit Circle

https://alexandertutoring.com/ This lesson is going to be on the unit circle. The unit circle is a subject that causes many students much grief in precalculus so I'm here to simplify the subject for you once and for all. It turns out if you really understand the fundamentals behind the unit circle you can solve almost every problem the teacher will throw at you. What I have drawn here is a standard x-y axis that you should be familiar with. I've drawn a circle right on the center with radius 1, thus a unit circle. I've drawn a right triangle with one vertex on the center and another on the circle. All angles on the unit circle will be measured from the x-axis. I’ll drop a line straight down from the vertex touching the circle such that it makes a 90 degree angle with the x-axis. We now have a right triangle inscribed in our unit circle. I’ll assume that your are comfortable with sohcahtoa, that's your sine, cosine and tangent equations from from a previous class. The s.o.h. of sohcahtoa says that the sine of the angle theta is equal to the length of the opposite side of the triangle divided by the length of its hypotenuse. Remember sine is a function, a function being a thing in which a number goes in, in this case the angle theta, and some number between -1 and 1 pops out. The number to pops out and in this case will be the length of the opposite side. So on the unit circle the length of this side is equivalent to the sine of theta. Similarly we will apply the same argument with the cosine. The cosine of theta is equal to the length of the adjacent side of the right triangle divided by its hypotenuse. We're talking about the same triangle so the hypotenuse is still one. Therefore in the case of the unit circle the cosine of theta is equal to the length of the adjacent side of the triangle. The big takeaway here is that on the unit circle the opposite side of the triangle is always equal to the sine of theta and the adjacent side of the triangle is always equal to the cosine of theta. If you understand this fact you can solve almost any problem related to the unit circle. This will make your precalculus experience WAY better.