TEF - The Unit Circle and Trigonometric Ratios Lesson

The Unit Circle

Recall: Rationalizing the Denominator

When a radical is left in the denominator you must rationalize!

Example: $\frac{5}{\sqrt[]{2}}:\frac{5}{\sqrt[]{2}}\left(\frac{\sqrt[]{2}}{\sqrt[]{2}}\right)=\frac{5\sqrt[]{2}}{2}$

In Precalculus, we often refer to the Unit Circle. A Unit Circle is a circle with the radius equal to 1 and the center at the origin.

We use our special right triangles and our Unit Circle to place important angles and coordinates on the Unit Circle.  Let's review the ratios of the two special right triangles in this presentation below.   

Now let's see how special right triangles are used in the unit circle.

**SPECIAL NOTE** In the video below, the angle in the 3rd quadrant is labeled incorrectly. It should be labeled 225 degrees NOT 215 degrees.

Defining the Ratios

Now, recall sine, cosine and tangent from Geometry. When you learned about them, we only used acute angles, but now we want to be able to apply those concepts to any angle on the plane. So, we are going to generalize the definitions to those below:

Sine: the ratio of y to r: $\sin\theta=\frac{y}{r}$

Cosine: the ratio of x to r: $\cos\theta=\frac{x}{r}$

Tangent: the ratio of y to x: $\tan\theta=\frac{y}{x}$

Where r represents the radius of the circle r > 0: $x^2+y^2=r^2$

Watch the video below to learn about how to use the trig ratios:

The reason the Unit Circle is so important is because the radius is 1. So our new ratios become:    and 

 

 

 

 

 

 

 

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