Unit Circle and Trigonometric Functions

1. Describe tangent in terms of sine and cosine as $${\frac{\mathrm{sin{\theta}}}{\mathrm{cos{\theta}}}}$$. 
2. Derive tangent from similar triangles in the unit circle and also as the slope of the radius at a given degree of $${\theta}$$. 
3. Identify when the value of the tangent is undefined and describe why this is the case.

• Be prepared for students to get stuck between the ideas that there are two different definitions of tangent—one that is a ratio and one that is a length in the unit circle. 
• This is the first time students are seeing a circle that is NOT a unit circle. Feel free to extend the work to non-unit circles, but be prepared to describe how the value of the tangent is scaled by the value of the radius.
• Here is a Desmos tool with the unit circle. It is helpful for demonstration purposes.
• This GeoGebra tool, The Reciprocal Trig Functions on the Unit Circle, is helpful for identifying the lengths of the trigonometric ratios and how they relate in a unit circle.

Draw and label a figure on the circle that illustrates the relationship of the trigonometric tangent functions $${\mathrm{tan}=\frac{\mathrm{sin}(\theta^\circ)}{\mathrm{cos}(\theta^\circ)}}$$ and the geometric tangent line to a circle through the point $${(1,0)}$$ when $${\theta=60}$$. Explain the relationship, labeling the figure as needed.