The purpose of this lesson is for students to make sense of what must be true about the tangent function using what they know about cosine, sine, and the unit circle. Like cosine and sine, the tangent function is periodic. Unlike the cosine and sine functions, the tangent function is not defined for all real numbers. Since it is a quotient of the sine and cosine, the tangent function has vertical asymptotes where the denominator takes the value 0. In the last activity, students apply what they learned about rational functions and periodic functions to predict the location of the asymptotes of the tangent function and reason about the value of its period.
Throughout this lesson, students use the structure of the unit circle to make sense of the periodic behavior of the different trigonometric functions (MP7), including identifying where tangent is positive, negative, 0, or does not exist.
Technology isn’t required for this lesson, but there are opportunities for students to choose to use appropriate technology to solve problems. We recommend making technology available.
- Compare and contrast (in writing) the features of the cosine, sine, and tangent functions.
- Describe (orally and in writing) features of the graph of the tangent function, such as period, domain, and asymptotes.
- Let’s learn more about tangent.
- I can explain why the tangent function has a period of $\pi$.
- I understand why the graph of tangent has asymptotes.
A function whose values repeat at regular intervals. If \(f\) is a periodic function then there is a number \(p\), called the period, so that \(f(x + p) = f(x)\) for all inputs \(x\).