Trigonometric Functions

Coming into this unit, students will already be familiar with right triangle trigonometry. They will have used sine, cosine and tangent to solve the angles and sides of a right triangle. Students will also have already seen that similar triangles preserve angles, as well as side ratios. For instance, they will have seen that a 3-4-5 right triangle has the same angles as a 6-8-10 triangle. They may have seen the “special right triangles” in the context of similar triangles. Students will also be familiar with Pythagorean Theorem, and will have used it to solve sides of right triangles. They will be familiar with the graphs of more than one type of function, including linear, quadratic, and exponential. Depending on timing, they may also have seen graphs of rational and polynomial functions.

In this unit, students make a big transition to thinking of trigonometric ratios as functions rather than a relationship between angles and side ratios in a right triangle. There are two important steps in this transition. First, students re-envision trigonometric ratios on the unit circle, and then use the idea of an angle as a movement around a circle to extend the definition to angles of any measure. Second, they use the unit circle to understand a new way of measuring angles, radian measure, which uses distance around the circumference of the circle to measure an angle, rather than an arbitrary division of the circle into 360 degrees. From now on students will be thinking of sine and cosine both as functions with numerical inputs, as well as ratios related to angles.

Students learn the basic shape of the graph of a trigonometric function, and then examine graphs of functions with parameters controlling the period, amplitude, and phase shift. They study the effect of varying these parameters and fit trigonometric functions to data. Addressing some (+) standards, students explore further the consequences of the unit circle definition of sine and cosine. They make a connection to the Pythagorean theorem. Then, they see how the symmetries of the circle give rise to symmetries of the graphs of sine and cosine, and represent these symmetries as identities.

In later math courses, students may see more complex uses of trigonometry. For instance, they may go on to learn that the derivative of the sin$\theta$ is the cos$\theta$, and use radians to prove some of the properties of trigonometric derivatives.