Trigonometric Functions
Lessons in this chapter
About Trigonometric Functions
Trigonometric Functions moves trigonometry from right triangles to the unit circle, covering every trig function and its graph.
Measuring Angles in Radians introduces radian measure as an alternative to degrees, before Finding the Arc Length and Area of a Sector applies it to circle measurements.
Sine and Cosine as Trigonometric Functions and Equivalent Trigonometric Expressions define sine and cosine for any angle using the unit circle, before Graphs and Features of Sine and Cosine Curves and Transformation on Sine and Cosine Functions cover graphing and transforming both curves, and Modeling with Cosine and Sine Functions applies them to real periodic situations.
The Tangent Function and Other Trigonometric Functions close the chapter with the tangent, cotangent, secant, and cosecant functions and their own graphs.
Nine lessons, each a practice worksheet in the page, run from Measuring Angles in Radians through Transformation on Sine and Cosine Functions to Other Trigonometric Functions. Problems are scored as you answer them, and they are set up to be reasoned through on the unit circle rather than typed into a trig calculator. Free to try, with no login needed for your first 3.
Frequently asked questions
Measuring Angles in Radians and Finding the Arc Length and Area of a Sector set up radian measure. Sine and Cosine as Trigonometric Functions and Equivalent Trigonometric Expressions define the functions on the unit circle, Graphs and Features of Sine and Cosine Curves and Transformation on Sine and Cosine Functions cover amplitude, period, phase shift, and midline, and Modeling with Cosine and Sine Functions fits them to periodic data. The Tangent Function and Other Trigonometric Functions close with the remaining ratios and their asymptotes. This sits in Functions (F) in the high school Common Core, with arc length in Geometry (G), and in Unit 3 of the AP Precalculus course framework, Trigonometric and Polar Functions.
A radian measures an angle by the arc length it cuts off on a unit circle, giving a natural, unit-free way to measure angles alongside degrees.
Arc length is the radius times the angle in radians, and a sector's area is one-half the radius squared times the angle in radians: both formulas require the angle in radians, not degrees.
On the unit circle, an angle's cosine is the x-coordinate and its sine is the y-coordinate of the point where the angle's terminal side meets the circle. This extends sine and cosine to any angle, not just those in a right triangle.
Reference angles and the unit circle’s symmetry let you rewrite a trigonometric value for a larger angle as an equivalent value for a smaller, more familiar angle.
The amplitude (height), period (length of one cycle), and midline (vertical center) describe the key features of a sine or cosine curve.