High School: Functions
Trigonometric Functions HSF-TF.A.4
4. Use the unit circle to explain symmetry (odd and even) and periodicity of trigonometric functions.
Students should be able to determine the symmetry of trigonometric functions. They should also know that when we say "symmetry," we aren't talking mirrors and palindromes.
By comparing the values of trigonometric functions in quadrants I and IV, students can determine whether the function is odd or even. To do that, though, they should really know what even and odd functions are, first.
Students should know that cosine and secant are even functions and are symmetric with respect to the y-axis. We know this is true because of the negative angle identities for cosine and secant.
cos(-θ) = cosθ
sec(-θ) = secθ
As expected, the rest of 'em (sine, cosecant, tangent, and cotangent) are odd functions and are symmetric to the origin. These also have negative angle identities.
sin(-θ) = -sinθ
csc(-θ) = -cscθ
tan(-θ) = -tanθ
cot(-θ) = -cotθ
Students might get these identities confused with All Students Take Calculus (ASTC). Remind them that these identities apply to angles, while ASTC applies to quadrants. Either way, they both still apply!
You might also want to take this opportunity to talk about these functions as lines on the unit circle. It's not necessary, but it may be helpful for students to see the trigonometric functions represented in this way so they can better understand the relationships of these functions and make predictions (about what might happen to the tangent function when θ = π⁄2, for example).
Students should also know that the six trigonometric functions are periodic functions. Specifically, the periods of sine, cosecant, cosine, and secant are 2π, and the periods of tangent and cotangent are π. To understand why, move θ from 0 all the way to 2π on the unit circle and check how each function changes.
After every period, the function repeats itself, which means that sin(θ + 2π) = sinθ. The same applies to the five other trigonometric functions and their respective periods. That's just the nature of periodicity.