AI Tutor
What happens if you plug a negative angle into a trigonometric function? Does cos⁑(βˆ’60∘)\cos(-60^\circ) become negative, or does the minus sign simply vanish?
To answer that, we look at function symmetry β€” how graphs behave when you flip your input from positive to negative.

Algebraic Definition of Parity

By the algebraic definition of parity, a function ff is even if f(βˆ’x)=f(x)f(-x) = f(x) for all xx in its domain, exhibiting reflectional symmetry across the yy-axis.
A function is odd if f(βˆ’x)=βˆ’f(x)f(-x) = -f(x) for all xx in its domain, exhibiting 180∘180^\circ rotational symmetry about the origin.
How do these symmetry rules apply to trigonometric functions on the unit circle?

Unit Circle Foundation

For an angle θ\theta in standard position, the terminal point on the unit circle is (x,y)=(cos⁑θ,sin⁑θ)(x, y) = (\cos\theta, \sin\theta).
For the opposite angle βˆ’ΞΈ-\theta, the terminal point reflects across the xx-axis to (x,βˆ’y)=(cos⁑(βˆ’ΞΈ),sin⁑(βˆ’ΞΈ))(x, -y) = (\cos(-\theta), \sin(-\theta)).
Because the xx-coordinates are equal, cos⁑(βˆ’ΞΈ)=cos⁑(ΞΈ)\cos(-\theta) = \cos(\theta), geometrically establishing that cosine is even.
Because the yy-coordinates flip signs, sin⁑(βˆ’ΞΈ)=βˆ’sin⁑(ΞΈ)\sin(-\theta) = -\sin(\theta), geometrically establishing that sine is odd.