Problem 66 - Precalculus: A Concise Course

Question

Use the unit circle to verify that the cosine and secant functions are even and that the sine, cosecant, tangent, and cotangent functions are odd.

Step-by-Step Solution

Answer

Cosine and Secant functions are even; and Sine, Cosecant, Tangent, and Cotangent functions are odd.

Step 1: Identifying Even Functions

Step 1: Start with the cosine function. Cos(θ) on the unit circle is defined as the x-coordinate of the point, and Cos(-θ) is the x-coordinate of the opposite point -- which is the same. Therefore, Cos(-θ) = Cos(θ), so Cos is an even function.\nNext, consider the secant function, Sec(θ) = 1/Cos(θ), is an even function because it is the reciprocal of an even function.

Step 2: Identifying Odd Functions

Step 2: Now, consider sine function, Sin(θ) on the unit circle is defined as the y-coordinate of the point, and Sin(-θ) is the y-coordinate of the opposite point, which is the same. So, Sin(-θ) = -Sin(θ), indicating Sin is an odd function. The cosecant function Csc(θ) = 1/Sin(θ) is odd because it is the reciprocal of an odd function.\nTangent function, Tan(θ) = Sin(θ)/Cos(θ), and Cotangent function, Cot(θ) = Cos(θ)/Sin(θ), are both odd functions since the numerator is an odd function and the denominator is an even function, and the quotient of an odd function and an even function is odd.