How do you find the domain and range for sec θ?

To determine the domain and range of the secant function, sec θ, we first need to understand its relationship with the cosine function.

Domain of sec θ

The secant function is defined as:

• sec θ = 1/cos θ

Since division by zero is undefined, we need to find where cos θ equals zero. The cosine function is zero at:

• θ = (2n + 1)π/2, where n is any integer.

Thus, the domain of sec θ excludes these values:

• Domain: θ ∈ ℝ, θ ≠ (2n + 1)π/2

Range of sec θ

The secant function outputs values based on the cosine function. Since sec θ is the reciprocal of cos θ, it can take on values that are either:

• Greater than or equal to 1 (sec θ ≥ 1)
• Less than or equal to -1 (sec θ ≤ -1)

Therefore, the range of sec θ is:

• Range: (-∞, -1] ∪ [1, ∞)

Summary

In summary, the domain of sec θ excludes odd multiples of π/2, while its range consists of values less than or equal to -1 and greater than or equal to 1.