How do you evaluate sec135?

To evaluate sec(135°), we can follow these steps:

Recall the definition of secant: Secant is the reciprocal of cosine. That is, sec(θ) = 1/cos(θ).

Find cos(135°): To evaluate cos(135°), we note that 135° is in the second quadrant, where cosine is negative. We can subtract 135° from 180° to find the reference angle: Reference angle = 180° - 135° = 45°.

Therefore, cos(135°) = -cos(45°), because cosine is negative in the second quadrant.

We know that cos(45°) = √2/2. So, cos(135°) = -√2/2.

Find sec(135°): Using the definition of secant, we have: sec(135°) = 1/cos(135°) = 1/(-√2/2).

Simplify the expression: To simplify 1/(-√2/2), multiply the numerator and denominator by 2: sec(135°) = 1/(-√2/2) * (2/2) = -2/√2.

Rationalize the denominator: To rationalize the denominator, multiply both the numerator and denominator by √2: sec(135°) = (-2/√2) * (√2/√2) = -2√2/2.

Simplify the result: Simplifying the fraction: sec(135°) = -√2.

Thus, sec(135°) = -√2.