Evaluate Sec 135 Degrees Without Calculator

The secant function, sec(θ), is the reciprocal of the cosine function, cos(θ). Evaluating sec(135°) without a calculator requires understanding trigonometric identities and reference angles. This guide explains how to determine the exact value of sec(135°) using fundamental trigonometric principles.

Understanding the Secant Function

The secant function is one of the six primary trigonometric functions. It is defined as the reciprocal of the cosine function:

sec(θ) = 1 / cos(θ)

This means that for any angle θ, the secant of that angle is equal to one divided by the cosine of that angle. The secant function has a period of 360°, meaning it repeats its values every full rotation.

Like cosine, the secant function is an even function, which means that sec(-θ) = sec(θ). This property can be useful when evaluating secant at negative angles.

Calculating sec(135°)

To evaluate sec(135°), we need to first determine the value of cos(135°). The angle 135° is in the second quadrant of the unit circle, where cosine values are negative. The reference angle for 135° is calculated as:

Reference angle = 180° - 135° = 45°

The cosine of the reference angle (45°) is a well-known value:

cos(45°) = √2/2 ≈ 0.7071

Since 135° is in the second quadrant where cosine is negative, we have:

cos(135°) = -cos(45°) = -√2/2

Now, we can find sec(135°) by taking the reciprocal of cos(135°):

sec(135°) = 1 / cos(135°) = 1 / (-√2/2) = -2/√2

This can be simplified further by rationalizing the denominator:

sec(135°) = -2/√2 × √2/√2 = -2√2/2 = -√2

Therefore, the exact value of sec(135°) is -√2.

Step-by-Step Calculation

Identify the quadrant of the angle: 135° is in the second quadrant (90° to 180°).
Determine the reference angle: 180° - 135° = 45°.
Recall the cosine of the reference angle: cos(45°) = √2/2.
Apply the sign rule for cosine in the second quadrant: cos(135°) = -cos(45°) = -√2/2.
Calculate the secant by taking the reciprocal: sec(135°) = 1 / (-√2/2) = -2/√2.
Rationalize the denominator: -2/√2 × √2/√2 = -2√2/2 = -√2.

Remember that the secant function is undefined when the cosine function equals zero, which occurs at 90° and 270°.

Verification

To ensure our calculation is correct, let's verify using the unit circle definition of trigonometric functions. At 135°:

The x-coordinate (cosine) is -√2/2.
The y-coordinate (sine) is √2/2.
The secant is the reciprocal of the x-coordinate: 1 / (-√2/2) = -√2.

This matches our previous result, confirming that sec(135°) = -√2.

Common Mistakes

When evaluating trigonometric functions without a calculator, several common errors can occur:

Forgetting to apply the sign rule based on the quadrant: Cosine is negative in the second quadrant, so it's easy to forget the negative sign when calculating secant.
Using the wrong reference angle: Always subtract the angle from 180° for angles between 90° and 180°.
Not rationalizing the denominator: Leaving the answer as -2/√2 is acceptable, but -√2 is the simplified form.
Confusing secant with cosecant: Remember that secant is the reciprocal of cosine, while cosecant is the reciprocal of sine.

FAQ

What is the exact value of sec(135°)?

The exact value of sec(135°) is -√2. This is derived from the cosine of 135° being -√2/2, and taking its reciprocal.

Why is sec(135°) negative?

The secant function is negative in the second quadrant (90° to 180°) because the cosine function, of which secant is the reciprocal, is negative in this range.

How do I calculate secant without a calculator?

Use trigonometric identities and reference angles. For sec(θ), find cos(θ) first, then take its reciprocal.

What is the difference between sec and cos?

The secant function is the reciprocal of the cosine function. So, sec(θ) = 1 / cos(θ).