How to Evaluate Sec 135 Degrees Without Calculator

Evaluating trigonometric functions like secant (sec) without a calculator requires understanding of trigonometric identities and reference angles. This guide explains how to calculate sec 135 degrees using fundamental trigonometric principles.

Understanding the Secant Function

The secant function, sec(θ), is the reciprocal of the cosine function: sec(θ) = 1/cos(θ). It represents the ratio of the hypotenuse to the adjacent side in a right-angled triangle.

Formula: sec(θ) = 1 / cos(θ)

To evaluate sec(135°), we first need to determine cos(135°). Since 135° is in the second quadrant where cosine values are negative, we'll use the reference angle concept.

Calculating Sec 135 Degrees

The reference angle for 135° is calculated as 180° - 135° = 45°. We know that cos(45°) = √2/2 ≈ 0.7071. However, since 135° is in the second quadrant where cosine is negative:

cos(135°) = -cos(45°) = -√2/2

Now, we can find sec(135°):

sec(135°) = 1 / cos(135°) = 1 / (-√2/2) = -2/√2 = -√2 ≈ -1.4142

Step-by-Step Calculation

Identify the quadrant: 135° is in the second quadrant (90° to 180°).
Find the reference angle: 180° - 135° = 45°.
Recall that cos(45°) = √2/2 ≈ 0.7071.
Determine the sign of cosine in the second quadrant: Negative.
Calculate cos(135°): -√2/2.
Find sec(135°): 1 / (-√2/2) = -2/√2 = -√2 ≈ -1.4142.

Verification of Results

To verify our calculation, we can use the unit circle representation. At 135° on the unit circle:

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

This confirms our earlier result.

Common Mistakes to Avoid

When calculating sec(135°), common errors include:

Forgetting to account for the negative cosine value in the second quadrant.
Using the wrong reference angle (should be 45°, not 135°).
Incorrectly simplifying the reciprocal expression (-2/√2 should be rationalized to -√2).

Frequently Asked Questions

Why is sec(135°) negative?: Because 135° is in the second quadrant where cosine values are negative, and secant is the reciprocal of cosine.
Can I use a calculator to verify this result?: Yes, most scientific calculators have a secant function. Enter 135° and press the sec button to verify the result.
What's the difference between sec and cos?: Secant is the reciprocal of cosine. While cos(θ) gives the ratio of adjacent/hypotenuse, sec(θ) gives the ratio of hypotenuse/adjacent.
How precise should my answer be?: The exact value is -√2. For practical purposes, you might use -1.4142 (rounded to 4 decimal places).