Evaluate sec(135) Without Using a Calculator

An interactive tool and guide to understanding trigonometric values.

Interactive Trigonometric Evaluator

What Does “Evaluate sec(135) without a calculator” Mean?

Evaluating sec(135) without a calculator is a classic trigonometry problem that tests your understanding of the unit circle, reference angles, and the properties of trigonometric functions. The goal is to find the exact value of the secant function for an angle of 135 degrees, not just a decimal approximation. The secant function, abbreviated as ‘sec’, is the reciprocal of the cosine function. Therefore, finding sec(135°) is the same as finding 1 divided by cos(135°).

This process is crucial for students of mathematics and physics, as it builds a foundational understanding of how angles in different quadrants relate to one another and how to use special triangles (like the 45-45-90 triangle) to derive exact values.

The Formula to Evaluate sec(135°)

The core of this problem relies on a few key formulas and concepts. The primary formula is the reciprocal identity connecting secant and cosine.

Primary Formula: sec(θ) = 1 / cos(θ)

Because 135° is in the second quadrant (between 90° and 180°), we need to find its reference angle. A reference angle is the acute angle that the terminal side of the given angle makes with the x-axis.

Reference Angle Formula (Quadrant II): θ' = 180° - θ

For our case, θ = 135°, so the reference angle is 180° – 135° = 45°. The value of a trigonometric function for an angle is the same as the value for its reference angle, but the sign (+ or -) depends on the quadrant. In Quadrant II, the x-coordinate (cosine) is negative, so cos(135°) will be negative.

Trigonometric Evaluation Variables for sec(135°)

Variable	Meaning	Unit	Value
θ	The original angle	Degrees	135°
Quadrant	The location of the angle on the Cartesian plane	Unitless	II
θ’	The reference angle	Degrees	45°
cos(θ’)	Cosine of the reference angle	Ratio	√2 / 2
cos(θ)	Cosine of the original angle (with correct sign)	Ratio	-√2 / 2
sec(θ)	Secant of the original angle	Ratio	-√2

Practical Examples

Example 1: Evaluating sec(135°) (The Primary Case)

• Input Angle: 135°
• Step 1: Identify the quadrant. 135° is in Quadrant II.
• Step 2: Determine the sign. Secant is negative in Quadrant II.
• Step 3: Find the reference angle. 180° – 135° = 45°.
• Step 4: Evaluate the function for the reference angle. sec(45°) = √2.
• Result: Apply the sign. sec(135°) = -√2.

Example 2: Evaluating sec(210°)

• Input Angle: 210°
• Step 1: Identify the quadrant. 210° is in Quadrant III.
• Step 2: Determine the sign. Secant is also negative in Quadrant III.
• Step 3: Find the reference angle. 210° – 180° = 30°.
• Step 4: Evaluate the function for the reference angle. sec(30°) = 2/√3.
• Result: Apply the sign. sec(210°) = -2/√3.

How to Use This sec(135) Calculator

This interactive tool is more than a calculator; it’s a learning guide designed to help you understand how to evaluate sec(135) and other trigonometric values.

1. Enter an Angle: Type any angle in degrees into the input field. The calculator is preset to 135°.
2. View the Results: The tool instantly calculates the exact value and its decimal approximation.
3. Follow the Steps: The “Step-by-Step Breakdown” shows the entire logical process, from finding the quadrant and reference angle to the final calculation.
4. Analyze the Chart: The Unit Circle Visualization dynamically updates to show a graphical representation of the angle you entered, helping you connect the abstract numbers to a geometric shape.

Key Factors That Affect the Value

Understanding these factors is key to mastering the evaluation of trigonometric functions without a calculator.

• The Angle’s Quadrant: This is the most critical factor as it determines the sign (positive or negative) of the result. For secant, values are positive in Quadrants I and IV and negative in II and III.
• The Reference Angle: This acute angle is what you use to look up the base value from the special triangles (30-60-90 or 45-45-90). All angles with the same reference angle have the same absolute trigonometric value.
• The Function Itself: Whether you are calculating sin, cos, tan, csc, sec, or cot determines which sides of the reference triangle you are comparing. For secant, it’s hypotenuse over adjacent.
• Special Triangles: Knowing the side ratios for 30-60-90 (1, √3, 2) and 45-45-90 (1, 1, √2) triangles is essential for finding exact values.
• Reciprocal Identities: Knowing that sec(θ) = 1/cos(θ), csc(θ) = 1/sin(θ), and cot(θ) = 1/tan(θ) allows you to solve for functions you may not have memorized.
• Units (Degrees vs. Radians): While this calculator uses degrees, all these principles apply to radians. You must know how to convert between them (e.g., 135° = 3π/4 radians).

Frequently Asked Questions (FAQ)

What is the exact value of sec(135°)?
The exact value is -√2.

Why is sec(135°) negative?
The angle 135° is in the second quadrant. In this quadrant, the x-coordinates are negative. Since sec(θ) = 1/cos(θ) and cos(θ) corresponds to the x-coordinate on the unit circle, the secant value is also negative.

How do you find the reference angle for 135°?
For any angle in Quadrant II, the reference angle is found by subtracting the angle from 180°. So, 180° – 135° = 45°.

What is the value in radians?
To convert degrees to radians, you multiply by π/180. So, 135 * (π/180) = 3π/4 radians. The value is sec(3π/4) = -√2.

What is the CAST rule?
CAST is a mnemonic to remember which trig functions are positive in which quadrant. C (Quadrant IV) for Cosine, A (Quadrant I) for All, S (Quadrant II) for Sine, and T (Quadrant III) for Tangent. Secant follows the same rule as Cosine.

Can this method be used for any angle?
This manual method works best for angles that have reference angles of 30°, 45°, or 60°, as those correspond to special, well-known triangles. For other angles, a calculator is typically required for a precise value.

What is cos(135°)?
Since sec(135°) = 1/cos(135°), we can find cos(135°) by taking the reciprocal of -√2. Therefore, cos(135°) = 1/(-√2) = -√2/2.

What about csc(135°)?
Csc(135°) is the reciprocal of sin(135°). Since 135° is in Quadrant II, sine is positive. The reference angle is 45°, and sin(45°) = √2/2. Therefore, sin(135°) = √2/2, and csc(135°) = 2/√2 = √2.