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Csc 225 Degrees Without Calculator

Reviewed by Calculator Editorial Team

Calculating the cosecant of 225 degrees without a calculator requires understanding the trigonometric functions and their relationships. This guide explains the formula, provides a step-by-step method, and includes a worked example to help you understand and perform the calculation manually.

What is CSC?

The cosecant (csc) is a trigonometric function that is the reciprocal of the sine function. It is defined as:

Definition

csc(θ) = 1 / sin(θ)

The cosecant function is periodic with a period of 360 degrees, meaning csc(θ) = csc(θ + 360° × n) for any integer n. It is also an odd function, meaning csc(-θ) = -csc(θ).

Formula

To calculate csc(225°), we can use the following steps:

  1. First, find the reference angle for 225 degrees.
  2. Determine the sine of the reference angle.
  3. Take the reciprocal of the sine value to get the cosecant.

Reference Angle

225° is in the third quadrant. The reference angle (θ') is calculated as:

θ' = θ - 180° = 225° - 180° = 45°

Sine of Reference Angle

sin(45°) = √2 / 2 ≈ 0.7071

Final Calculation

csc(225°) = 1 / sin(225°) = 1 / (-sin(45°)) = -1 / (√2 / 2) = -2/√2 = -√2 ≈ -1.4142

Step-by-Step Calculation

  1. Identify the quadrant: 225° is in the third quadrant where both sine and cosine are negative.
  2. Find the reference angle: Subtract 180° from 225° to get 45°.
  3. Calculate sin(45°): sin(45°) = √2 / 2 ≈ 0.7071.
  4. Apply the sign based on the quadrant: Since 225° is in the third quadrant, sin(225°) = -sin(45°).
  5. Calculate csc(225°): csc(225°) = 1 / sin(225°) = 1 / (-0.7071) ≈ -1.4142.

Note

The exact value of csc(225°) is -√2, which is approximately -1.4142. The negative sign indicates the direction in the third quadrant.

Worked Example

Let's calculate csc(225°) step by step:

  1. 225° is in the third quadrant.
  2. Reference angle: 225° - 180° = 45°.
  3. sin(45°) = √2 / 2 ≈ 0.7071.
  4. sin(225°) = -sin(45°) ≈ -0.7071.
  5. csc(225°) = 1 / (-0.7071) ≈ -1.4142.

The exact value is csc(225°) = -√2.

FAQ

What is the difference between CSC and SEC?
CSC is the reciprocal of sine, while SEC is the reciprocal of cosine. Both are periodic functions with a period of 360 degrees.
Why is csc(225°) negative?
225° is in the third quadrant where both sine and cosine are negative. Therefore, csc(225°) is negative.
How do I calculate csc(θ) for any angle?
First, find the reference angle, then determine the sign based on the quadrant, and finally take the reciprocal of the sine value.
What is the exact value of csc(225°)?
The exact value is -√2, which is approximately -1.4142.