Cal11 calculator
Search calculator...

Tan 225 Without Calculator

Reviewed by Calculator Editorial Team

Calculating the tangent of 225 degrees without a calculator requires understanding trigonometric identities and reference angles. This guide explains the process step-by-step, including the formula, reference angle calculation, and final result.

How to calculate tan 225° without a calculator

The tangent of 225 degrees can be found using trigonometric identities and reference angles. Since 225° is in the third quadrant of the unit circle, we can use the properties of tangent in that quadrant to find the value.

Formula: tan(θ) = sin(θ)/cos(θ)

For angles in the third quadrant, both sine and cosine values are negative.

To find tan(225°), we'll follow these steps:

  1. Identify the reference angle for 225°
  2. Find the sine and cosine of the reference angle
  3. Apply the quadrant's sign rules to get the final values
  4. Divide sine by cosine to get the tangent

Formula used

The tangent function can be expressed as the ratio of sine to cosine:

tan(θ) = sin(θ)/cos(θ)

For 225°, which is in the third quadrant, both sine and cosine values are negative. The reference angle for 225° is calculated as:

Reference angle = θ - 180° = 225° - 180° = 45°

We know that:

  • sin(45°) = √2/2 ≈ 0.7071
  • cos(45°) = √2/2 ≈ 0.7071

Applying the third quadrant rules:

  • sin(225°) = -sin(45°) = -√2/2
  • cos(225°) = -cos(45°) = -√2/2

Therefore:

tan(225°) = sin(225°)/cos(225°) = (-√2/2)/(-√2/2) = 1

Step-by-step calculation

  1. Identify the reference angle

    225° is in the third quadrant. The reference angle is calculated by subtracting 180° from the angle:

    Reference angle = 225° - 180° = 45°

  2. Find sine and cosine of the reference angle

    For 45°:

    • sin(45°) = √2/2 ≈ 0.7071
    • cos(45°) = √2/2 ≈ 0.7071

  3. Apply quadrant rules

    In the third quadrant, both sine and cosine are negative:

    • sin(225°) = -sin(45°) = -√2/2
    • cos(225°) = -cos(45°) = -√2/2

  4. Calculate the tangent

    Divide sine by cosine:

    tan(225°) = sin(225°)/cos(225°) = (-√2/2)/(-√2/2) = 1

Worked example

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

  1. Reference angle: 225° - 180° = 45°

  2. sin(45°) = √2/2 ≈ 0.7071

    cos(45°) = √2/2 ≈ 0.7071

  3. sin(225°) = -0.7071

    cos(225°) = -0.7071

  4. tan(225°) = (-0.7071)/(-0.7071) = 1

The tangent of 225 degrees is exactly 1. This is because the sine and cosine values cancel out in the ratio, resulting in a positive value in the third quadrant.

FAQ

Why is tan(225°) equal to 1?
Because 225° is in the third quadrant where both sine and cosine are negative, and the reference angle is 45° where sin(45°) = cos(45°). The negatives cancel out in the ratio.
Can I use this method for other angles?
Yes, this method works for any angle by first finding its reference angle and then applying the appropriate quadrant rules for sine and cosine.
Is tan(225°) the same as tan(45°)?
No, tan(225°) is equal to tan(45°) because they have the same reference angle, but the signs are different based on the quadrant.
What's the difference between tan and cot?
The cotangent is the reciprocal of the tangent. So cot(θ) = 1/tan(θ). For 225°, cot(225°) would be 1/1 = 1.