Angle Input

Half Angle Results

Half Angle (θ/2)
30.0000 °
Sine sin(θ/2)
0.5000
Cosine cos(θ/2)
0.8660
Tangent tan(θ/2)
0.5774
Half Angle Quadrant
Quadrant I

A Half Angle Calculator is a mathematical tool used in trigonometry to find the exact sine, cosine, and tangent values of half of a given angle. This is particularly useful in geometry and physics when you only know the properties of the original angle and need to compute the fractional angle without losing numerical precision.

Understanding Half Angle Formulas

In standard trigonometry, the half angle formulas allow you to compute the trigonometric functions of half an angle (θ / 2) by using the cosine of the original full angle (θ).

  • Sine Half Angle: The sine of the half angle is calculated using the square root of (1 − cos θ) divided by 2.
  • Cosine Half Angle: The cosine of the half angle is calculated using the square root of (1 + cos θ) divided by 2.
  • Tangent Half Angle: The tangent is found by dividing the sine result by the cosine result, or by using the square root of (1 − cos θ) divided by (1 + cos θ).

How to Use This Calculator

  • Enter your original full angle into the "Original Angle" input field. You can use decimals, negative numbers, or numbers much larger than 360.
  • Select the correct unit of measurement (Degrees or Radians) from the dropdown list to ensure accurate math.
  • The calculator instantly divides your angle in half and computes the exact sine, cosine, and tangent values for that new half angle.
  • The quadrant indicator helps you visualize exactly where the new half angle physically rests on a standard coordinate circle.

Frequently Asked Questions

How do you know if the half angle result is positive or negative?

When solving half angle equations manually using the square root formulas, you must place a plus or minus sign in front of the equation. The correct sign is determined entirely by which quadrant the half angle falls into. This calculator automatically computes the correct positive or negative sign for you.

Why do we need half angle formulas?

Half angle formulas are critical in calculus when integrating certain trigonometric functions. They are also highly practical in mechanical engineering and architecture when calculating the forces acting on a bisected joint or discovering the exact lengths of sides in complex geometric shapes.

What happens if the tangent becomes undefined?

If the original angle causes the half angle to land exactly on a 90-degree or 270-degree axis, the cosine of the half angle will equal zero. Because tangent is calculated by dividing sine by cosine, this scenario mathematically forces a division by zero, making the tangent undefined.