Half Angle
Free online half angle calculator computes sin(theta/2), cos(theta/2), and tan(theta/2) using half-angle identities. Interactive graph for visualizing half-angle trigonometric functions.
About This Calculator
This half angle calculator evaluates all three half-angle trigonometric identities -- sin(theta/2), cos(theta/2), and tan(theta/2) -- from a single angle input in degrees. Half-angle formulas are derived from the double-angle identities and are essential for simplifying trigonometric expressions, particularly in calculus integration problems involving squared trig functions such as integral sin^2(x) dx = x/2 - sin(2x)/4 + C.
The calculator applies the standard half-angle formulas: sin(theta/2) = +/-sqrt((1 - cos theta)/2), cos(theta/2) = +/-sqrt((1 + cos theta)/2), and tan(theta/2) = +/-sqrt((1 - cos theta)/(1 + cos theta)). The sign (positive or negative) is automatically determined based on the quadrant of theta/2: quadrant I (all positive), quadrant II (sine positive), quadrant III (tangent positive), or quadrant IV (cosine positive). This quadrant-aware sign handling sets it apart from basic calculators that only return the positive principal root.
Half-angle identities are widely used in calculus for trigonometric substitution, reducing the power of integrands, and solving differential equations. In geometry, they help compute side ratios in subdivided triangles. Physics applications include wave interference analysis where half-angle formulas simplify phase-difference expressions.
How to use
Enter a full angle in degrees (e.g. 60, 90, 120, 270), click Calculate, and instantly see the correct signed values of sin(theta/2), cos(theta/2), and tan(theta/2). The interactive chart visualizes how half-angle sine and cosine behave across the full 0 deg to 360 deg range. Share specific calculations via the automatically updated URL -- just copy the page URL after calculating.
Regional notes
Half-angle trigonometry is universal across all educational systems. Students in India (CBSE/ICSE), the United States (Common Core, high school precalculus), the United Kingdom (A-level mathematics), and elsewhere encounter half-angle identities as part of their trigonometry curriculum. This calculator uses degrees (not radians) by default to match the convention used in most secondary-school mathematics courses worldwide.
Frequently Asked Questions
What is the half-angle formula for sine?
The half-angle formula for sine is sin(theta/2) = +/-sqrt((1 - cos theta)/2). The sign depends on the quadrant containing theta/2. For example, in quadrant I where all trig functions are positive, use the positive root; in quadrant II, sine is positive.
What is the half-angle formula for cosine?
The half-angle formula for cosine is cos(theta/2) = +/-sqrt((1 + cos theta)/2). The sign is positive when theta/2 lies in quadrant I or IV and negative in quadrants II or III. These formulas are derived from the double-angle identity cos(2φ) = 2 cos^2φ - 1 by setting φ = theta/2.
What is the half-angle formula for tangent?
The half-angle formula for tangent is tan(theta/2) = +/-sqrt((1 - cos theta)/(1 + cos theta)). Alternative forms that avoid sign ambiguity are tan(theta/2) = sin theta / (1 + cos theta) = (1 - cos theta) / sin theta, which automatically determine the correct sign based on the quadrant.
How do you determine the sign for half-angle formulas?
The sign is determined by the quadrant of theta/2. If theta is between 0 deg and 180 deg, theta/2 is between 0 deg and 90 deg (quadrant I) where all functions are positive. If theta is between 180 deg and 360 deg, theta/2 is between 90 deg and 180 deg (quadrant II) where sine is positive but cosine and tangent are negative.
How are half-angle formulas derived?
Half-angle formulas are derived from the double-angle identities. Starting from cos(2φ) = 1 - 2 sin^2φ, substitute φ = theta/2 to get cos theta = 1 - 2 sin^2(theta/2), then solve for sin(theta/2) = +/-sqrt((1 - cos theta)/2). Similarly, from cos(2φ) = 2 cos^2φ - 1, you get cos(theta/2) = +/-sqrt((1 + cos theta)/2).
Where are half-angle identities used in calculus?
Half-angle identities are crucial in calculus for integrating powers of sine and cosine, such as integral sin^2(x) dx = integral (1 - cos(2x))/2 dx. They also simplify trigonometric substitutions, reduce the order of differential equations, and are used in Fourier series expansions to convert squared trig functions into linear ones.