Half Angle Calculator
Calculate half angle instantly with our math tool. Shows detailed work, formulas used, and multiple solution methods. Get results you can export or share.
Formula
sin(A/2) = +/-sqrt((1-cosA)/2), cos(A/2) = +/-sqrt((1+cosA)/2), tan(A/2) = sinA/(1+cosA)
The half angle formulas express trig functions of A/2 in terms of cos(A). The sign depends on which quadrant A/2 falls in. The tangent formula avoids square roots and determines sign automatically.
Worked Examples
Example 1: Half Angle of 60 Degrees
Problem: Calculate sin(30), cos(30), and tan(30) using half angle formulas with A = 60 degrees.
Solution: cos(60) = 0.5\nsin(30) = sqrt((1 - cos(60))/2) = sqrt((1 - 0.5)/2) = sqrt(0.25) = 0.5\ncos(30) = sqrt((1 + cos(60))/2) = sqrt((1 + 0.5)/2) = sqrt(0.75) = 0.86602540\ntan(30) = sin(60)/(1 + cos(60)) = 0.86602540/1.5 = 0.57735027\nAll positive since 30 degrees is in Q1.\nVerification: sin(30) = 0.5, cos(30) = 0.86602540, tan(30) = 0.57735027 (all confirmed)
Result: sin(30) = 0.5 | cos(30) = 0.86602540 | tan(30) = 0.57735027
Example 2: Half Angle of 270 Degrees
Problem: Find sin(135), cos(135), tan(135) using half angle formulas with A = 270 degrees.
Solution: cos(270) = 0\nA/2 = 135 degrees (Quadrant II: sin positive, cos negative)\nsin(135) = +sqrt((1 - 0)/2) = +sqrt(0.5) = 0.70710678\ncos(135) = -sqrt((1 + 0)/2) = -sqrt(0.5) = -0.70710678\ntan(135) = sin(270)/(1 + cos(270)) = -1/1 = -1.0\nVerification: sin(135) = 0.70710678, cos(135) = -0.70710678, tan(135) = -1.0 (confirmed)
Result: sin(135) = 0.70710678 | cos(135) = -0.70710678 | tan(135) = -1.0
Frequently Asked Questions
What are the half angle formulas?
The half angle formulas express trigonometric functions of half an angle in terms of the cosine of the full angle. The three primary formulas are: sin(A/2) = plus or minus sqrt((1 - cos(A))/2), cos(A/2) = plus or minus sqrt((1 + cos(A))/2), and tan(A/2) = sin(A)/(1 + cos(A)) = (1 - cos(A))/sin(A). The plus or minus sign for sine and cosine depends on the quadrant in which A/2 falls. These formulas are derived from the double angle formulas by solving for the half-angle function. They are particularly useful in calculus for evaluating integrals involving square roots of trigonometric expressions and in deriving the Weierstrass substitution.
How are half angle formulas derived from double angle formulas?
The half angle formulas are derived by rearranging the double angle cosine formulas. Starting with cos(2B) = 1 - 2sin2(B), solve for sin(B): sin2(B) = (1 - cos(2B))/2, so sin(B) = plus or minus sqrt((1 - cos(2B))/2). Now substitute B = A/2: sin(A/2) = plus or minus sqrt((1 - cos(A))/2). Similarly, from cos(2B) = 2cos2(B) - 1, solve for cos(B): cos2(B) = (1 + cos(2B))/2, giving cos(A/2) = plus or minus sqrt((1 + cos(A))/2). The tangent half angle formula tan(A/2) = sin(A)/(1 + cos(A)) avoids the square root by using the identity tan(A/2) = sin(A/2)/cos(A/2) and algebraic manipulation.
How do you determine the sign in half angle formulas?
The sign (positive or negative) in the half angle formulas for sine and cosine depends on the quadrant in which the half angle A/2 falls, not the quadrant of A itself. If A/2 is in the first or second quadrant (0 to 180 degrees), sin(A/2) is positive. If A/2 is in the first or fourth quadrant (0 to 90 or 270 to 360 degrees), cos(A/2) is positive. For example, if A = 300 degrees, then A/2 = 150 degrees which is in the second quadrant. Therefore sin(150) is positive and cos(150) is negative. The tangent half angle formulas using sin(A)/(1+cos(A)) or (1-cos(A))/sin(A) automatically give the correct sign without needing to determine the quadrant separately.
What is the Weierstrass substitution and how does it use half angle formulas?
The Weierstrass substitution (also called the tangent half-angle substitution) is a powerful calculus technique that converts trigonometric integrals into rational function integrals. You set t = tan(x/2), then use the half angle identities to express: sin(x) = 2t/(1+t2), cos(x) = (1-t2)/(1+t2), and dx = 2/(1+t2) dt. This transforms any rational function of sin(x) and cos(x) into a rational function of t, which can be integrated using partial fractions. While not always the most efficient method, the Weierstrass substitution is universal and guaranteed to work for any rational trigonometric integral. It was developed by Karl Weierstrass in the 19th century and remains a standard technique in calculus courses.
What are common applications of half angle formulas?
Half angle formulas have important applications across mathematics and engineering. In calculus, they enable the power reduction technique: sin2(x) = (1 - cos(2x))/2 and cos2(x) = (1 + cos(2x))/2, which are essential for integrating even powers of sine and cosine. In physics, half angle formulas appear in the analysis of optical systems, specifically in calculations involving prism refraction and thin film interference patterns. In electrical engineering, they help simplify AC circuit analysis when dealing with rectified waveforms. In computer graphics, half angle vectors are used in specular reflection calculations (Blinn-Phong shading model). In surveying and navigation, half angle formulas appear in the haversine formula for calculating great circle distances.
How do half angle formulas help find exact trigonometric values?
Half angle formulas allow you to find exact values of trigonometric functions for angles that are half of known standard angles. For example, to find the exact value of sin(15 degrees), use sin(30/2) = sqrt((1 - cos(30))/2) = sqrt((1 - sqrt(3)/2)/2) = sqrt((2 - sqrt(3))/4). This gives an exact radical expression rather than a decimal approximation. Similarly, cos(22.5 degrees) = sqrt((1 + cos(45))/2) = sqrt((1 + sqrt(2)/2)/2) = sqrt((2 + sqrt(2))/4). By repeatedly applying half angle formulas, you can find exact values for angles like 7.5 degrees, 3.75 degrees, and so on. This technique is valuable in pure mathematics where exact symbolic expressions are preferred over numerical approximations.