Enter an angle to compute sin, cos, and tan half-angle values with step-by-step derivation
Angle (degrees)
Result
Half Angle:
sin(A/2)
-
cos(A/2)
-
tan(A/2)
-
Detailed Derivation
Half Angle Formulas
sin(A/2) = +/- sqrt((1 - cosA) / 2)
cos(A/2) = +/- sqrt((1 + cosA) / 2)
tan(A/2) = (1 - cosA) / sinA
tan(A/2) = sinA / (1 + cosA)
Half-angle formulas are derived from double-angle identities and are essential for finding exact trig values of common angles like 15, 22.5, and 75 degrees.
⚠The +/- sign depends on the quadrant of A/2. The tool uses positive values for standard angles in Quadrant I.
What Are Half-Angle Formulas?
Half-angle formulas allow you to compute trigonometric functions of half an angle using the cosine of the original angle. They are powerful tools for simplifying complex trigonometric expressions and finding exact values.
Sine Half-Angle
sin(A/2) = +/- sqrt((1-cosA)/2). Used to compute sin of 15, 22.5, and other common half-angles.
Cosine Half-Angle
cos(A/2) = +/- sqrt((1+cosA)/2). The sign depends on the quadrant of the half-angle.
Formulas that express trig functions of A/2 in terms of cosA: sin(A/2) = sqrt((1-cosA)/2), cos(A/2) = sqrt((1+cosA)/2).
What is the half-angle of 30 degrees?▼
Half of 30 is 15 degrees. sin15 = 0.259, cos15 = 0.966, tan15 = 0.268. Exact: sin15 = (sqrt(6)-sqrt(2))/4.
How are half-angle formulas derived?▼
From cos(2u) = 1-2sin^2(u). Set u = A/2, then cosA = 1-2sin^2(A/2). Rearrange to get sin^2(A/2) = (1-cosA)/2.
Why do half-angle formulas have +/- signs?▼
Because taking a square root gives both positive and negative values. Choose + or - based on the quadrant of the half-angle. In Quadrant I, both sin and cos are positive.
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