Enter an angle to compute sin(3A), cos(3A), and tan(3A) with step-by-step derivation
Angle A (degrees)
Result
Triple Angle: 3A =
sin(3A)
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cos(3A)
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tan(3A)
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Detailed Derivation
Triple Angle Formulas
sin(3A) = 3 sinA - 4 sin^3A
cos(3A) = 4 cos^3A - 3 cosA
tan(3A) = (3 tanA - tan^3A) / (1 - 3 tan^2A)
Triple-angle identities are derived by applying sum formulas to (2A + A). They reveal elegant patterns that connect trigonometry to algebra, particularly in cubic equation solving.
⚠tan(3A) is undefined when denominator 1 - 3tan^2A = 0, which occurs at tanA = +/- 1/sqrt(3) (A = 30, 150 degrees).
What Are Triple-Angle Identities?
Triple-angle identities express trigonometric functions of 3A in terms of functions of A. They extend the pattern of double-angle formulas and have elegant connections to cubic equations and Chebyshev polynomials.
Sine Triple
sin(3A) = 3sinA - 4sin^3A. Notice the cubic pattern: 3sinA minus 4sin^3A.
Cosine Triple
cos(3A) = 4cos^3A - 3cosA. The opposite sign pattern: 4cos^3A minus 3cosA.
Derivation Path
Use sin(2A+A) with sin2A = 2sinAcosA and cos2A = 2cos^2A-1. Apply Pythagorean identity to simplify.
Cubic Connection
Let x = cosA. Then cos(3A) = 4x^3 - 3x. Solving 4x^3 - 3x = c gives the trigonometric solution of cubics.
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