Trigonometric Identities
Apr 19 2016
Verify Trigonometric Identities
We verify several important trigonometric identities using knowledge of Euler's formula.
We first verify the angle addition formulas.
`cos(alpha + beta) = cos(alpha) cos(beta) - sin(alpha) sin(beta)``sin(alpha + beta) = cos(alpha) sin(beta) + sin(alpha) cos(beta)`
Next, we verify the formulas on multiplications of circular functions.
`cos(alpha) cos(beta) = 1 / 2 (cos(alpha + beta) + cos(alpha - beta))``sin(alpha) cos(beta) = 1 / 2 (sin(alpha + beta) + sin(alpha - beta))`
`sin(alpha) sin(beta) = -1 / 2 (cos(alpha + beta) - cos(alpha - beta))`
`cos(alpha) sin(beta) = 1 / 2 (sin(alpha + beta) - sin(alpha - beta))`
Finally, we verify the triple-angle formulas.
`cos(3 alpha) = 4 cos(alpha)^3 - 3 cos(alpha)``sin(3 alpha) = -4 sin(alpha)^3 + 3 sin(alpha)`