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How the fundamental and Pythagorean identities were derived, along with a few identity problems.

Trigonometric identities.

Date Created:Sunday January 21st, 2007 12:32 AM
Date Modified:Saturday August 02nd, 2008 11:42 AM

Pythagorean Identites:

start out with the unit circle x2 + y2 = 12 x = cosθ and y = sin&theta cos2θ + sin2θ = 1 (1 of 3 pythagorean identities) divide by cos2θ cos2θ/cos2θ + sin2/cos2θ = 1/cos2θ 1 + sin2θ/cos2θ = 1/cos2θ tan2θ + 1 = sec2θ (2 of 3 pythagorean identities) divide unit circle equation by sin2θ cos2θ/sin2θ + sin2θ/sin2θ = 1/sin2θ cos2θ/sin2θ + 1 = 1/sin2θ cot2θ + 1 = csc2θ (3 of 3 pythagorean identities)

Fundamental Identities

tanθ = sinθ/cosθ cotθ = cosθ/sinθ cotθ = 1/tanθ cscθ = 1/sinθ secθ = 1/cosθ cos2θ + sin2θ = 1 tan2θ + 1 = sec2θ cot2θ + 1 = csc2θ

Using Identities to generate new equations

start with the pythagorean identity: cos2θ + sin2θ = 1 subtact sin2θ from unit circle equation and take square root

cosθ = ±√(1 - sin2θ)

subtact cos2θ from unit circle equation and take square root

sinθ = ±√(1- cos2θ)

divide by cos2θ from unit circle equation and take square root cos2θ + sin2θ = 1 cos2θ/cos2θ + sin2θ/cos2θ = 1/cos2θ 1 + tan2θ = sec2θ ±√(1 + tan2θ) = secθ

secθ = ±√(1 + tan2θ)

divide by sin2θ from unit circle equation and take square root cos2θ + sin2θ = 1 cos2θ/sin2θ + sin2θ/sin2θ = 1/sin2θ cot2θ + 1 = csc2θ ±√(cot2θ + 1) = cscθ

cscθ = ±√(cot2θ + 1)

Identities by Dan Lynch
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