Derivation of Trigonometric Functions

Pythagorean Identities

400

cos²x + sin²x = 1 (cos²x + sin²x)/sin²x = 1/sin²x cos²x/sin²x + 1 = 1/sin²x cot²x + 1 = csc²x

What is the derivation of cot²x + 1 = csc²x?

500

Cos²x + sin²x=1 (Cos²x + sin²x)/cos²x = 1/cos²x 1 + sin²x/cos²x = 1/cos²x 1+tan²x= sec²x

What is the derivation of 1+tan²x= sec²x?

Derivation for cosx

400

cos2x = cos(x+x) Cosxcosx-sinxsinx Cos²x- sin²x Cos²x- (1-cos²x) Cos²x-1-cos²x 2cos²x-1 cos2x = 2cos²x-1 let x=x/2 cos x = 2cos²(x/2)-1 +1 to both sides 1+cosx = 2cos²(x/2) divide by 2 (1+cosx)/2 = cos²(x/2) square root ±√((1+cosx)/2) = cos(x/2)

What is the half-angle derivation of cos(x/2)?

500

cos(x+y) = cosxcosy - sinxsiny Let y=x cosxcosx-sinxsinx Cos²x-sin²x Cos2x Cosxcosx-sinxsinx Cos²x- sin²x Cos²x- (1-cos²x) Cos²x-1-cos²x 2cos²x-1 cos2x=2cos²x-1 (Cos2x+1)/2 = (2cos²x)/2 (cos2x+1)/2 = Cos²x

What is the power-reducing derivation of Cos²x?

Derivations for sinx

300

cos2x = cos(x+x) Cosxcosx-sinxsinx Cos²x-sin²x sin²x-(1-sin²x) Sin²x -1-sin²x 1-2sin²x cos2x=1-2sin²x -1 from both sides cos2x - 1 = -2sin²x divide both sides by -2 (1-cos2x)/2 = sin²x

What is power-reducing identity of Sin²x?

400

cos2x=cos(x+x) Cosxcosx-sinxsinx Cos²x-sin²x sin²x-(1-sin²x) Sin²x -1-sin²x 1-2sin²x cos2x = 1-2sin²x Let x=x/2 cos x = 1-2sin²(x/2) +2sin²(x/2) to both sides -cos x to both sides 2sin²x(x/2) = 1-cosx divide both sides by 2 and square root sin(x/2) = ±√((1-cosx)/2)

What is the half-angle derivation of sin(x/2)?

500

Sin2x = sin(x+x) sin(x+y) = sinxcosy + cosxsiny Let y=x Sinxcosx+cosxsinx =2sinxcosx

What is the double-angle derivation of sin2x?

Derivations for tanx

400

tanx = sinx/cosx tan²x = sin²x/cos²x tan²x = ((1-cos2x)/2)/((cos2x+1)/2) same, dot, flip tan²x = (1-cos2x)/2*2/(cos2x+1) tan²x = (1-cos2x)/(cos2x+1)

What is power-reducing derivation for tan²x?

500

tan2x = tan(x+x) tan(x+y) = (tanx + tany)/(1 - tanxtany) Let y=x (Tanx+tanx)/(1- tanxtanx) (2tanx)/(1-tan²x)

What is the Double-Angle derivation of tan2x?

Functions with 3 Identities

400

1. tanx = sinx/cosx tan²x = sin²x/cos²x tan²x = ((1-cos2x)/2)/((cos2x+1)/2) same, dot, flip tan²x = (1-cos2x)/2*2/(cos2x+1) tan²x = (1-cos2x)/(cos2x+1) square root tanx = ±√((1-cos2x)/(cos2x+1)) let x=x/2 tan(x/2) = ±√((1-cosx)/(cosx+1)) 2. tan(x/2) = ±√((1-cosx)/(cosx+1)) multiplied by (cosx+1)/(cosx+1) tan(x/2) = ±√((1-cos²x)/(1+cosx)²) tan(x/2) = ±√(Sin²x/(1+cosx)²) tan(x/2) = sinx/(1+cosx) 3. tan(x/2) = ±√((1-cosx)/(cosx+1)) multiplied by (cosx-1)/(cosx-1) tan(x/2) = ±√((1-cosx)²/(1-cos²x)) tan(x/2) = ±√((1-cosx)²/Sin²x) tan(x/2) = (1-cosx)/sinx

Why/How are there three half-angle identities of tan(u/2)?

500

1. cos(x+y) = cosxcosy - sinxsiny Let y=x cosxcosx-sinxsinx Cos²x-sin²x 2. Cosxcosx-sinxsinx Cos²x-sin²x sin²x-(1-sin²x) Sin²x -1-sin²x 1-2sin²x 3. Cosxcosx-sinxsinx Cos²x- sin²x Cos²x- (1-cos²x) Cos²x-1-cos²x 2cos²x-1

Why/How are there three double angle identities of cos2x?