TRIGONOMETRIC IDENTITIES
Reciprocal identities
- sin(u) = 1/csc(u) ; cos(u) = 1/sec(u)
- tan(u) = 1/cot(u) ; cot(u) = 1/tan(u)
- csc(u) = 1/sin(u) ; sec(u) = 1/cos(u)
Pythagorean Identities
- sin²(u) + cos²(u) = 1
- 1 + tan²(u) = sec²(u)
- 1 + cot²(u) = csc²(u)
Quotient Identities
- tan(u) = sin(u)/cos(u)
- cot(u) = cos(u)/sin(u)
Co-Function Identities
- sind(x), cosd(x), tand(x)
- asind(x), acosd(x), atand(x)
Co-Function Identities
- sin((pi/2)-u) = cos(u) ; cos((pi/2)-u) = sin(u)
- tan((pi/2)-u) = cot(u) ; cot((pi/2)-u) = tan(u)
- csc((pi/2)-u) = sec(u) ; sec((pi/2)-u) = csc(u)
Parity Identities (Even & Odd)
- sin(-u) = -sin(u) ; cos(-u) = cos(u)
- tan(-u) = -tan(u) ; cot(-u) = -cot(u)
- csc(-u) = -csc(u) ; sec(-u) = sec(u)
Sum & Difference Formulas
- sin(u ± v) = sin(u)cos(v) ± cos(u)sin(v)
- cos(u ± v) = cos(u)cos(v) ∓ sin(u)sin(v)
- tan(u ± v) = (tan(u) ± tan(v)) / (1 ∓ tan(u)tan(v))
Double Angle Formulas
- sin(2u) = 2sin(u)cos(u)
- cos(2u) = cos²(u) + sin²(u)
= 2cos²(u) - 1
= 1 - 2sin²(u)
- tan(2u) = (2tan(u)) / (1 - tan²(u))
Power-Reducing/Half Angle Formulas
- sin²(u) = (1 - cos(2u)) / 2
- cos²(u) = (1 + cos(2u)) / 2
- tan²(u) = (1 - cos(2u)) / (1 + cos(2u))
Sum-to-Product Formulas
- sin(u) + sin(v) = 2sin((u+v)/2) * cos((u-v)/2)
- sin(u) - sin(v) = 2cos((u+v)/2) * sin((u-v)/2)
- cos(u) + cos(v) = 2cos((u+v)/2) * cos((u-v)/2)
- cos(u) - cos(v) = 2sin((u+v)/2) * sin((u-v)/2)
Product-to-Sum Formulas
- sin(u)sin(v) = ½[cos(u-v) - cos(u+v)]
- cos(u)cos(v) = ½[cos(u-v) + cos(u+v)]
- sin(u)cos(v) = ½[sin(u+v) + sin(u-v)]
- cos(u)sin(v) = ½[sin(u+v) - sin(u-v)]