Trigonometry Summary - rFronteddu/general_wiki GitHub Wiki

  • cos(a) = ADJ/HYP
  • sin(a) = OPP/HYP

Law of sines

  • The law of sines states that in any triangle, the ratio of the sine of an angle to the opposite side length is equal for all three angles and sides.
  • sin(A)/a = sin(B)/b = sin(C)/c
  • Where A, B, C are the angle measures of the triangle, and a, b, and c, are the opposite side lengths.

Law of cosine

  • The law of cosines states that in any triangle, the square of a side is equal to the sum of the squares of the other two sides minus twice the product of the other two sides and the cosine of the opposite angle.

Reciprocal and quotient identities

  • sec(a) = 1/cos(a)
  • csc(a) = 1/sin(a)
  • cot(a) = 1/tan(a)
  • tan(a) = sin(a)/cos(a)
  • cot(a) = cos(a)/sin(a)

Pythagorean identities

  • $$sin^2(a) + cos^2(a) = 1$$
  • $$tan^2(a) + 1 = sec^2(a)$$
  • $$cot^2(a) + 1 = csc^2(a)$$

Identities from sums, differences, multiples and fractions of angles

Angle sum and difference identities

  • sin(a+b)=sin(a)cos(b)+sin(b)cos(a)

  • sin(a-c)=sin(a)cos(c)-sin(c)cos(a)

  • cos(a+b)=cos(a)cos(b)-sin(a)sin(b)

  • cos(a-b)=cos(a)cos(b)+sin(a)sin(b)

  • tan(x+y)=(tan(x) + tan(y))/(1-tan(x)tan(y)).

  • tan(x-y)=(tan(x) - tan(y))/(1+tan(x)tan(y)).

Double angle identities

  • $$sin(2a)=2sin(a)cos(a)$$
  • $$cos(2a)=cos^2(a)-sin^2(a)$$
  • $$tan(2a)=2tan(a)/(1-tan^2(a))$$

Half angle identities

  • sin(a/2) = +- sqrt((1-cos(a)/2))
  • cos(a/2) = +- sqrt((1+cos(a)/2))
  • tan(a/2) = +- sqrt((1-cos(a))/(1+cos(a))) = (1-cos(a))/sin(a)=sin(a)/(1+cos(a))

Symmetry and periodicity identities

  • sin(-a) = -sin(a)
  • cos(-a) = cos(a)
  • tan(-a) = -tan(a)
  • sin(a+2pi) = sin(a)
  • cos(a+2pi) = cos(a)
  • tan(a+pi) = tan(a)

Cofunction identities

  • sin(a) = cos(pi/2-a)
  • cos(a) = sin(pi/2-a)
  • tan(a) = cot(pi/2 - a)
  • cot(a) = tan(pi/2 - a)
  • sec(a) = csc(pi/2-a)
  • csc(a) = sec(pi/2-a)

More

  • $$2*cos^2(a)-1=cos(2a)$$
  • $$cos^2(a)=(1/2) * (1+cos(2a))$$
  • $$cos^2(a)+sin^2(a)=1$$
  • $$cos(2a)=1-2sin^2(a)$$
  • $$sin^2(a)=(1/2)*(1-cos(2a))$$
  • The word Sine originates from the Latin word Sinus, and it means curve or bay. Here it is used to mean: The line formed by the inside triangle, that is closest to the curve.
  • The word Secant, is from the Latin word secare, and it means to cut. Here it is used to mean: The line formed by the inside triangle, that when extended, cuts the curve.
  • The word Tangent, in Latin, means touching. Here it is used to mean: The line that touches the curve, at the same point where the inside triangle touches the curve.
  • The "co-" prefix comes from Latin, it literally means "complimentary", or as I like to think of it: "at 90 degrees to something"
    • Cosine for: The line complementary to the Sine. Or stated another way: The line at 90 degrees to the line of the triangle, closest to the curve.
    • Cosecant for: The line complementary to the Secant. Or rather: The line at 90 degrees to Secant that, when extended, also cuts the curve.
    • Cotangent for: The line complementary to the Tangent. Or perhaps better understood as: The line-segment of the extended touching-line.