Trigonometric Identities and Formulas

Trig Identities

Unit Circle

The unit circle is represented by coordinates (x,y)=(cosθ,sinθ)(x, y) = (cos \theta, sin \theta).

Reciprocal Identities

  • Cosecant: csc(x)=1sin(x)csc(x) = \frac{1}{sin(x)} which implies sin(x)⋅csc(x)=1sin(x) \cdot csc(x) = 1
  • Secant: sec(x)=1cos(x)sec(x) = \frac{1}{cos(x)} which implies cos(x)⋅sec(x)=1cos(x) \cdot sec(x) = 1

Pythagorean Identities

  • sin2(x)+cos2(x)=1sin^2(x) + cos^2(x) = 1
  • 1+tan2(x)=sec2(x)1 + tan^2(x) = sec^2(x)
  • cot2(x)+1=csc2(x)cot^2(x) + 1 = csc^2(x)

Double Angle Formulas

  • Sine: sin(2x)=2sin(x)cos(x)sin(2x) = 2 sin(x) cos(x)
  • Cosine: cos(2x)=cos2(x)−sin2(x)=2cos2(x)−1=1−2sin2(x)cos(2x) = cos^2(x) - sin^2(x) = 2 cos^2(x) - 1 = 1 - 2 sin^2(x)

Domain Restrictions on Inverse Trig Functions

  • y=sin−1(x)y = sin^{-1}(x)
  • y=cos−1(x)y = cos^{-1}(x)
  • y=tan−1(x)y = tan^{-1}(x)

Sum Formulas

  • Sine: sin(A+B)=sin(A)cos(B)+cos(A)sin(B)sin(A + B) = sin(A)cos(B) + cos(A)sin(B)
  • Cosine: cos(A+B)=cos(A)cos(B)−sin(A)sin(B)cos(A + B) = cos(A)cos(B) - sin(A)sin(B)

Difference Formulas

  • Sine: sin(A−B)=sin(A)cos(B)−cos(A)sin(B)sin(A - B) = sin(A)cos(B) - cos(A)sin(B)
  • Cosine: cos(A−B)=cos(A)cos(B)+sin(A)sin(B)cos(A - B) = cos(A)cos(B) + sin(A)sin(B)