Trigonometric Identities and Formulas

Trig Identities

Unit Circle

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

Reciprocal Identities

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

Pythagorean Identities

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

Double Angle Formulas

• Sine: sin(2x) = 2 sin(x) cos(x)
• Cosine: 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 = cos^{-1}(x)
• y = tan^{-1}(x)

Sum Formulas

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