Trigonometry Identities

Trigonometric Identities

Pythagorean Identities

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

Sum and Difference Identities

• sin(A + B) = sin(A)cos(B) + cos(A)sin(B)
• sin(A - B) = sin(A)cos(B) - cos(A)sin(B)
• cos(A + B) = cos(A)cos(B) - sin(A)sin(B)
• cos(A - B) = cos(A)cos(B) + sin(A)sin(B)
• tan(A + B) = (tan(A) + tan(B)) / (1 - tan(A)tan(B))
• tan(A - B) = (tan(A) - tan(B)) / (1 + tan(A)tan(B))

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) = ±√((1 - cos(A)) / 2)
• cos(A/2) = ±√((1 + cos(A)) / 2)
• tan(A/2) = ±√((1 - cos(A)) / (1 + cos(A)))

Product Identities

• sin(A)cos(B) = (1/2)[sin(A + B) + sin(A - B)]
• cos(A)sin(B) = (1/2)[sin(A + B) - sin(A - B)]
• cos(A)cos(B) = (1/2)[cos(A + B) + cos(A - B)]
• sin(A)sin(B) = (1/2)[cos(A - B) - cos(A + B)]

Co-Function Identities

• sin(A) = cos(π/2 - A)
• cos(A) = sin(π/2 - A)
• tan(A) = cot(π/2 - A)
• cot(A) = tan(π/2 - A)
• sec(A) = csc(π/2 - A)
• csc(A) = sec(π/2 - A)

Trigonometric Identities

Pythagorean Identities

• The sum of sin^2(A) and cos^2(A) is always 1
• The sum of tan^2(A) and 1 is equal to sec^2(A)
• 1 plus cot^2(A) is equal to csc^2(A)

Sum and Difference Identities

• The sine of the sum of two angles A and B is sin(A)cos(B) + cos(A)sin(B)
• The sine of the difference of two angles A and B is sin(A)cos(B) - cos(A)sin(B)
• The cosine of the sum of two angles A and B is cos(A)cos(B) - sin(A)sin(B)
• The cosine of the difference of two angles A and B is cos(A)cos(B) + sin(A)sin(B)
• The tangent of the sum of two angles A and B is (tan(A) + tan(B)) / (1 - tan(A)tan(B))
• The tangent of the difference of two angles A and B is (tan(A) - tan(B)) / (1 + tan(A)tan(B))

Double Angle Identities

• The sine of 2A is 2sin(A)cos(A)
• The cosine of 2A is cos^2(A) - sin^2(A)
• The tangent of 2A is 2tan(A) / (1 - tan^2(A))

Half Angle Identities

• The sine of A/2 is ±√((1 - cos(A)) / 2)
• The cosine of A/2 is ±√((1 + cos(A)) / 2)
• The tangent of A/2 is ±√((1 - cos(A)) / (1 + cos(A)))

Product Identities

• The product of sin(A) and cos(B) is (1/2)[sin(A + B) + sin(A - B)]
• The product of cos(A) and sin(B) is (1/2)[sin(A + B) - sin(A - B)]
• The product of cos(A) and cos(B) is (1/2)[cos(A + B) + cos(A - B)]
• The product of sin(A) and sin(B) is (1/2)[cos(A - B) - cos(A + B)]

Co-Function Identities

• The sine of A is equal to the cosine of π/2 - A
• The cosine of A is equal to the sine of π/2 - A
• The tangent of A is equal to the cotangent of π/2 - A
• The cotangent of A is equal to the tangent of π/2 - A
• The secant of A is equal to the cosecant of π/2 - A
• The cosecant of A is equal to the secant of π/2 - A