Math: List of Trigonometric Identities

Basic Identities : Pythagorean and Ratios

• sin² A + cos² A = 1
• tan² A + 1 = sec² A
• 1 + cot² A = cosec² A
• tanA = sinA / cosA
• cosecA = 1 / sinA
• secA = 1 / cosA
• cotA = cosA / sinA

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  cotA = 1 / tanA

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Co-function Identities: Shifts

• sin[(½)π - A] = cosA

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  sin(90° - A) = cosA
• cos[(½)π - A] = sinA

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  cos(90° - A) = sinA
• tan[(½)π - A] = cotA

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  tan(90° - A) = cotA

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Odd-Even Identities : Reflection

• sin(-A) = - sinA
• cos(-A) = cosA
• tan(-A) = - tanA

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Addition Formulas : Angle Sum and Difference

• sin(A + B) = sinA·cosB + cosA·sinB
• sin(A - B) = sinA·cosB - cosA·sinB
• cos(A + B) = cosA·cosB - sinA·sinB
• cos(A - B) = cosA·cosB + sinA·sinB
• tan(A + B) = ( tanA + tanB ) / ( 1 - tanA·tanB )
• tan(A - B) = ( tanA - tanB ) / ( 1 + tanA·tanB )

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Double-Angle Formulas

• sin2A = 2·sinA·cosA
• cos2A = cos² A - sin² A

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  cos2A = 1 - 2·sin² A

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  cos2A = 2·cos² A - 1
• tan2A = ( 2·tanA ) / ( 1 - tan² A )

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Half-Angle Formulas

• sin²(½A) = (1 - cosA)/2

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  sin(½A) = √[( 1 - cosA )/2]
• cos²(½A) = ( 1 + cosA )/2

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  cos(½A) = √[( 1 + cosA )/2]
• tan(½A) = ( 1 - cosA ) / sinA

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  tan(½A) = sinA / ( 1 + cosA )

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Product Formulas

• 2·sinA·cosB = sin(A + B) + sin(A - B)
• 2·cosA·cosB = cos(A + B) + cos(A - B)
• 2·cosA·sinB = sin(A + B) - sin(A - B)
• 2·sinA·sinB = cos(A - B) - cos(A + B)

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Factoring Formulas

• sinA + sinB = 2·cos[ (A - B)/2 ]·sin[ (A + B)/2 ]
• sinA - sinB = 2·cos[ (A + B)/2 ]·sin[ (A - B)/2 ]
• cosA + cosB = 2·cos[ (A + B)/2 ]·cos[ (A - B)/2 ]
• cosA - cosB = - 2·sin[ (A + B)/2 ]·sin[ (A - B)/2 ]