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45 results for “identity”

Formulae (43)

Difference of Squares

a^2 - b^2 = (a+b)(a-b)

Factors the difference of two perfect squares.

AlgebraBasic

Perfect Square Trinomial

(a \pm b)^2 = a^2 \pm 2ab + b^2

Expands the square of a binomial.

AlgebraIntermediate

Sum of Cubes

a^3 + b^3 = (a+b)(a^2 - ab + b^2)

Factors the sum of two cubes.

AlgebraIntermediate

Difference of Cubes

a^3 - b^3 = (a-b)(a^2 + ab + b^2)

Factors the difference of two cubes.

AlgebraIntermediate

Cube of a Binomial

(a+b)^3 = a^3 + 3a^2b + 3ab^2 + b^3

Expands the cube of a binomial expression.

Geometry & TrigonometryIntermediate

Sine Double Angle

\sin(2\theta) = 2\sin\theta\cos\theta

Double-angle identity for sine.

Geometry & TrigonometryIntermediate

Cosine Double Angle

\cos(2\theta) = \cos^2\theta - \sin^2\theta

Double-angle identity for cosine.

Geometry & TrigonometryIntermediate

Pythagorean Identity

\sin^2\theta + \cos^2\theta = 1

The fundamental trigonometric identity.

Geometry & TrigonometryIntermediate

Tangent Identity

\tan\theta = \frac{\sin\theta}{\cos\theta}

Defines tangent in terms of sine and cosine.

Geometry & TrigonometryIntermediate

Sine Angle Sum

\sin(A \pm B) = \sin A\cos B \pm \cos A\sin B

Angle addition formula for sine.

Geometry & TrigonometryIntermediate

Cosine Angle Sum

\cos(A \pm B) = \cos A\cos B \mp \sin A\sin B

Angle addition formula for cosine.

Geometry & TrigonometryIntermediate

Cotangent Identity

\cot\theta = \frac{\cos\theta}{\sin\theta}

Cotangent as the ratio of cosine to sine.

Geometry & TrigonometryIntermediate

Reciprocal Trig Identities

\sec\theta = \tfrac{1}{\cos\theta}, \ \csc\theta = \tfrac{1}{\sin\theta}

Secant and cosecant as reciprocals.

Geometry & TrigonometryAdvanced

Tangent Double Angle

\tan(2\theta) = \frac{2\tan\theta}{1 - \tan^2\theta}

Double-angle identity for tangent.

Geometry & TrigonometryAdvanced

Tangent Half-Angle

\tan\frac{\theta}{2} = \frac{1-\cos\theta}{\sin\theta}

Expresses the tangent of a half angle.

Geometry & TrigonometryBasic

Cofunction Identity

\sin\theta = \cos(90\degree - \theta)

Sine and cosine of complementary angles are equal.

TrigonometryBasic

Tangent Identity

\tan\theta = \frac{\sin\theta}{\cos\theta}

Tangent equals sine divided by cosine.

TrigonometryBasic

Pythagorean Identity

\sin^2\theta + \cos^2\theta = 1

The fundamental trig identity from the unit circle.

TrigonometryIntermediate

Identity: 1 + tan² = sec²

1 + \tan^2\theta = \sec^2\theta

Derived Pythagorean identity dividing by cos².

TrigonometryIntermediate

Identity: 1 + cot² = csc²

1 + \cot^2\theta = \csc^2\theta

Derived Pythagorean identity dividing by sin².

TrigonometryIntermediate

Sine of a Sum

\sin(A+B) = \sin A\cos B + \cos A\sin B

Angle addition formula for sine.

TrigonometryIntermediate

Sine of a Difference

\sin(A-B) = \sin A\cos B - \cos A\sin B

Angle subtraction formula for sine.

TrigonometryIntermediate

Cosine of a Sum

\cos(A+B) = \cos A\cos B - \sin A\sin B

Angle addition formula for cosine.

TrigonometryIntermediate

Cosine of a Difference

\cos(A-B) = \cos A\cos B + \sin A\sin B

Angle subtraction formula for cosine.

TrigonometryAdvanced

Tangent of a Sum

\tan(A+B) = \frac{\tan A + \tan B}{1 - \tan A\tan B}

Angle addition formula for tangent.

TrigonometryAdvanced

Tangent of a Difference

\tan(A-B) = \frac{\tan A - \tan B}{1 + \tan A\tan B}

Angle subtraction formula for tangent.

TrigonometryIntermediate

Double Angle: Sine

\sin 2\theta = 2\sin\theta\cos\theta

Sine of twice an angle.

TrigonometryIntermediate

Double Angle: Cosine

\cos 2\theta = \cos^2\theta - \sin^2\theta

Cosine of twice an angle (also 2cos²θ−1 or 1−2sin²θ).

TrigonometryAdvanced

Double Angle: Tangent

\tan 2\theta = \frac{2\tan\theta}{1 - \tan^2\theta}

Tangent of twice an angle.

TrigonometryAdvanced

Half Angle: Sine

\sin\frac{\theta}{2} = \pm\sqrt{\frac{1-\cos\theta}{2}}

Sine of half an angle.

TrigonometryAdvanced

Half Angle: Cosine

\cos\frac{\theta}{2} = \pm\sqrt{\frac{1+\cos\theta}{2}}

Cosine of half an angle.

TrigonometryAdvanced

Half Angle: Tangent

\tan\frac{\theta}{2} = \frac{1-\cos\theta}{\sin\theta}

Tangent of half an angle.

TrigonometryAdvanced

Product: sin A cos B

\sin A\cos B = \tfrac{1}{2}[\sin(A+B) + \sin(A-B)]

Product-to-sum identity.

TrigonometryAdvanced

Product: cos A cos B

\cos A\cos B = \tfrac{1}{2}[\cos(A-B) + \cos(A+B)]

Product-to-sum identity.

TrigonometryAdvanced

Product: sin A sin B

\sin A\sin B = \tfrac{1}{2}[\cos(A-B) - \cos(A+B)]

Product-to-sum identity.

TrigonometryAdvanced

Sum to Product: sin A + sin B

\sin A + \sin B = 2\sin\tfrac{A+B}{2}\cos\tfrac{A-B}{2}

Sum-to-product identity.

TrigonometryAdvanced

Sum to Product: sin A − sin B

\sin A - \sin B = 2\cos\tfrac{A+B}{2}\sin\tfrac{A-B}{2}

Sum-to-product identity.

TrigonometryAdvanced

Sum to Product: cos A + cos B

\cos A + \cos B = 2\cos\tfrac{A+B}{2}\cos\tfrac{A-B}{2}

Sum-to-product identity.

TrigonometryAdvanced

Sum to Product: cos A − cos B

\cos A - \cos B = -2\sin\tfrac{A+B}{2}\sin\tfrac{A-B}{2}

Sum-to-product identity.

TrigonometryAdvanced

Euler's Formula

e^{i\theta} = \cos\theta + i\sin\theta

Links complex exponentials to trigonometry.

Linear AlgebraBasic

Identity Matrix

AI = IA = A

The identity matrix leaves any matrix unchanged under multiplication.

Linear AlgebraBasic

Identity Matrix Property

AI = IA = A

Multiplying by the identity leaves a matrix unchanged.

AccountingBasic

Accounting Equation

Assets = Liabilities + Equity

The fundamental balance-sheet identity.

Math Tricks (2)

Squaring Numbers Near 50

Quick method to square any number near 50 using the identity (50±d)² = 2500 ± 100d + d².

View trick

Square Any Number (a±b Method)

Use the identity (a+b)(a-b) + b² = a² to square numbers quickly by going to the nearest easy number.

View trick