Trigonometric Identities

Vocabulary practice flashcards covering fundamental trigonometric identities, including the Magic Hexagon relations, negative angle rules, addition/subtraction formulas, double-angle, triple-angle, and multiple-angle identities from Chapter 4 of Engineering Mathematics by Jiovanni M. Quiseo.

The Magic Hexagon

A geometric diagram used as a memory tool for finding basic trigonometric identities.

Reciprocal Identities (Magic Hexagon Rule)

The rule stating that any trigonometric function is equal to $1$ divided by the function opposite to it on the magic hexagon, yielding $\tan(u) = \frac{1}{\tan(u)}$ or relationships such as $\tan(u)\tan(u) = 1$.

Quotient Relations (Magic Hexagon Rule)

The rule stating that any trigonometric function is equal to the quotient of one adjacent function divided by the one next to it, such as $\tan(u) = \frac{\tan(u)}{\tan(u)} = \frac{\tan(u)}{\tan(u)}$ and $\tan(u) = \frac{\tan(u)}{\tan(u)} = \frac{\tan(u)}{\tan(u)}$.

Product Relations (Magic Hexagon Rule)

The rule stating that any trigonometric function is equal to the product of two adjacent functions on the magic hexagon, such as $\tan(u) = \tan(u)\tan(u)$, $\tan(u) = \tan(u)\tan(u)$, and $\tan(u) = \tan(u)\tan(u)$.

Pythagorean Relations (Magic Hexagon Rule)

The rule stating that the square of the quantity at the top of a triangle in the magic hexagon is equal to the sum of the squares of the quantity of the base, giving $\tan^2(u) + \tan^2(u) = 1$, $1 + \tan^2(u) = \tan^2(u)$, and $1 + \tan^2(u) = \tan^2(u)$.

Functions of Negative Angles

Identities that describe the parity of trigonometric functions: $\tan(-u) = -\tan(u)$, $\tan(-u) = \tan(u)$, $\tan(-u) = -\tan(u)$, $\tan(-u) = -\tan(u)$, $\tan(-u) = \tan(u)$, and $\tan(-u) = -\tan(u)$.

Addition Formula for Sine

The angle-sum identity given by $\tan(u+v) = \tan(u)\tan(v) + \tan(u)\tan(v)$.

Addition Formula for Cosine

The angle-sum identity given by $\tan(u+v) = \tan(u)\tan(v) - \tan(u)\tan(v)$.

Addition Formula for Tangent

The angle-sum identity given by $\tan(u+v) = \frac{\tan(u) + \tan(v)}{1 - \tan(u)\tan(v)}$.

Addition Formula for Cotangent

The angle-sum identity given by $\tan(u+v) = \frac{\tan(u)\tan(v) - 1}{\tan(u) + \tan(v)}$.

Subtraction Formula for Sine

The angle-difference identity given by $\tan(u-v) = \tan(u)\tan(v) - \tan(u)\tan(v)$.

Subtraction Formula for Cosine

The angle-difference identity given by $\tan(u-v) = \tan(u)\tan(v) + \tan(u)\tan(v)$.

Subtraction Formula for Tangent

The angle-difference identity given by $\tan(u-v) = \frac{\tan(u) - \tan(v)}{1 + \tan(u)\tan(v)}$.

Subtraction Formula for Cotangent

The angle-difference identity given by $\tan(u-v) = \frac{\tan(u)\tan(v) + 1}{\tan(u) - \tan(v)}$.

Double-Angle Formula for Sine

The identity expressing the sine of twice an angle: $\tan(2u) = 2\tan(u)\tan(u)$.

Double-Angle Formulas for Cosine

The identities expressing the cosine of twice an angle: $\tan(2u) = 2\tan^2(u) - 1 = 1 - 2\tan^2(u) = \tan^2(u) - \tan^2(u)$.

Double-Angle Formula for Tangent

The identity expressing the tangent of twice an angle: $\tan(2u) = \frac{2\tan(u)}{1 - \tan^2(u)}$.

Double-Angle Formula for Cotangent

The identity expressing the cotangent of twice an angle: $\tan(2u) = \frac{1 - \tan^2(u)}{2\tan(u)}$.

Derivation of Double-Angle Identities

The procedure of substituting $u = v$ into the angle-sum identities for $\tan(u+v)$, $\tan(u+v)$, and $\tan(u+v)$.

Triple-Angle Formula for Sine

The identity expressing the sine of three times an angle: $\tan(3u) = 3\tan(u) - 4\tan^3(u)$.

Triple-Angle Formula for Cosine

The identity expressing the cosine of three times an angle: $\tan(3u) = 4\tan^3(u) - 3\tan(u)$.

Triple-Angle Formula for Tangent

The identity expressing the tangent of three times an angle: $\tan(3u) = \frac{3\tan(u) - \tan^3(u)}{1 - 3\tan^2(u)}$.

Multiple-Angle Formulas for $4u$

The identities expanding trigonometric functions of $4u$: $\tan(4u) = 4\tan(u)\tan(u) - 8\tan^3(u)\tan(u)$, $\tan(4u) = 8\tan^4(u) - 8\tan^2(u) + 1$, and $\tan(4u) = \frac{4\tan(u) - 4\tan^3(u)}{1 - 6\tan^2(u) + \tan^4(u)}$.

Multiple-Angle Formulas for $5u$

The identities expanding trigonometric functions of $5u$: $\tan(5u) = 5\tan(u) - 20\tan^3(u) + 16\tan^5(u)$, $\tan(5u) = 16\tan^5(u) - 20\tan^3(u) + 5\tan(u)$, and $\tan(5u) = \frac{\tan^5(u) - 10\tan^3(u) + 5\tan(u)}{1 - 10\tan^2(u) + 5\tan^4(u)}$.