Fundamental Trigonometric Identities: Reciprocal, Quotient, and Pythagorean

Fundamental Trigonometric Identities

Reciprocal Identities

  • Definition of Trigonometric Functions for Acute Angle hetaheta:

    • extSine(extsinheta)ext{Sine } ( ext{sin} heta): Ratio of the side opposite to the hypotenuse.
    • extCosecant(extcscheta)ext{Cosecant } ( ext{csc} heta): Ratio of the hypotenuse to the opposite side. (Notice: These are reciprocals).
    • extCosine(extcosheta)ext{Cosine } ( ext{cos} heta): Ratio of the side adjacent to the hypotenuse.
    • extSecant(extsecheta)ext{Secant } ( ext{sec} heta): Ratio of the hypotenuse to the adjacent side. (Notice: These are reciprocals).
    • extTangent(exttanheta)ext{Tangent } ( ext{tan} heta): Ratio of the side opposite to the adjacent side.
    • extCotangent(extcotheta)ext{Cotangent } ( ext{cot} heta): Ratio of the adjacent side to the opposite side. (Notice: These are reciprocals).

  • Formal Reciprocal Identities:

    • extcscheta=1extsinhetaext{csc} heta = \frac{1}{ ext{sin} heta}
    • extsecheta=1extcoshetaext{sec} heta = \frac{1}{ ext{cos} heta}
    • extcotheta=1exttanhetaext{cot} heta = \frac{1}{ ext{tan} heta}

Quotient Identities

  • Derivation of exttanhetaext{tan} heta:
    • Consider the ratio extsinhetaextcosheta\frac{ ext{sin} heta}{ ext{cos} heta}.
    • Substitute definitions: extsinhetaextcosheta=extoppositeexthypotenuseextadjacentexthypotenuse\frac{ ext{sin} heta}{ ext{cos} heta} = \frac{\frac{ ext{opposite}}{ ext{hypotenuse}}}{\frac{ ext{adjacent}}{ ext{hypotenuse}}}.
    • Simplify the compound fraction by