Using Axiological Identities

 29 July 03:40   Some of the axiological trigometric identities are those acquired from the Pythagorean Theorem. These are authentic using a appropriate triangle:

    By the Pythagorean Theorem,

    :$A^2+B^2=C^2$

    Dividing through by C² gives

    :$left(frac$
ight)^2+left(frac
ight)^2=left(frac
ight)^2=1

    We accept already the sine of a in this case as A/C and the cosine of a as B/C. Appropriately we can acting these into to get

    :$sin^2\; a\; +\; cos^2\; a\; =\; 1$

    Related identities include:

    :$sin^2\; a\; =\; 1\; -\; cos^2\; amboxcos^2\; a\; =\; 1-sin^2\; a$

    :$an^2\; a\; +\; 1\; =\; sec^2\; ambox\; an^2\; a\; =\; sec^2\; a\; -1$

    :$1\; +\; cot^2\; a\; =\; csc^2\; amboxcot^2\; a\; =\; csc^2\; a\; -1$

    Other Axiological Identities cover the Reciprocal, Ratio, and Co-function identities

    Reciprocal identities

    :$csc\; a\; =\; fracquadsec\; a\; =\; fracquadcot\; a\; =\; frac$

    Ratio identities

    :$an\; a\; =\; fracquadcot\; a\; =\; frac$

    Co-function identities (in radians)

    :$cos\; a\; =\; sinleft(frac-a$
ight)quad csc a = secleft(frac-a
ight)quad cot a = anleft(frac-a
ight)