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Advanced trigonometry deals with all the deeper topics and they are again categorized as inverse trigonometric functions, trigonometric forms, graphs, and formulas.

Inverse trigonometric functions:

The inverse trigonometric functionsare partial inverse functionsfor the trigonometric functions. For example, the inverse function for the sine, known as the inverse sine (sin?1) or arcsine (arcsin or asin), satisfies

sin (arcsin x) = x

and

arcsin (sin ?) = ? for –?/2 ? ? ? ?/2

Inverse trigonometric functions

Inverse of sin = arcsin, inverse of cos = arccos, inverse of tan = arctan, inverse of cosec = arccosec, inverse of sec = arcsec and inverse of cot = arccot.

Trigonometric form of complex numbers

The general form of a complex number is x + iy; any complex number can be expressed in trigonometry except 0.

z = r (cos ? + i•sin ?)

where z = ?cos2 ? + sin2 ?

z/|z| = cos ? + i•sin ?

? can be formed by using the formula tan ? = y/x

If z = r (cos ? + i•sin ?) and w = s (cos ? + i•sin ?), then

zw = rs (cos ? + i•sin ?) (cos ? + i•sin ?)

= rs (cos (?+?) + i. sin (? + ?))

(cos ? + i•sin ?)n = cos (n?) + i•sin (n?)

[r (cos ? + i•sin ?)]n = rn cos (n?) + i•sin (n?)

Graphing Functions

Graphing functions based on trigonometry is mainly based up on sine, cosine and tangent waves they are represented as shown below

Formulas for Sines and Cosines

There are few sine and cosine formulas that are rep[resented below

sin(?/2 – ?) = cos ?

cos(?/2 – ?) = sin?

sin(q) = sin(q + 2pk)

cos(q) = cos(q + 2pk),

where k is an integer.

sin(-q) = -sin(q) (if sine function is odd)

cos(-q) = cos(q) (if cosine function is even)

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

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

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

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

sin(2x) = 2sin(x)cos(x)

cos(2x) = cos2(x) - sin2(x)

sin2(x) = 1 - cos2(x)

cos2(x) = 1 - sin2(x)