Composites - Trig Functions and Inverse Trig Functions

EVALUATING COMPOSITES - TRIG FUNCTIONS AND INVERSE TRIG FUNCTIONS paul vaz The table below lists some composites of trig functions and inverse trig functions:

sin 1 cos 1 tan 1 sin sin(sin 1 x) sin(cos 1 x) sin( tan 1 x) sin 1 (sin x) cos 1 (sin x) tan 1 (sin x) cos cos(sin 1 x) cos(cos 1 x) cos( tan 1 x) sin 1 (cos x) cos 1 (cos x) tan 1 (cos x) tan tan(sin 1 x) tan( cos 1 x) tan(tan 1 x) sin 1 (tan x) cos 1 (tan x) tan 1 (tan x) csc csc(sin 1 x) csc(cos 1 x) csc( tan 1 x) sec sec(sin 1 x) sec(cos 1 x) sec( tan 1 x) cot cot(sin 1 x) cot( cos 1 x) cot( tan 1 x)

We will refer to any composite as: outside function(inside function) Example: In tan(sin 1 x), 'tan' is the outside function, and 'sin 1 ' is the inside function.

Inside function is an inverse: Generally, for all composites in which the inside function is an inverse, we evaluate the composite for a given 'x' using a substitution for the inside function. The substitution and the definition of the inverse, leads to the construction of a right triangle that helps evaluate the composite easily. Example: Evaluate tan(sin 1 0.8) Solution: Let p = sin 1 0.8 (Substitution) Therefore, sin p = 0.8 (Definition) Construction of right triangle:

angle p 0.6

( Using the Pythagorean, the third side of the triangle is 1 0.82  0.6 ) Therefore, tan(sin 1 0.8) = tan(p) = 0.8/0.6 = 1.33.

Useful Results: 1. sin(sin 1 x) = x if  1  x  1 2. cos(cos 1 x) = x if  1  x  1 3. tan(tan 1 x) = x if x is any real number.

Example: Evaluate sin(sin 1 0.8)

Since  1  0.8  1, sin(sin 1 0.8) = 0.8.

Inside function is not an inverse: In this case, work the inside function first, and then apply the outside function to the result.

 E xample: Evaluate sin 1 (tan ) 3  Solution: tan  3  1.732 3  Therefore, sin 1 (tan ) = sin 1 (1.732) (Not defined) 3

Useful Results:   4. sin 1 (sin x) = x if   x  2 2 5. cos 1 (cos x) = x if 0  x     6. tan 1 (tan x) = x if   x  2 2  Example: Evaluate tan 1 (tan ) 3      Since    , tan 1 (tan ) = 2 3 2 3 3