Subjective Type

What is the value of $$ sin -1 (cos x) $$?

Solution

Assuming you mistyped and meant $$ sin^{-1}( cos (x)) $$ or simply arcsin $$ ( cos (x)) $$ we can easily solve this by putting it on terms of the sine function .
We know that the cosine function , is nothing more than the sine $$ \frac { \pi}{2} $$ radians out of phase as proved below
$$ cos ( \theta - \frac { \pi}{2}) = cos ( \theta) cos ( - \frac { \pi }{2}) - sin ( \theta )sin ( - \frac { \pi }{2} ) $$
$$ cos ( \theta - \frac { \pi}{2}) = cos ( \theta). 0 - (-sin ( \theta )sin ( \frac { \pi}{})) $$
$$ cos ( \theta - \frac { \pi}{2}) =sin ( \theta ) .1 = sin ( \theta ) $$
So we can say that the sine function, 90 degrees ahead, is the cosine function.
$$ arcsin ( cos (x)) = arcsin ( sin ( x + \frac { \pi }{2} )) $$
Using the property of inverse functions that $$ f^{-1}(f(x)) = x $$ we have
$$ arcsin (cos (x)) = x + \frac { \pi}{2} $$
if you must use degrees just convert those $$ \frac { \pi}{2} $$ radians to $$ 90^0 $$ degrees.

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