How would I go about solving $\displaystyle |\cos x|^{1+\sin x+\cos x} \geq 1$? All help is welcome.

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- Nov 20th 2010, 04:22 AMatreyyuTrig inequality?
How would I go about solving $\displaystyle |\cos x|^{1+\sin x+\cos x} \geq 1$? All help is welcome.

- Nov 20th 2010, 04:47 AMHallsofIvy
By THINKING, not using "formulas"!

You know that $\displaystyle |cos(x)|$ is**always less**than or equal to 1. You also know that powers of a number between 0 and 1 are always smaller than the number itself: if $\displaystyle 0< y< 1$ then $\displaystyle y^n< y$.

Saying that an absolute value of a power of cos(x) is "greater than or equal to 1" is exactly the same as saying it is**equal**to 1. And so the power itself is equal to either 1 or -1. That can only happen when the base is 1 or -1. For what x is cos(x)= 1 or cos(x)= -1? - Nov 20th 2010, 04:56 AMAlso sprach Zarathustra
- Nov 20th 2010, 04:59 AMPlato
Actually there are more values of x than Prof Halls gave.

If $\displaystyle 1+\sin(x)+\cos(x)<0$ then the given inequality holds.