Find-the-set-of-values-of-x-0-2pi-which-satisfy-sin-x-gt-cos-x-1-pi-4-3pi-4-5pi-4-2pi-2-0-pi-4-5pi-4-2pi-3-pi-4-5pi-4-4-0-3pi-4-5pi-4-2pi-

Question Number 16092 by Tinkutara last updated on 17/Jun/17

$$\mathrm{Find}\:\mathrm{the}\:\mathrm{set}\:\mathrm{of}\:\mathrm{values}\:\mathrm{of}\:{x}\:\in\:\left[\mathrm{0},\:\mathrm{2}\pi\right] \\ $$$$\mathrm{which}\:\mathrm{satisfy}\:\mathrm{sin}\:{x}\:>\:\mathrm{cos}\:{x}. \\ $$$$\left(\mathrm{1}\right)\:\left(\frac{\pi}{\mathrm{4}},\:\frac{\mathrm{3}\pi}{\mathrm{4}}\right)\:\cup\:\left(\frac{\mathrm{5}\pi}{\mathrm{4}},\:\mathrm{2}\pi\right) \\ $$$$\left(\mathrm{2}\right)\:\left(\mathrm{0},\:\frac{\pi}{\mathrm{4}}\right)\:\cup\:\left(\frac{\mathrm{5}\pi}{\mathrm{4}},\:\mathrm{2}\pi\right) \\ $$$$\left(\mathrm{3}\right)\:\left(\frac{\pi}{\mathrm{4}},\:\frac{\mathrm{5}\pi}{\mathrm{4}}\right) \\ $$$$\left(\mathrm{4}\right)\:\left(\mathrm{0},\:\frac{\mathrm{3}\pi}{\mathrm{4}}\right)\:\cup\:\left(\frac{\mathrm{5}\pi}{\mathrm{4}},\:\mathrm{2}\pi\right) \\ $$

Commented by Tinkutara last updated on 19/Jun/17

$$\mathrm{Thanks}\:\mathrm{Sir}! \\ $$

Commented by mrW1 last updated on 17/Jun/17

$$\left(\mathrm{3}\right) \\ $$

Commented by Tinkutara last updated on 18/Jun/17

$$\mathrm{Can}\:\mathrm{you}\:\mathrm{explain}\:\mathrm{please}? \\ $$

Commented by mrW1 last updated on 18/Jun/17

$$\mathrm{sin}\:\mathrm{x}−\mathrm{cos}\:\mathrm{x}>\mathrm{0} \\ $$$$\sqrt{\mathrm{2}}\left(\frac{\mathrm{1}}{\:\sqrt{\mathrm{2}}}\mathrm{sin}\:\mathrm{x}−\frac{\mathrm{1}}{\:\sqrt{\mathrm{2}}}\mathrm{cos}\:\mathrm{x}\right)>\mathrm{0} \\ $$$$\sqrt{\mathrm{2}}\mathrm{sin}\:\left(\mathrm{x}−\frac{\pi}{\mathrm{4}}\right)>\mathrm{0} \\ $$$$\mathrm{sin}\:\left(\mathrm{x}−\frac{\pi}{\mathrm{4}}\right)>\mathrm{0} \\ $$$$\Rightarrow\mathrm{0}<\mathrm{x}−\frac{\pi}{\mathrm{4}}<\pi \\ $$$$\frac{\pi}{\mathrm{4}}<\mathrm{x}<\frac{\mathrm{5}\pi}{\mathrm{4}} \\ $$

Commented by mrW1 last updated on 18/Jun/17

$$\mathrm{or}\:\mathrm{you}\:\mathrm{just}\:\mathrm{have}\:\mathrm{a}\:\mathrm{look}\:\mathrm{at}\:\mathrm{the}\:\mathrm{graphs} \\ $$$$\mathrm{from}\:\mathrm{sin}\:\mathrm{x}\:\mathrm{and}\:\mathrm{cos}\:\mathrm{x}. \\ $$

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