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

Find the set of values of x ∈ [0, 2π] which satisfy sin x > cos x. (1) ((π/4), ((3π)/4)) ∪ (((5π)/4), 2π) (2) (0, (π/4)) ∪ (((5π)/4), 2π) (3) ((π/4), ((5π)/4)) (4) (0, ((3π)/4)) ∪ (((5π)/4), 2π)

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

Commented byTinkutara last updated on 19/Jun/17

Thanks Sir!

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

Commented bymrW1 last updated on 17/Jun/17

(3)

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

Commented byTinkutara last updated on 18/Jun/17

Can you explain please?

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

Commented bymrW1 last updated on 18/Jun/17

sin x−cos x>0 (√2)((1/(√2))sin x−(1/(√2))cos x)>0 (√2)sin (x−(π/4))>0 sin (x−(π/4))>0 ⇒0<x−(π/4)<π (π/4)<x<((5π)/4)

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

Commented bymrW1 last updated on 18/Jun/17

or you just have a look at the graphs from sin x and cos x.

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

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