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Popular >

cos(x)sin(x)<0.451

Solution

cos (x) sin (x) < 0.451

Solution

\frac{- π - 1.12437 \dots}{2} + πn < x < 0.56218 \dots + πn

+2

Interval Notation

(\frac{- π - 1.12437 \dots}{2} + πn, 0.56218 \dots + πn)

Decimal Notation

- 2.13298 \dots + πn < x < 0.56218 \dots + πn

Solution steps

cos (x) sin (x) < 0.451

Use the following identity:

2 cos (x) sin (x) = sin (2 x)

Therefore

cos (x) sin (x) = \frac{sin ( 2 x )}{2}

\frac{sin ( 2 x )}{2} < 0.451

Multiply both sides by

2

\frac{sin ( 2 x )}{2} < 0.451

Multiply both sides by

2

\frac{2 sin ( 2 x )}{2} < 0.451 \cdot 2

Simplify

sin (2 x) < 0.902

sin (2 x) < 0.902

For

sin (x) < a

, if

- 1 < a \leq 1

then

- π - arcsin (a) + 2 πn < x < arcsin (a) + 2 πn

- π - arcsin (0.902) + 2 πn < 2 x < arcsin (0.902) + 2 πn

If

a < u < b

then

a < u and u < b

- π - arcsin (0.902) + 2 πn < 2 x and 2 x < arcsin (0.902) + 2 πn

- π - arcsin (0.902) + 2 πn < 2 x : x > \frac{- π - arcsin ( 0.902 )}{2} + πn

- π - arcsin (0.902) + 2 πn < 2 x

Switch sides

2 x > - π - arcsin (0.902) + 2 πn

Divide both sides by

2

2 x > - π - arcsin (0.902) + 2 πn

Divide both sides by

2

\frac{2 x}{2} > - \frac{π}{2} - \frac{arcsin ( 0.902 )}{2} + \frac{2 πn}{2}

Simplify

x > - \frac{π}{2} - \frac{arcsin ( 0.902 )}{2} + πn

Simplify

- \frac{π}{2} - \frac{arcsin ( 0.902 )}{2} : \frac{- π - arcsin ( 0.902 )}{2}

- \frac{π}{2} - \frac{arcsin ( 0.902 )}{2}

Apply rule

\frac{a}{c} \pm \frac{b}{c} = \frac{a \pm b}{c}

= \frac{- π - arcsin ( 0.902 )}{2}

x > \frac{- π - arcsin ( 0.902 )}{2} + πn

x > \frac{- π - arcsin ( 0.902 )}{2} + πn

2 x < arcsin (0.902) + 2 πn : x < \frac{arcsin ( 0.902 )}{2} + πn

2 x < arcsin (0.902) + 2 πn

Divide both sides by

2

2 x < arcsin (0.902) + 2 πn

Divide both sides by

2

\frac{2 x}{2} < \frac{arcsin ( 0.902 )}{2} + \frac{2 πn}{2}

Simplify

x < \frac{arcsin ( 0.902 )}{2} + πn

x < \frac{arcsin ( 0.902 )}{2} + πn

Combine the intervals

x > \frac{- π - arcsin ( 0.902 )}{2} + πn and x < \frac{arcsin ( 0.902 )}{2} + πn

Merge Overlapping Intervals

\frac{- π - 1.12437 \dots}{2} + πn < x < 0.56218 \dots + πn

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