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Popular Trigonometry >

(sin(2θ))/2 <= 0.451

Solution

\frac{sin ( 2 θ )}{2} \leq 0.451

Solution

\frac{- π - 1.12437 \dots}{2} + πn \leq θ \leq 0.56218 \dots + πn

+2

Interval Notation

[\frac{- π - 1.12437 \dots}{2} + πn, 0.56218 \dots + πn]

Decimal

- 2.13298 \dots + πn \leq θ \leq 0.56218 \dots + πn

Solution steps

\frac{sin ( 2 θ )}{2} \leq 0.451

Multiply both sides by

2

\frac{sin ( 2 θ )}{2} \leq 0.451

Multiply both sides by

2

\frac{2 sin ( 2 θ )}{2} \leq 0.451 \cdot 2

Simplify

sin (2 θ) \leq 0.902

sin (2 θ) \leq 0.902

For

sin (x) \leq a

, if

- 1 < a < 1

then

- π - arcsin (a) + 2 πn \leq x \leq arcsin (a) + 2 πn

- π - arcsin (0.902) + 2 πn \leq 2 θ \leq arcsin (0.902) + 2 πn

If

a \leq u \leq b

then

a \leq u and u \leq b

- π - arcsin (0.902) + 2 πn \leq 2 θ and 2 θ \leq arcsin (0.902) + 2 πn

- π - arcsin (0.902) + 2 πn \leq 2 θ : θ \geq \frac{- π - arcsin ( 0.902 )}{2} + πn

- π - arcsin (0.902) + 2 πn \leq 2 θ

Switch sides

2 θ \geq - π - arcsin (0.902) + 2 πn

Divide both sides by

2

2 θ \geq - π - arcsin (0.902) + 2 πn

Divide both sides by

2

\frac{2 θ}{2} \geq - \frac{π}{2} - \frac{arcsin ( 0.902 )}{2} + \frac{2 πn}{2}

Simplify

θ \geq - \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}

θ \geq \frac{- π - arcsin ( 0.902 )}{2} + πn

θ \geq \frac{- π - arcsin ( 0.902 )}{2} + πn

2 θ \leq arcsin (0.902) + 2 πn : θ \leq \frac{arcsin ( 0.902 )}{2} + πn

2 θ \leq arcsin (0.902) + 2 πn

Divide both sides by

2

2 θ \leq arcsin (0.902) + 2 πn

Divide both sides by

2

\frac{2 θ}{2} \leq \frac{arcsin ( 0.902 )}{2} + \frac{2 πn}{2}

Simplify

θ \leq \frac{arcsin ( 0.902 )}{2} + πn

θ \leq \frac{arcsin ( 0.902 )}{2} + πn

Combine the intervals

θ \geq \frac{- π - arcsin ( 0.902 )}{2} + πn and θ \leq \frac{arcsin ( 0.902 )}{2} + πn

Merge Overlapping Intervals

\frac{- π - 1.12437 \dots}{2} + πn \leq θ \leq 0.56218 \dots + πn

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