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Popular >

2sin^2(x)+5sin(x)>2-2sin^2(x)

Solution

2 sin^{2} (x) + 5 sin (x) > 2 - 2 sin^{2} (x)

Solution

arcsin (\frac{57 - 5}{8}) + 2 πn < x < π - arcsin (\frac{57 - 5}{8}) + 2 πn

Interval Notation

(arcsin (\frac{57 - 5}{8}) + 2 πn, π - arcsin (\frac{57 - 5}{8}) + 2 πn)

Decimal Notation

0.32438 \dots + 2 πn < x < 2.81720 \dots + 2 πn

Solution steps

2 sin^{2} (x) + 5 sin (x) > 2 - 2 sin^{2} (x)

Let:

u = sin (x)

2 u^{2} + 5 u > 2 - 2 u^{2}

2 u^{2} + 5 u > 2 - 2 u^{2} : u < \frac{- 57 - 5}{8} or u > \frac{57 - 5}{8}

2 u^{2} + 5 u > 2 - 2 u^{2}

Rewrite in standard form

2 u^{2} + 5 u > 2 - 2 u^{2}

Add

2 u^{2}

to both sides

2 u^{2} + 5 u + 2 u^{2} > 2 - 2 u^{2} + 2 u^{2}

Simplify

4 u^{2} + 5 u > 2

Subtract

2

from both sides

4 u^{2} + 5 u - 2 > 2 - 2

Simplify

4 u^{2} + 5 u - 2 > 0

4 u^{2} + 5 u - 2 > 0

Complete the square

4 u^{2} + 5 u - 2 : 4 (u + \frac{5}{8})^{2} - \frac{57}{16}

4 u^{2} + 5 u - 2

Write

4 u^{2} + 5 u - 2

in the form:

x^{2} + 2 a x + a^{2}

Factor out

4

4 (u^{2} + \frac{5 u}{4} - \frac{1}{2})

2 a = \frac{5}{4} : a = \frac{5}{8}

2 a = \frac{5}{4}

Divide both sides by

2

2 a = \frac{5}{4}

Divide both sides by

2

\frac{2 a}{2} = \frac{\frac{5}{4}}{2}

Simplify

\frac{2 a}{2} = \frac{\frac{5}{4}}{2}

Simplify

\frac{2 a}{2} : a

\frac{2 a}{2}

Divide the numbers:

\frac{2}{2} = 1

= a

Simplify

\frac{\frac{5}{4}}{2} : \frac{5}{8}

\frac{\frac{5}{4}}{2}

Apply the fraction rule:

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

= \frac{5}{4 \cdot 2}

Multiply the numbers:

4 \cdot 2 = 8

= \frac{5}{8}

a = \frac{5}{8}

a = \frac{5}{8}

a = \frac{5}{8}

Add and subtract

(\frac{5}{8})^{2}

4 (u^{2} + \frac{5 u}{4} - \frac{1}{2} + (\frac{5}{8})^{2} - (\frac{5}{8})^{2})

x^{2} + 2 a x + a^{2} = (x + a)^{2}

u^{2} + \frac{5}{4} u + (\frac{5}{8})^{2} = (u + \frac{5}{8})^{2}

4 ((u + \frac{5}{8})^{2} - \frac{1}{2} - (\frac{5}{8})^{2})

Simplify

4 (u + \frac{5}{8})^{2} - \frac{57}{16}

4 (u + \frac{5}{8})^{2} - \frac{57}{16} > 0

Move

\frac{57}{16}

to the right side

4 (u + \frac{5}{8})^{2} - \frac{57}{16} > 0

Add

\frac{57}{16}

to both sides

4 (u + \frac{5}{8})^{2} - \frac{57}{16} + \frac{57}{16} > 0 + \frac{57}{16}

Simplify

4 (u + \frac{5}{8})^{2} > \frac{57}{16}

4 (u + \frac{5}{8})^{2} > \frac{57}{16}

Divide both sides by

4

4 (u + \frac{5}{8})^{2} > \frac{57}{16}

Divide both sides by

4

\frac{4 ( u + \frac{5}{8} ) ^{2}}{4} > \frac{\frac{57}{16}}{4}

Simplify

\frac{4 ( u + \frac{5}{8} ) ^{2}}{4} > \frac{\frac{57}{16}}{4}

Simplify

\frac{4 ( u + \frac{5}{8} ) ^{2}}{4} : (u + \frac{5}{8})^{2}

\frac{4 ( u + \frac{5}{8} ) ^{2}}{4}

Divide the numbers:

\frac{4}{4} = 1

= (u + \frac{5}{8})^{2}

Simplify

\frac{\frac{57}{16}}{4} : \frac{57}{64}

\frac{\frac{57}{16}}{4}

Apply the fraction rule:

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

= \frac{57}{16 \cdot 4}

Multiply the numbers:

16 \cdot 4 = 64

= \frac{57}{64}

(u + \frac{5}{8})^{2} > \frac{57}{64}

(u + \frac{5}{8})^{2} > \frac{57}{64}

(u + \frac{5}{8})^{2} > \frac{57}{64}

For

u^{n} > a

, if

n

is even then

u + \frac{5}{8} < - \frac{57}{64} or u + \frac{5}{8} > \frac{57}{64}

u + \frac{5}{8} < - \frac{57}{64} : u < \frac{- 57 - 5}{8}

u + \frac{5}{8} < - \frac{57}{64}

Simplify

\frac{57}{64} : \frac{57}{8}

\frac{57}{64}

Apply radical rule:

assuming

a \geq 0, b \geq 0

= \frac{57}{64}

64 = 8

64

Factor the number:

64 = 8^{2}

= 8^{2}

Apply radical rule:

8^{2} = 8

= 8

= \frac{57}{8}

u + \frac{5}{8} < - \frac{57}{8}

Move

\frac{5}{8}

to the right side

u + \frac{5}{8} < - \frac{57}{8}

Subtract

\frac{5}{8}

from both sides

u + \frac{5}{8} - \frac{5}{8} < - \frac{57}{8} - \frac{5}{8}

Simplify

u + \frac{5}{8} - \frac{5}{8} < - \frac{57}{8} - \frac{5}{8}

Simplify

u + \frac{5}{8} - \frac{5}{8} : u

u + \frac{5}{8} - \frac{5}{8}

Add similar elements:

\frac{5}{8} - \frac{5}{8} < 0

= u

Simplify

- \frac{57}{8} - \frac{5}{8} : \frac{- 57 - 5}{8}

- \frac{57}{8} - \frac{5}{8}

Apply rule

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

= \frac{- 57 - 5}{8}

u < \frac{- 57 - 5}{8}

u < \frac{- 57 - 5}{8}

u < \frac{- 57 - 5}{8}

u + \frac{5}{8} > \frac{57}{64} : u > \frac{57 - 5}{8}

u + \frac{5}{8} > \frac{57}{64}

Apply radical rule:

assuming

a \geq 0, b \geq 0

u + \frac{5}{8} > \frac{57}{64}

64 = 8

64

Factor the number:

64 = 8^{2}

= 8^{2}

Apply radical rule:

8^{2} = 8

= 8

u + \frac{5}{8} > \frac{57}{8}

Move

\frac{5}{8}

to the right side

u + \frac{5}{8} > \frac{57}{8}

Subtract

\frac{5}{8}

from both sides

u + \frac{5}{8} - \frac{5}{8} > \frac{57}{8} - \frac{5}{8}

Simplify

u + \frac{5}{8} - \frac{5}{8} > \frac{57}{8} - \frac{5}{8}

Simplify

u + \frac{5}{8} - \frac{5}{8} : u

u + \frac{5}{8} - \frac{5}{8}

Add similar elements:

\frac{5}{8} - \frac{5}{8} > 0

= u

Simplify

\frac{57}{8} - \frac{5}{8} : \frac{57 - 5}{8}

\frac{57}{8} - \frac{5}{8}

Apply rule

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

= \frac{57 - 5}{8}

u > \frac{57 - 5}{8}

u > \frac{57 - 5}{8}

u > \frac{57 - 5}{8}

Combine the intervals

u < \frac{- 57 - 5}{8} or u > \frac{57 - 5}{8}

u < \frac{- 57 - 5}{8} or u > \frac{57 - 5}{8}

Substitute back

u = sin (x)

sin (x) < \frac{- 57 - 5}{8} or sin (x) > \frac{57 - 5}{8}

sin (x) < \frac{- 57 - 5}{8} :

False for all

x \in R

sin (x) < \frac{- 57 - 5}{8}

Range of

sin (x) : - 1 \leq sin (x) \leq 1

Function range definition

The range of the basic

sin

function is

- 1 \leq sin (x) \leq 1

- 1 \leq sin (x) \leq 1

sin (x) < \frac{- 57 - 5}{8} and - 1 \leq sin (x) \leq 1 :

False

Let

y = sin (x)

Combine the intervals

y < \frac{- 57 - 5}{8} and - 1 \leq y \leq 1

Merge Overlapping Intervals

y < \frac{- 57 - 5}{8} and - 1 \leq y \leq 1

The intersection of two intervals is the set of numbers which are in both intervals

y < \frac{- 57 - 5}{8}

and

- 1 \leq y \leq 1

False for all y \in R

False for all y \in R

No Solution for x \in R

False for all x \in R

sin (x) > \frac{57 - 5}{8} : arcsin (\frac{57 - 5}{8}) + 2 πn < x < π - arcsin (\frac{57 - 5}{8}) + 2 πn

sin (x) > \frac{57 - 5}{8}

For

sin (x) > a

, if

- 1 \leq a < 1

then

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

arcsin (\frac{57 - 5}{8}) + 2 πn < x < π - arcsin (\frac{57 - 5}{8}) + 2 πn

Combine the intervals

False for all x \in R or arcsin (\frac{57 - 5}{8}) + 2 πn < x < π - arcsin (\frac{57 - 5}{8}) + 2 πn

Merge Overlapping Intervals

arcsin (\frac{57 - 5}{8}) + 2 πn < x < π - arcsin (\frac{57 - 5}{8}) + 2 πn

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