English

Español

Português

Français

Deutsch

Italiano

Русский

中文(简体)

한국어

日本語

Tiếng Việt

עברית

العربية

Popular >

sin^2(x)> 15/40

Solution

sin^{2} (x) > \frac{15}{40}

Solution

arcsin (\frac{6}{4}) + 2 πn < x < π - arcsin (\frac{6}{4}) + 2 πn or - π + arcsin (\frac{6}{4}) + 2 πn < x < - arcsin (\frac{6}{4}) + 2 πn

Interval Notation

(arcsin (\frac{6}{4}) + 2 πn, π - arcsin (\frac{6}{4}) + 2 πn) \cup (- π + arcsin (\frac{6}{4}) + 2 πn, - arcsin (\frac{6}{4}) + 2 πn)

Decimal Notation

0.65905 \dots + 2 πn < x < 2.48253 \dots + 2 πn or - 2.48253 \dots + 2 πn < x < - 0.65905 \dots + 2 πn

Solution steps

sin^{2} (x) > \frac{15}{40}

For

u^{n} > a

, if

n

is even then

sin (x) < - \frac{15}{40} or sin (x) > \frac{15}{40}

\frac{15}{40} = \frac{3}{2 2}

\frac{15}{40}

\frac{15}{40} = \frac{3}{8}

\frac{15}{40}

Cancel the common factor:

5

= \frac{3}{8}

= \frac{3}{8}

Apply radical rule:

assuming

a \geq 0, b \geq 0

= \frac{3}{8}

8 = 22

8

Prime factorization of

8 : 2^{3}

8

8

divides by

28 = 4 \cdot 2

= 2 \cdot 4

4

divides by

24 = 2 \cdot 2

= 2 \cdot 2 \cdot 2

2

is a prime number, therefore no further factorization is possible

= 2 \cdot 2 \cdot 2

= 2^{3}

= 2^{3}

Apply exponent rule:

a^{b + c} = a^{b} \cdot a^{c}

= 2^{2} \cdot 2

Apply radical rule:

= 2 2^{2}

Apply radical rule:

2^{2} = 2

= 22

= \frac{3}{2 2}

sin (x) < - \frac{3}{2 2} or sin (x) > \frac{3}{2 2}

sin (x) < - \frac{3}{2 2} : - π + arcsin (\frac{6}{4}) + 2 πn < x < - arcsin (\frac{6}{4}) + 2 πn

sin (x) < - \frac{3}{2 2}

Simplify

- \frac{3}{2 2} : - \frac{6}{4}

- \frac{3}{2 2}

Multiply by the conjugate

\frac{2}{2}

= - \frac{3 2}{2 2 2}

32 = 6

32

Apply radical rule:

a b = a \cdot b

32 = 3 \cdot 2

= 3 \cdot 2

Multiply the numbers:

3 \cdot 2 = 6

= 6

222 = 4

222

Apply exponent rule:

a^{b} \cdot a^{c} = a^{b + c}

222 = 2 \cdot 2^{\frac{1}{2}} \cdot 2^{\frac{1}{2}} = 2^{1 + \frac{1}{2} + \frac{1}{2}}

= 2^{1 + \frac{1}{2} + \frac{1}{2}}

Add similar elements:

\frac{1}{2} + \frac{1}{2} = 2 \cdot \frac{1}{2}

= 2^{1 + 2 \cdot \frac{1}{2}}

2 \cdot \frac{1}{2} = 1

2 \cdot \frac{1}{2}

Multiply fractions:

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

= \frac{1 \cdot 2}{2}

Cancel the common factor:

2

= 1

= 2^{1 + 1}

Add the numbers:

1 + 1 = 2

= 2^{2}

2^{2} = 4

= 4

= - \frac{6}{4}

sin (x) < - \frac{6}{4}

For

sin (x) < a

, if

- 1 < a \leq 1

then

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

- π - arcsin (- \frac{6}{4}) + 2 πn < x < arcsin (- \frac{6}{4}) + 2 πn

Simplify

- π - arcsin (- \frac{6}{4}) : - π + arcsin (\frac{6}{4})

- π - arcsin (- \frac{6}{4})

Use the following property:

arcsin (- x) = - arcsin (x)

arcsin (- \frac{6}{4}) = - arcsin (\frac{6}{4})

= - π - (- arcsin (\frac{6}{4}))

Apply rule

- (- a) = a

= - π + arcsin (\frac{6}{4})

Simplify

arcsin (- \frac{6}{4}) : - arcsin (\frac{6}{4})

arcsin (- \frac{6}{4})

Use the following property:

arcsin (- x) = - arcsin (x)

arcsin (- \frac{6}{4}) = - arcsin (\frac{6}{4})

= - arcsin (\frac{6}{4})

- π + arcsin (\frac{6}{4}) + 2 πn < x < - arcsin (\frac{6}{4}) + 2 πn

sin (x) > \frac{3}{2 2} : arcsin (\frac{6}{4}) + 2 πn < x < π - arcsin (\frac{6}{4}) + 2 πn

sin (x) > \frac{3}{2 2}

Simplify

\frac{3}{2 2} : \frac{6}{4}

\frac{3}{2 2}

Multiply by the conjugate

\frac{2}{2}

= \frac{3 2}{2 2 2}

32 = 6

32

Apply radical rule:

a b = a \cdot b

32 = 3 \cdot 2

= 3 \cdot 2

Multiply the numbers:

3 \cdot 2 = 6

= 6

222 = 4

222

Apply exponent rule:

a^{b} \cdot a^{c} = a^{b + c}

222 = 2 \cdot 2^{\frac{1}{2}} \cdot 2^{\frac{1}{2}} = 2^{1 + \frac{1}{2} + \frac{1}{2}}

= 2^{1 + \frac{1}{2} + \frac{1}{2}}

Add similar elements:

\frac{1}{2} + \frac{1}{2} = 2 \cdot \frac{1}{2}

= 2^{1 + 2 \cdot \frac{1}{2}}

2 \cdot \frac{1}{2} = 1

2 \cdot \frac{1}{2}

Multiply fractions:

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

= \frac{1 \cdot 2}{2}

Cancel the common factor:

2

= 1

= 2^{1 + 1}

Add the numbers:

1 + 1 = 2

= 2^{2}

2^{2} = 4

= 4

= \frac{6}{4}

sin (x) > \frac{6}{4}

For

sin (x) > a

, if

- 1 \leq a < 1

then

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

arcsin (\frac{6}{4}) + 2 πn < x < π - arcsin (\frac{6}{4}) + 2 πn

Combine the intervals

- π + arcsin (\frac{6}{4}) + 2 πn < x < - arcsin (\frac{6}{4}) + 2 πn or arcsin (\frac{6}{4}) + 2 πn < x < π - arcsin (\frac{6}{4}) + 2 πn

Merge Overlapping Intervals

arcsin (\frac{6}{4}) + 2 πn < x < π - arcsin (\frac{6}{4}) + 2 πn or - π + arcsin (\frac{6}{4}) + 2 πn < x < - arcsin (\frac{6}{4}) + 2 πn

Popular Examples

(1+2sin(x))/((cos(2x)))>0

\frac{1 + 2 sin ( x )}{( cos ( 2 x ) )} > 0

(cos^2(x)-1/2)/(tan(x)-sqrt(3))<0

\frac{cos ^{2} ( x ) - \frac{1}{2}}{tan ( x ) - 3} < 0

tan(θ)>0,cot(θ)>0

tan (θ) > 0, cot (θ) > 0

6sin(θ)>= 0

6 sin (θ) \geq 0

tan(x)>\sqrt[4]{5},-pi<= x<= pi