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Popular Trigonometry >

(sin(x)-1/2)(sin(x)-7/2)<= 0

Solution

(sin (x) - \frac{1}{2}) (sin (x) - \frac{7}{2}) \leq 0

Solution

\frac{π}{6} + 2 πn \leq x \leq \frac{5 π}{6} + 2 πn

Interval Notation

[\frac{π}{6} + 2 πn, \frac{5 π}{6} + 2 πn]

Decimal

0.52359 \dots + 2 πn \leq x \leq 2.61799 \dots + 2 πn

Solution steps

(sin (x) - \frac{1}{2}) (sin (x) - \frac{7}{2}) \leq 0

Let:

u = sin (x)

(u - \frac{1}{2}) (u - \frac{7}{2}) \leq 0

(u - \frac{1}{2}) (u - \frac{7}{2}) \leq 0 : \frac{1}{2} \leq u \leq \frac{7}{2}

(u - \frac{1}{2}) (u - \frac{7}{2}) \leq 0

Rewrite in standard form

(u - \frac{1}{2}) (u - \frac{7}{2}) \leq 0

Expand

(u - \frac{1}{2}) (u - \frac{7}{2}) : u^{2} - 4 u + \frac{7}{4}

(u - \frac{1}{2}) (u - \frac{7}{2})

Apply FOIL method:

(a + b) (c + d) = a c + a d + b c + b d

a = u, b = - \frac{1}{2}, c = u, d = - \frac{7}{2}

= uu + u (- \frac{7}{2}) + (- \frac{1}{2}) u + (- \frac{1}{2}) (- \frac{7}{2})

Apply minus-plus rules

+ (- a) = - a, (- a) (- b) = ab

= uu - \frac{7}{2} u - \frac{1}{2} u + \frac{1}{2} \cdot \frac{7}{2}

Simplify

uu - \frac{7}{2} u - \frac{1}{2} u + \frac{1}{2} \cdot \frac{7}{2} : u^{2} - 4 u + \frac{7}{4}

uu - \frac{7}{2} u - \frac{1}{2} u + \frac{1}{2} \cdot \frac{7}{2}

Add similar elements:

- \frac{7}{2} u - \frac{1}{2} u = - 4 u

- \frac{7}{2} u - \frac{1}{2} u

Factor out common term

u

= u (- \frac{7}{2} - \frac{1}{2})

- \frac{7}{2} - \frac{1}{2} = - 4

- \frac{7}{2} - \frac{1}{2}

Apply rule

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

= \frac{- 7 - 1}{2}

Subtract the numbers:

- 7 - 1 = - 8

= \frac{- 8}{2}

Apply the fraction rule:

\frac{- a}{b} = - \frac{a}{b}

= - \frac{8}{2}

Divide the numbers:

\frac{8}{2} = 4

= - 4

= - 4 u

= uu - 4 u + \frac{1}{2} \cdot \frac{7}{2}

uu = u^{2}

uu

Apply exponent rule:

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

uu = u^{1 + 1}

= u^{1 + 1}

Add the numbers:

1 + 1 = 2

= u^{2}

\frac{1}{2} \cdot \frac{7}{2} = \frac{7}{4}

\frac{1}{2} \cdot \frac{7}{2}

Multiply fractions:

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

= \frac{1 \cdot 7}{2 \cdot 2}

Refine

= \frac{7}{4}

= u^{2} - 4 u + \frac{7}{4}

= u^{2} - 4 u + \frac{7}{4}

u^{2} - 4 u + \frac{7}{4} \leq 0

Multiply both sides by

4

u^{2} \cdot 4 - 4 u \cdot 4 + \frac{7}{4} \cdot 4 \leq 0 \cdot 4

4 u^{2} - 16 u + 7 \leq 0

4 u^{2} - 16 u + 7 \leq 0

Factor

4 u^{2} - 16 u + 7 : (2 u - 1) (2 u - 7)

4 u^{2} - 16 u + 7

Break the expression into groups

4 u^{2} - 16 u + 7

Definition

Factors of

28 : 1, 2, 4, 7, 14, 28

28

Divisors (Factors)

Find the Prime factors of

28 : 2, 2, 7

28

28

divides by

228 = 14 \cdot 2

= 2 \cdot 14

14

divides by

214 = 7 \cdot 2

= 2 \cdot 2 \cdot 7

2, 7

are all prime numbers, therefore no further factorization is possible

= 2 \cdot 2 \cdot 7

Multiply the prime factors of

28 : 4, 14

2 \cdot 2 = 4

2 \cdot 7 = 14

4, 14

4, 14

Add the prime factors:

2, 7

Add 1 and the number

28

itself

1, 28

The factors of

28

1, 2, 4, 7, 14, 28

Negative factors of

28 : - 1, - 2, - 4, - 7, - 14, - 28

Multiply the factors by

- 1

to get the negative factors

- 1, - 2, - 4, - 7, - 14, - 28

For every two factors such that

u * v = 28,

check if

u + v = - 16

Check

u = 1, v = 28 : u * v = 28, u + v = 29 \Rightarrow

FalseCheck

u = 2, v = 14 : u * v = 28, u + v = 16 \Rightarrow

False

u = - 2, v = - 14

Group into

(a x^{2} + ux) + (vx + c)

(4 u^{2} - 2 u) + (- 14 u + 7)

= (4 u^{2} - 2 u) + (- 14 u + 7)

Factor out

2 u

from

4 u^{2} - 2 u : 2 u (2 u - 1)

4 u^{2} - 2 u

Apply exponent rule:

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

u^{2} = uu

= 4 uu - 2 u

Rewrite

4

2 \cdot 2

= 2 \cdot 2 uu - 2 u

Factor out common term

2 u

= 2 u (2 u - 1)

Factor out

- 7

from

- 14 u + 7 : - 7 (2 u - 1)

- 14 u + 7

Rewrite

14

7 \cdot 2

= - 7 \cdot 2 u + 7

Factor out common term

- 7

= - 7 (2 u - 1)

= 2 u (2 u - 1) - 7 (2 u - 1)

Factor out common term

2 u - 1

= (2 u - 1) (2 u - 7)

(2 u - 1) (2 u - 7) \leq 0

Identify the intervals

Find the signs of the factors of

(2 u - 1) (2 u - 7)

Find the signs of

2 u - 1

2 u - 1 = 0 : u = \frac{1}{2}

2 u - 1 = 0

Move

1

to the right side

2 u - 1 = 0

Add

1

to both sides

2 u - 1 + 1 = 0 + 1

Simplify

2 u = 1

2 u = 1

Divide both sides by

2

2 u = 1

Divide both sides by

2

\frac{2 u}{2} = \frac{1}{2}

Simplify

u = \frac{1}{2}

u = \frac{1}{2}

2 u - 1 < 0 : u < \frac{1}{2}

2 u - 1 < 0

Move

1

to the right side

2 u - 1 < 0

Add

1

to both sides

2 u - 1 + 1 < 0 + 1

Simplify

2 u < 1

2 u < 1

Divide both sides by

2

2 u < 1

Divide both sides by

2

\frac{2 u}{2} < \frac{1}{2}

Simplify

u < \frac{1}{2}

u < \frac{1}{2}

2 u - 1 > 0 : u > \frac{1}{2}

2 u - 1 > 0

Move

1

to the right side

2 u - 1 > 0

Add

1

to both sides

2 u - 1 + 1 > 0 + 1

Simplify

2 u > 1

2 u > 1

Divide both sides by

2

2 u > 1

Divide both sides by

2

\frac{2 u}{2} > \frac{1}{2}

Simplify

u > \frac{1}{2}

u > \frac{1}{2}

Find the signs of

2 u - 7

2 u - 7 = 0 : u = \frac{7}{2}

2 u - 7 = 0

Move

7

to the right side

2 u - 7 = 0

Add

7

to both sides

2 u - 7 + 7 = 0 + 7

Simplify

2 u = 7

2 u = 7

Divide both sides by

2

2 u = 7

Divide both sides by

2

\frac{2 u}{2} = \frac{7}{2}

Simplify

u = \frac{7}{2}

u = \frac{7}{2}

2 u - 7 < 0 : u < \frac{7}{2}

2 u - 7 < 0

Move

7

to the right side

2 u - 7 < 0

Add

7

to both sides

2 u - 7 + 7 < 0 + 7

Simplify

2 u < 7

2 u < 7

Divide both sides by

2

2 u < 7

Divide both sides by

2

\frac{2 u}{2} < \frac{7}{2}

Simplify

u < \frac{7}{2}

u < \frac{7}{2}

2 u - 7 > 0 : u > \frac{7}{2}

2 u - 7 > 0

Move

7

to the right side

2 u - 7 > 0

Add

7

to both sides

2 u - 7 + 7 > 0 + 7

Simplify

2 u > 7

2 u > 7

Divide both sides by

2

2 u > 7

Divide both sides by

2

\frac{2 u}{2} > \frac{7}{2}

Simplify

u > \frac{7}{2}

u > \frac{7}{2}

Summarize in a table:

2 u - 1 2 u - 7 (2 u - 1) (2 u - 7) u < \frac{1}{2} - - + u = \frac{1}{2} 0 - 0 \frac{1}{2} < u < \frac{7}{2} + - - u = \frac{7}{2} + 00 u > \frac{7}{2} + + +

Identify the intervals that satisfy the required condition:

\leq 0

u = \frac{1}{2} or \frac{1}{2} < u < \frac{7}{2} or u = \frac{7}{2}

Merge Overlapping Intervals

\frac{1}{2} \leq u < \frac{7}{2} or u = \frac{7}{2}

The union of two intervals is the set of numbers which are in either interval

u = \frac{1}{2}

\frac{1}{2} < u < \frac{7}{2}

\frac{1}{2} \leq u < \frac{7}{2}

The union of two intervals is the set of numbers which are in either interval

\frac{1}{2} \leq u < \frac{7}{2}

u = \frac{7}{2}

\frac{1}{2} \leq u \leq \frac{7}{2}

\frac{1}{2} \leq u \leq \frac{7}{2}

\frac{1}{2} \leq u \leq \frac{7}{2}

\frac{1}{2} \leq u \leq \frac{7}{2}

Substitute back

u = sin (x)

\frac{1}{2} \leq sin (x) \leq \frac{7}{2}

a \leq u \leq b

then

a \leq u and u \leq b

\frac{1}{2} \leq sin (x) and sin (x) \leq \frac{7}{2}

\frac{1}{2} \leq sin (x) : \frac{π}{6} + 2 πn \leq x \leq \frac{5 π}{6} + 2 πn

\frac{1}{2} \leq sin (x)

Switch sides

sin (x) \geq \frac{1}{2}

For

sin (x) \geq a

, if

- 1 < a < 1

then

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

arcsin (\frac{1}{2}) + 2 πn \leq x \leq π - arcsin (\frac{1}{2}) + 2 πn

Simplify

arcsin (\frac{1}{2}) : \frac{π}{6}

arcsin (\frac{1}{2})

Use the following trivial identity:

arcsin (\frac{1}{2}) = \frac{π}{6}

x 0 \frac{1}{2} \frac{2}{2} \frac{3}{2} 1 arcsin (x) 0 \frac{π}{6} \frac{π}{4} \frac{π}{3} \frac{π}{2} arcsin (x) 0^{\circ} 3 0^{\circ} 4 5^{\circ} 6 0^{\circ} 9 0^{\circ}

= \frac{π}{6}

Simplify

π - arcsin (\frac{1}{2}) : \frac{5 π}{6}

π - arcsin (\frac{1}{2})

Use the following trivial identity:

arcsin (\frac{1}{2}) = \frac{π}{6}

x 0 \frac{1}{2} \frac{2}{2} \frac{3}{2} 1 arcsin (x) 0 \frac{π}{6} \frac{π}{4} \frac{π}{3} \frac{π}{2} arcsin (x) 0^{\circ} 3 0^{\circ} 4 5^{\circ} 6 0^{\circ} 9 0^{\circ}

= π - \frac{π}{6}

Simplify

π - \frac{π}{6}

Convert element to fraction:

π = \frac{π 6}{6}

= \frac{π 6}{6} - \frac{π}{6}

Since the denominators are equal, combine the fractions:

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

= \frac{π 6 - π}{6}

Add similar elements:

6 π - π = 5 π

= \frac{5 π}{6}

= \frac{5 π}{6}

\frac{π}{6} + 2 πn \leq x \leq \frac{5 π}{6} + 2 πn

sin (x) \leq \frac{7}{2} :

True for all

x \in R

sin (x) \leq \frac{7}{2}

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) \leq \frac{7}{2} and - 1 \leq sin (x) \leq 1 : - 1 \leq sin (x) \leq 1

Let

y = sin (x)

Combine the intervals

y \leq \frac{7}{2} and - 1 \leq y \leq 1

Merge Overlapping Intervals

y \leq \frac{7}{2} and - 1 \leq y \leq 1

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

y \leq \frac{7}{2}

and

- 1 \leq y \leq 1

- 1 \leq y \leq 1

- 1 \leq y \leq 1

True for all x

True for all x \in R

Combine the intervals

\frac{π}{6} + 2 πn \leq x \leq \frac{5 π}{6} + 2 πn and True for all x \in R

Merge Overlapping Intervals

\frac{π}{6} + 2 πn \leq x \leq \frac{5 π}{6} + 2 πn

Popular Examples

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tan(1/x)<= tan(1/(x+1))

cos(x^4)+sin(x^4)>= 0.5

1-tan(x)<2

2sin(x)-1>=-3