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Popular Trigonometry >

cos(2θ)-3sin(θ)-2>0

Solution

cos (2 θ) - 3 sin (θ) - 2 > 0

Solution

\frac{7 π}{6} + 2 πn < θ < \frac{3 π}{2} + 2 πn or \frac{3 π}{2} + 2 πn < θ < \frac{11 π}{6} + 2 πn

Interval Notation

(\frac{7 π}{6} + 2 πn, \frac{3 π}{2} + 2 πn) \cup (\frac{3 π}{2} + 2 πn, \frac{11 π}{6} + 2 πn)

Decimal

3.66519 \dots + 2 πn < θ < 4.71238 \dots + 2 πn or 4.71238 \dots + 2 πn < θ < 5.75958 \dots + 2 πn

Solution steps

cos (2 θ) - 3 sin (θ) - 2 > 0

Use the following identity:

cos (2 x) = 1 - 2 sin^{2} (x)

- 2 + 1 - 2 sin^{2} (θ) - 3 sin (θ) > 0

Simplify

- 2 sin^{2} (θ) - 3 sin (θ) - 1 > 0

Let:

u = sin (θ)

- 2 u^{2} - 3 u - 1 > 0

- 2 u^{2} - 3 u - 1 > 0 : - 1 < u < - \frac{1}{2}

- 2 u^{2} - 3 u - 1 > 0

Factor

- 2 u^{2} - 3 u - 1 : - (2 u + 1) (u + 1)

- 2 u^{2} - 3 u - 1

Factor out common term

- 1

= - (2 u^{2} + 3 u + 1)

Factor

2 u^{2} + 3 u + 1 : (2 u + 1) (u + 1)

2 u^{2} + 3 u + 1

Break the expression into groups

2 u^{2} + 3 u + 1

Definition

Factors of

2 : 1, 2

2

Divisors (Factors)

Find the Prime factors of

2 : 2

2

2

is a prime number, therefore no factorization is possible

= 2

Add 1

1

The factors of

2

1, 2

For every two factors such that

u * v = 2,

check if

u + v = 3

Check

u = 1, v = 2 : u * v = 2, u + v = 3 \Rightarrow

True

u = 1, v = 2

Group into

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

(2 u^{2} + u) + (2 u + 1)

= (2 u^{2} + u) + (2 u + 1)

Factor out

u

from

2 u^{2} + u : u (2 u + 1)

2 u^{2} + u

Apply exponent rule:

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

u^{2} = uu

= 2 uu + u

Factor out common term

u

= u (2 u + 1)

= u (2 u + 1) + (2 u + 1)

Factor out common term

2 u + 1

= (2 u + 1) (u + 1)

= - (2 u + 1) (u + 1)

- (2 u + 1) (u + 1) > 0

Multiply both sides by

- 1

(reverse the inequality)

(- (2 u + 1) (u + 1)) (- 1) < 0 \cdot (- 1)

Simplify

(2 u + 1) (u + 1) < 0

Identify the intervals

Find the signs of the factors of

(2 u + 1) (u + 1)

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

Subtract

1

from 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

Subtract

1

from 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

Subtract

1

from 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

u + 1

u + 1 = 0 : u = - 1

u + 1 = 0

Move

1

to the right side

u + 1 = 0

Subtract

1

from both sides

u + 1 - 1 = 0 - 1

Simplify

u = - 1

u = - 1

u + 1 < 0 : u < - 1

u + 1 < 0

Move

1

to the right side

u + 1 < 0

Subtract

1

from both sides

u + 1 - 1 < 0 - 1

Simplify

u < - 1

u < - 1

u + 1 > 0 : u > - 1

u + 1 > 0

Move

1

to the right side

u + 1 > 0

Subtract

1

from both sides

u + 1 - 1 > 0 - 1

Simplify

u > - 1

u > - 1

Summarize in a table:

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

Identify the intervals that satisfy the required condition:

< 0

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

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

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

Substitute back

u = sin (θ)

- 1 < sin (θ) < - \frac{1}{2}

a < u < b

then

a < u and u < b

- 1 < sin (θ) and sin (θ) < - \frac{1}{2}

- 1 < sin (θ) : - \frac{π}{2} + 2 πn < θ < \frac{3 π}{2} + 2 πn

- 1 < sin (θ)

Switch sides

sin (θ) > - 1

For

sin (x) > a

, if

- 1 \leq a < 1

then

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

arcsin (- 1) + 2 πn < θ < π - arcsin (- 1) + 2 πn

Simplify

arcsin (- 1) : - \frac{π}{2}

arcsin (- 1)

Use the following property:

arcsin (- x) = - arcsin (x)

arcsin (- 1) = - arcsin (1)

= - arcsin (1)

Use the following trivial identity:

arcsin (1) = \frac{π}{2}

arcsin (1)

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}{2}

= - \frac{π}{2}

Simplify

π - arcsin (- 1) : \frac{3 π}{2}

π - arcsin (- 1)

arcsin (- 1) = - \frac{π}{2}

arcsin (- 1)

Use the following property:

arcsin (- x) = - arcsin (x)

arcsin (- 1) = - arcsin (1)

= - arcsin (1)

Use the following trivial identity:

arcsin (1) = \frac{π}{2}

arcsin (1)

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}{2}

= - \frac{π}{2}

= π - (- \frac{π}{2})

Simplify

π - (- \frac{π}{2})

Apply rule

- (- a) = a

= π + \frac{π}{2}

Convert element to fraction:

π = \frac{π 2}{2}

= \frac{π 2}{2} + \frac{π}{2}

Since the denominators are equal, combine the fractions:

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

= \frac{π 2 + π}{2}

Add similar elements:

2 π + π = 3 π

= \frac{3 π}{2}

= \frac{3 π}{2}

- \frac{π}{2} + 2 πn < θ < \frac{3 π}{2} + 2 πn

sin (θ) < - \frac{1}{2} : - \frac{5 π}{6} + 2 πn < θ < - \frac{π}{6} + 2 πn

sin (θ) < - \frac{1}{2}

For

sin (x) < a

, if

- 1 < a \leq 1

then

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

- π - arcsin (- \frac{1}{2}) + 2 πn < θ < arcsin (- \frac{1}{2}) + 2 πn

Simplify

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

- π - arcsin (- \frac{1}{2})

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

arcsin (- \frac{1}{2})

Use the following property:

arcsin (- x) = - arcsin (x)

arcsin (- \frac{1}{2}) = - arcsin (\frac{1}{2})

= - arcsin (\frac{1}{2})

Use the following trivial identity:

arcsin (\frac{1}{2}) = \frac{π}{6}

arcsin (\frac{1}{2})

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}

= - \frac{π}{6}

= - π - (- \frac{π}{6})

Simplify

- π - (- \frac{π}{6})

Apply rule

- (- a) = a

= - 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}

Apply the fraction rule:

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

= - \frac{5 π}{6}

= - \frac{5 π}{6}

Simplify

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

arcsin (- \frac{1}{2})

Use the following property:

arcsin (- x) = - arcsin (x)

arcsin (- \frac{1}{2}) = - arcsin (\frac{1}{2})

= - arcsin (\frac{1}{2})

Use the following trivial identity:

arcsin (\frac{1}{2}) = \frac{π}{6}

arcsin (\frac{1}{2})

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}

= - \frac{π}{6}

- \frac{5 π}{6} + 2 πn < θ < - \frac{π}{6} + 2 πn

Combine the intervals

- \frac{π}{2} + 2 πn < θ < \frac{3 π}{2} + 2 πn and - \frac{5 π}{6} + 2 πn < θ < - \frac{π}{6} + 2 πn

Merge Overlapping Intervals

\frac{7 π}{6} + 2 πn < θ < \frac{3 π}{2} + 2 πn or \frac{3 π}{2} + 2 πn < θ < \frac{11 π}{6} + 2 πn

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