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Popular >

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

Solution

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

Solution

2 πn < x < \frac{π}{6} + 2 πn or \frac{5 π}{6} + 2 πn < x < π + 2 πn

Interval Notation

(2 πn, \frac{π}{6} + 2 πn) \cup (\frac{5 π}{6} + 2 πn, π + 2 πn)

Decimal Notation

2 πn < x < 0.52359 \dots + 2 πn or 2.61799 \dots + 2 πn < x < 3.14159 \dots + 2 πn

Solution steps

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

Use the following identity:

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

1 - 2 sin^{2} (x) + sin (x) > 1

Let:

u = sin (x)

1 - 2 u^{2} + u > 1

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

1 - 2 u^{2} + u > 1

Rewrite in standard form

1 - 2 u^{2} + u > 1

Subtract

1

from both sides

1 - 2 u^{2} + u - 1 > 1 - 1

Simplify

- 2 u^{2} + u > 0

- 2 u^{2} + u > 0

Factor

- 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) > 0

Multiply both sides by

- 1

(reverse the inequality)

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

Simplify

u (2 u - 1) < 0

Identify the intervals

Find the signs of the factors of

u (2 u - 1)

Find the signs of

u

u = 0

u < 0

u > 0

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}

Summarize in a table:

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

Identify the intervals that satisfy the required condition:

< 0

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

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

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

Substitute back

u = sin (x)

0 < sin (x) < \frac{1}{2}

a < u < b

then

a < u and u < b

0 < sin (x) and sin (x) < \frac{1}{2}

0 < sin (x) : 2 πn < x < π + 2 πn

0 < sin (x)

Switch sides

sin (x) > 0

For

sin (x) > a

, if

- 1 \leq a < 1

then

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

arcsin (0) + 2 πn < x < π - arcsin (0) + 2 πn

Simplify

arcsin (0) : 0

arcsin (0)

Use the following trivial identity:

arcsin (0) = 0

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}

= 0

Simplify

π - arcsin (0) : π

π - arcsin (0)

Use the following trivial identity:

arcsin (0) = 0

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 - 0

π - 0 = π

= π

0 + 2 πn < x < π + 2 πn

Simplify

2 πn < x < π + 2 πn

sin (x) < \frac{1}{2} : - \frac{7 π}{6} + 2 πn < x < \frac{π}{6} + 2 πn

sin (x) < \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 < x < arcsin (\frac{1}{2}) + 2 πn

Simplify

- π - arcsin (\frac{1}{2}) : - \frac{7 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 = - 7 π

= \frac{- 7 π}{6}

Apply the fraction rule:

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

= - \frac{7 π}{6}

= - \frac{7 π}{6}

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}

- \frac{7 π}{6} + 2 πn < x < \frac{π}{6} + 2 πn

Combine the intervals

2 πn < x < π + 2 πn and - \frac{7 π}{6} + 2 πn < x < \frac{π}{6} + 2 πn

Merge Overlapping Intervals

2 πn < x < \frac{π}{6} + 2 πn or \frac{5 π}{6} + 2 πn < x < π + 2 πn

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