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Popular >

4-tan(θ/2)>3

Solution

4 - tan (\frac{θ}{2}) > 3

Solution

- π + 2 πn < θ < \frac{π}{2} + 2 πn

Interval Notation

(- π + 2 πn, \frac{π}{2} + 2 πn)

Decimal Notation

- 3.14159 \dots + 2 πn < θ < 1.57079 \dots + 2 πn

Solution steps

4 - tan (\frac{θ}{2}) > 3

Move

4

to the right side

4 - tan (\frac{θ}{2}) > 3

Subtract

4

from both sides

4 - tan (\frac{θ}{2}) - 4 > 3 - 4

Simplify

- tan (\frac{θ}{2}) > - 1

- tan (\frac{θ}{2}) > - 1

Multiply both sides by

- 1

- tan (\frac{θ}{2}) > - 1

Multiply both sides by -1 (reverse the inequality)

(- tan (\frac{θ}{2})) (- 1) < (- 1) (- 1)

Simplify

tan (\frac{θ}{2}) < 1

tan (\frac{θ}{2}) < 1

tan (x) < a

then

- \frac{π}{2} + πn < x < arctan (a) + πn

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

a < u < b

then

a < u and u < b

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

- \frac{π}{2} + πn < \frac{θ}{2} : θ > - π + 2 πn

- \frac{π}{2} + πn < \frac{θ}{2}

Switch sides

\frac{θ}{2} > - \frac{π}{2} + πn

Multiply both sides by

2

\frac{θ}{2} > - \frac{π}{2} + πn

Multiply both sides by

2

\frac{2 θ}{2} > - 2 \cdot \frac{π}{2} + 2 πn

Simplify

\frac{2 θ}{2} > - 2 \cdot \frac{π}{2} + 2 πn

Simplify

\frac{2 θ}{2} : θ

\frac{2 θ}{2}

Divide the numbers:

\frac{2}{2} = 1

= θ

Simplify

- 2 \cdot \frac{π}{2} + 2 πn : - π + 2 πn

- 2 \cdot \frac{π}{2} + 2 πn

2 \cdot \frac{π}{2} = π

2 \cdot \frac{π}{2}

Multiply fractions:

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

= \frac{π 2}{2}

Cancel the common factor:

2

= π

= - π + 2 πn

θ > - π + 2 πn

θ > - π + 2 πn

θ > - π + 2 πn

\frac{θ}{2} < arctan (1) + πn : θ < \frac{π}{2} + 2 πn

\frac{θ}{2} < arctan (1) + πn

Simplify

arctan (1) + πn : \frac{π}{4} + πn

arctan (1) + πn

Use the following trivial identity:

arctan (1) = \frac{π}{4}

x 0 \frac{3}{3} 13 arctan (x) 0 \frac{π}{6} \frac{π}{4} \frac{π}{3} arctan (x) 0^{\circ} 3 0^{\circ} 4 5^{\circ} 6 0^{\circ}

= \frac{π}{4} + πn

\frac{θ}{2} < \frac{π}{4} + πn

Multiply both sides by

2

\frac{θ}{2} < \frac{π}{4} + πn

Multiply both sides by

2

\frac{2 θ}{2} < 2 \cdot \frac{π}{4} + 2 πn

Simplify

\frac{2 θ}{2} < 2 \cdot \frac{π}{4} + 2 πn

Simplify

\frac{2 θ}{2} : θ

\frac{2 θ}{2}

Divide the numbers:

\frac{2}{2} = 1

= θ

Simplify

2 \cdot \frac{π}{4} + 2 πn : \frac{π}{2} + 2 πn

2 \cdot \frac{π}{4} + 2 πn

2 \cdot \frac{π}{4} = \frac{π}{2}

2 \cdot \frac{π}{4}

Multiply fractions:

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

= \frac{π 2}{4}

Cancel the common factor:

2

= \frac{π}{2}

= \frac{π}{2} + 2 πn

θ < \frac{π}{2} + 2 πn

θ < \frac{π}{2} + 2 πn

θ < \frac{π}{2} + 2 πn

Combine the intervals

θ > - π + 2 πn and θ < \frac{π}{2} + 2 πn

Merge Overlapping Intervals

- π + 2 πn < θ < \frac{π}{2} + 2 πn

Popular Examples

sin(x)>-(sqrt(2))/2

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

((2cos(x)+1))/(2sin(x)-sqrt(3))>0

\frac{( 2 cos ( x ) + 1 )}{2 sin ( x ) - 3} > 0

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

cos (x^{4}) + sin (x^{4}) \leq 0.5

sin(3X-15)>-(sqrt(2))/2

sin (3 X - 15) > - \frac{2}{2}

2(tan(x)+1)+3(1-tan(x))<-2(tan(x)-3)

2 (tan (x) + 1) + 3 (1 - tan (x)) < - 2 (tan (x) - 3)