What is the general solution for sin(x)tan(x)-(sqrt(2))/2 tan(x)=0 ?

The general solution for sin(x)tan(x)-(sqrt(2))/2 tan(x)=0 is x=pin,x= pi/4+2pin,x=(3pi)/4+2pin

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sin(x)tan(x)-(sqrt(2))/2 tan(x)=0

Solution

sin (x) tan (x) - \frac{2}{2} tan (x) = 0

Solution

x = πn, x = \frac{π}{4} + 2 πn, x = \frac{3 π}{4} + 2 πn

Degrees

x = 0^{\circ} + 18 0^{\circ} n, x = 4 5^{\circ} + 36 0^{\circ} n, x = 13 5^{\circ} + 36 0^{\circ} n

Solution steps

sin (x) tan (x) - \frac{2}{2} tan (x) = 0

Factor

sin (x) tan (x) - \frac{2}{2} tan (x) : tan (x) (sin (x) - \frac{1}{2})

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

Factor out common term

tan (x)

= tan (x) (sin (x) - \frac{2}{2})

Simplify

tan (x) (sin (x) - \frac{2}{2}) : tan (x) (sin (x) - \frac{1}{2})

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

Cancel

\frac{2}{2} : \frac{1}{2}

\frac{2}{2}

Cancel

\frac{2}{2} : \frac{1}{2}

\frac{2}{2}

Apply radical rule:

n a = a^{\frac{1}{n}}

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

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

Apply exponent rule:

\frac{x ^{a}}{x ^{b}} = \frac{1}{x ^{b - a}}

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

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

Subtract the numbers:

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

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

Apply radical rule:

a^{\frac{1}{n}} = n a

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

= \frac{1}{2}

= \frac{1}{2}

= tan (x) (sin (x) - \frac{1}{2})

= tan (x) (sin (x) - \frac{1}{2})

tan (x) (sin (x) - \frac{1}{2}) = 0

Solving each part separately

tan (x) = 0 or sin (x) - \frac{1}{2} = 0

tan (x) = 0 : x = πn

tan (x) = 0

General solutions for

tan (x) = 0

tan (x)

periodicity table with

πn

cycle:

x 0 \frac{π}{6} \frac{π}{4} \frac{π}{3} \frac{π}{2} \frac{2 π}{3} \frac{3 π}{4} \frac{5 π}{6} tan (x) 0 \frac{3}{3} 13 \pm \infty - 3 - 1 - \frac{3}{3}

x = 0 + πn

x = 0 + πn

Solve

x = 0 + πn : x = πn

x = 0 + πn

0 + πn = πn

x = πn

x = πn

sin (x) - \frac{1}{2} = 0 : x = \frac{π}{4} + 2 πn, x = \frac{3 π}{4} + 2 πn

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

Move

\frac{1}{2}

to the right side

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

Add

\frac{1}{2}

to both sides

sin (x) - \frac{1}{2} + \frac{1}{2} = 0 + \frac{1}{2}

Simplify

sin (x) - \frac{1}{2} + \frac{1}{2} = 0 + \frac{1}{2}

Simplify

sin (x) - \frac{1}{2} + \frac{1}{2} : sin (x)

sin (x) - \frac{1}{2} + \frac{1}{2}

Add similar elements:

- \frac{1}{2} + \frac{1}{2} = 0

= sin (x)

Simplify

0 + \frac{1}{2} : \frac{2}{2}

0 + \frac{1}{2}

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

= \frac{1}{2}

Rationalize

\frac{1}{2} : \frac{2}{2}

\frac{1}{2}

Multiply by the conjugate

\frac{2}{2}

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

1 \cdot 2 = 2

22 = 2

22

Apply radical rule:

a a = a

22 = 2

= 2

= \frac{2}{2}

= \frac{2}{2}

sin (x) = \frac{2}{2}

sin (x) = \frac{2}{2}

sin (x) = \frac{2}{2}

General solutions for

sin (x) = \frac{2}{2}

sin (x)

periodicity table with

2 πn

cycle:

x 0 \frac{π}{6} \frac{π}{4} \frac{π}{3} \frac{π}{2} \frac{2 π}{3} \frac{3 π}{4} \frac{5 π}{6} sin (x) 0 \frac{1}{2} \frac{2}{2} \frac{3}{2} 1 \frac{3}{2} \frac{2}{2} \frac{1}{2} x π \frac{7 π}{6} \frac{5 π}{4} \frac{4 π}{3} \frac{3 π}{2} \frac{5 π}{3} \frac{7 π}{4} \frac{11 π}{6} sin (x) 0 - \frac{1}{2} - \frac{2}{2} - \frac{3}{2} - 1 - \frac{3}{2} - \frac{2}{2} - \frac{1}{2}

x = \frac{π}{4} + 2 πn, x = \frac{3 π}{4} + 2 πn

x = \frac{π}{4} + 2 πn, x = \frac{3 π}{4} + 2 πn

Combine all the solutions

x = πn, x = \frac{π}{4} + 2 πn, x = \frac{3 π}{4} + 2 πn

Graph

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Frequently Asked Questions (FAQ)

What is the general solution for sin(x)tan(x)-(sqrt(2))/2 tan(x)=0 ?
The general solution for sin(x)tan(x)-(sqrt(2))/2 tan(x)=0 is x=pin,x= pi/4+2pin,x=(3pi)/4+2pin