What is the general solution for sqrt(2)sec(x)-3tan(x)=-tan(x) ?

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

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sqrt(2)sec(x)-3tan(x)=-tan(x)

Solution

2 sec (x) - 3 tan (x) = - tan (x)

Solution

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

Degrees

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

Solution steps

2 sec (x) - 3 tan (x) = - tan (x)

Subtract

- tan (x)

from both sides

2 sec (x) - 2 tan (x) = 0

Express with sin, cos

- 2 tan (x) + sec (x) 2

Use the basic trigonometric identity:

tan (x) = \frac{sin ( x )}{cos ( x )}

= - 2 \cdot \frac{sin ( x )}{cos ( x )} + sec (x) 2

Use the basic trigonometric identity:

sec (x) = \frac{1}{cos ( x )}

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

Simplify

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

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

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

2 \cdot \frac{sin ( x )}{cos ( x )}

Multiply fractions:

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

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

\frac{1}{cos ( x )} 2 = \frac{2}{cos ( x )}

\frac{1}{cos ( x )} 2

Multiply fractions:

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

= \frac{1 \cdot 2}{cos ( x )}

Multiply:

1 \cdot 2 = 2

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

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

Apply rule

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

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

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

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

\frac{f ( x )}{g ( x )} = 0 \Rightarrow f (x) = 0

2 - 2 sin (x) = 0

Move

2

to the right side

2 - 2 sin (x) = 0

Subtract

2

from both sides

2 - 2 sin (x) - 2 = 0 - 2

Simplify

- 2 sin (x) = - 2

- 2 sin (x) = - 2

Divide both sides by

- 2

- 2 sin (x) = - 2

Divide both sides by

- 2

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

Simplify

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

Graph

Popular Examples

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cot (θ) + csc (θ) = - 3

3tan^2(θ)+1= 2/(tan^2(θ))

3 tan^{2} (θ) + 1 = \frac{2}{tan ^{2} ( θ )}

4tan(x)=2sec^2(x)

4 tan (x) = 2 sec^{2} (x)

sin(x)+cos(x)+cot(x)=csc(x)

sin (x) + cos (x) + cot (x) = csc (x)

cos^2(x)+cos(x)=cos(2x)

cos^{2} (x) + cos (x) = cos (2 x)

Frequently Asked Questions (FAQ)

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