What is the general solution for sqrt(3)tan(x)+2sec(x)=1 ?

The general solution for sqrt(3)tan(x)+2sec(x)=1 is x=(5pi)/3+2pin

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Popular Trigonometry >

sqrt(3)tan(x)+2sec(x)=1

Solution

3 tan (x) + 2 sec (x) = 1

Solution

x = \frac{5 π}{3} + 2 πn

Degrees

x = 30 0^{\circ} + 36 0^{\circ} n

Solution steps

3 tan (x) + 2 sec (x) = 1

Subtract

1

from both sides

3 tan (x) + 2 sec (x) - 1 = 0

Express with sin, cos

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

Simplify

3 \frac{sin ( x )}{cos ( x )} + 2 \cdot \frac{1}{cos ( x )} - 1 : \frac{3 sin ( x ) + 2 - cos ( x )}{cos ( x )}

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

3 \frac{sin ( x )}{cos ( x )} = \frac{3 sin ( x )}{cos ( x )}

3 \frac{sin ( x )}{cos ( x )}

Multiply fractions:

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

= \frac{sin ( x ) 3}{cos ( x )}

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

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

Multiply fractions:

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

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

Multiply the numbers:

1 \cdot 2 = 2

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

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

Combine the fractions

\frac{3 sin ( x )}{cos ( x )} + \frac{2}{cos ( x )} : \frac{3 sin ( x ) + 2}{cos ( x )}

Apply rule

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

= \frac{3 sin ( x ) + 2}{cos ( x )}

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

Convert element to fraction:

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

= \frac{sin ( x ) 3 + 2}{cos ( x )} - \frac{1 \cdot cos ( x )}{cos ( x )}

Since the denominators are equal, combine the fractions:

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

= \frac{sin ( x ) 3 + 2 - 1 \cdot cos ( x )}{cos ( x )}

Multiply:

1 \cdot cos (x) = cos (x)

= \frac{3 sin ( x ) + 2 - cos ( x )}{cos ( x )}

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

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

3 sin (x) + 2 - cos (x) = 0

Add

cos (x)

to both sides

3 sin (x) + 2 = cos (x)

Square both sides

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

Subtract

cos^{2} (x)

from both sides

(3 sin (x) + 2)^{2} - cos^{2} (x) = 0

Rewrite using trig identities

(2 + sin (x) 3)^{2} - cos^{2} (x)

Use the Pythagorean identity:

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

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

= (2 + sin (x) 3)^{2} - (1 - sin^{2} (x))

Simplify

(2 + sin (x) 3)^{2} - (1 - sin^{2} (x)) : 4 sin^{2} (x) + 43 sin (x) + 3

(2 + sin (x) 3)^{2} - (1 - sin^{2} (x))

= (2 + 3 sin (x))^{2} - (1 - sin^{2} (x))

(2 + sin (x) 3)^{2} : 4 + 43 sin (x) + 3 sin^{2} (x)

Apply Perfect Square Formula:

(a + b)^{2} = a^{2} + 2 ab + b^{2}

a = 2, b = sin (x) 3

= 2^{2} + 2 \cdot 2 sin (x) 3 + (sin (x) 3)^{2}

Simplify

2^{2} + 2 \cdot 2 sin (x) 3 + (sin (x) 3)^{2} : 4 + 43 sin (x) + 3 sin^{2} (x)

2^{2} + 2 \cdot 2 sin (x) 3 + (sin (x) 3)^{2}

2^{2} = 4

2^{2}

2^{2} = 4

= 4

2 \cdot 2 sin (x) 3 = 43 sin (x)

2 \cdot 2 sin (x) 3

Multiply the numbers:

2 \cdot 2 = 4

= 43 sin (x)

(sin (x) 3)^{2} = 3 sin^{2} (x)

(sin (x) 3)^{2}

Apply exponent rule:

(a \cdot b)^{n} = a^{n} b^{n}

= (3)^{2} sin^{2} (x)

(3)^{2} : 3

Apply radical rule:

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

= (3^{\frac{1}{2}})^{2}

Apply exponent rule:

(a^{b})^{c} = a^{b c}

= 3^{\frac{1}{2} \cdot 2}

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

\frac{1}{2} \cdot 2

Multiply fractions:

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

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

Cancel the common factor:

2

= 1

= 3

= sin^{2} (x) \cdot 3

= 4 + 43 sin (x) + 3 sin^{2} (x)

= 4 + 43 sin (x) + 3 sin^{2} (x)

= 4 + 43 sin (x) + 3 sin^{2} (x) - (1 - sin^{2} (x))

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

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

Distribute parentheses

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

Apply minus-plus rules

- (- a) = a, - (a) = - a

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

= 4 + 43 sin (x) + 3 sin^{2} (x) - 1 + sin^{2} (x)

Simplify

4 + 43 sin (x) + 3 sin^{2} (x) - 1 + sin^{2} (x) : 4 sin^{2} (x) + 43 sin (x) + 3

4 + 43 sin (x) + 3 sin^{2} (x) - 1 + sin^{2} (x)

Group like terms

= 43 sin (x) + 3 sin^{2} (x) + sin^{2} (x) + 4 - 1

Add similar elements:

3 sin^{2} (x) + sin^{2} (x) = 4 sin^{2} (x)

= 43 sin (x) + 4 sin^{2} (x) + 4 - 1

Add/Subtract the numbers:

4 - 1 = 3

= 4 sin^{2} (x) + 43 sin (x) + 3

= 4 sin^{2} (x) + 43 sin (x) + 3

= 4 sin^{2} (x) + 43 sin (x) + 3

3 + 4 sin^{2} (x) + 4 sin (x) 3 = 0

Solve by substitution

3 + 4 sin^{2} (x) + 4 sin (x) 3 = 0

Let:

sin (x) = u

3 + 4 u^{2} + 4 u 3 = 0

3 + 4 u^{2} + 4 u 3 = 0 : u = - \frac{3}{2}

3 + 4 u^{2} + 4 u 3 = 0

Write in the standard form

a x^{2} + b x + c = 0

4 u^{2} + 43 u + 3 = 0

Solve with the quadratic formula

4 u^{2} + 43 u + 3 = 0

Quadratic Equation Formula:

For

a = 4, b = 43, c = 3

u_{1, 2} = \frac{- 4 3 \pm ( 4 3 ) ^{2} - 4 \cdot 4 \cdot 3}{2 \cdot 4}

u_{1, 2} = \frac{- 4 3 \pm ( 4 3 ) ^{2} - 4 \cdot 4 \cdot 3}{2 \cdot 4}

(43)^{2} - 4 \cdot 4 \cdot 3 = 0

(43)^{2} - 4 \cdot 4 \cdot 3

(43)^{2} = 4^{2} \cdot 3

(43)^{2}

Apply exponent rule:

(a \cdot b)^{n} = a^{n} b^{n}

= 4^{2} (3)^{2}

(3)^{2} : 3

Apply radical rule:

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

= (3^{\frac{1}{2}})^{2}

Apply exponent rule:

(a^{b})^{c} = a^{b c}

= 3^{\frac{1}{2} \cdot 2}

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

\frac{1}{2} \cdot 2

Multiply fractions:

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

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

Cancel the common factor:

2

= 1

= 3

= 4^{2} \cdot 3

4 \cdot 4 \cdot 3 = 48

4 \cdot 4 \cdot 3

Multiply the numbers:

4 \cdot 4 \cdot 3 = 48

= 48

= 4^{2} \cdot 3 - 48

4^{2} \cdot 3 = 48

4^{2} \cdot 3

4^{2} = 16

= 16 \cdot 3

Multiply the numbers:

16 \cdot 3 = 48

= 48

= 48 - 48

Subtract the numbers:

48 - 48 = 0

= 0

u_{1, 2} = \frac{- 4 3 \pm 0}{2 \cdot 4}

u = \frac{- 4 3}{2 \cdot 4}

\frac{- 4 3}{2 \cdot 4} = - \frac{3}{2}

\frac{- 4 3}{2 \cdot 4}

Multiply the numbers:

2 \cdot 4 = 8

= \frac{- 4 3}{8}

Apply the fraction rule:

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

= - \frac{4 3}{8}

Cancel the common factor:

4

= - \frac{3}{2}

u = - \frac{3}{2}

The solution to the quadratic equation is:

u = - \frac{3}{2}

Substitute back

u = sin (x)

sin (x) = - \frac{3}{2}

sin (x) = - \frac{3}{2}

sin (x) = - \frac{3}{2} : x = \frac{4 π}{3} + 2 πn, x = \frac{5 π}{3} + 2 πn

sin (x) = - \frac{3}{2}

General solutions for

sin (x) = - \frac{3}{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{4 π}{3} + 2 πn, x = \frac{5 π}{3} + 2 πn

x = \frac{4 π}{3} + 2 πn, x = \frac{5 π}{3} + 2 πn

Combine all the solutions

x = \frac{4 π}{3} + 2 πn, x = \frac{5 π}{3} + 2 πn

Verify solutions by plugging them into the original equation

Check the solutions by plugging them into

3 tan (x) + 2 sec (x) = 1

Remove the ones that don't agree with the equation.

Check the solution

\frac{4 π}{3} + 2 πn :

False

\frac{4 π}{3} + 2 πn

Plug in

n = 1

\frac{4 π}{3} + 2 π 1

For

3 tan (x) + 2 sec (x) = 1

plug in

x = \frac{4 π}{3} + 2 π 1

3 tan (\frac{4 π}{3} + 2 π 1) + 2 sec (\frac{4 π}{3} + 2 π 1) = 1

Refine

- 1 = 1

\Rightarrow False

Check the solution

\frac{5 π}{3} + 2 πn :

True

\frac{5 π}{3} + 2 πn

Plug in

n = 1

\frac{5 π}{3} + 2 π 1

For

3 tan (x) + 2 sec (x) = 1

plug in

x = \frac{5 π}{3} + 2 π 1

3 tan (\frac{5 π}{3} + 2 π 1) + 2 sec (\frac{5 π}{3} + 2 π 1) = 1

Refine

1 = 1

\Rightarrow True

x = \frac{5 π}{3} + 2 πn

Graph

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Frequently Asked Questions (FAQ)

What is the general solution for sqrt(3)tan(x)+2sec(x)=1 ?
The general solution for sqrt(3)tan(x)+2sec(x)=1 is x=(5pi)/3+2pin