What is the general solution for 3sec(x)-cos(x)-2=0 ?

The general solution for 3sec(x)-cos(x)-2=0 is x=2pin

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3sec(x)-cos(x)-2=0

Solution

3 sec (x) - cos (x) - 2 = 0

Solution

x = 2 πn

Degrees

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

Solution steps

3 sec (x) - cos (x) - 2 = 0

Rewrite using trig identities

- 2 - cos (x) + 3 sec (x)

Use the basic trigonometric identity:

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

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

- 2 - \frac{1}{sec ( x )} + 3 sec (x) = 0

Solve by substitution

- 2 - \frac{1}{sec ( x )} + 3 sec (x) = 0

Let:

sec (x) = u

- 2 - \frac{1}{u} + 3 u = 0

- 2 - \frac{1}{u} + 3 u = 0 : u = 1, u = - \frac{1}{3}

- 2 - \frac{1}{u} + 3 u = 0

Multiply both sides by

u

- 2 - \frac{1}{u} + 3 u = 0

Multiply both sides by

u

- 2 u - \frac{1}{u} u + 3 uu = 0 \cdot u

Simplify

- 2 u - \frac{1}{u} u + 3 uu = 0 \cdot u

Simplify

- \frac{1}{u} u : - 1

- \frac{1}{u} u

Multiply fractions:

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

= - \frac{1 \cdot u}{u}

Cancel the common factor:

u

= - 1

Simplify

3 uu : 3 u^{2}

3 uu

Apply exponent rule:

a^{b} \cdot a^{c} = a^{b + c}

uu = u^{1 + 1}

= 3 u^{1 + 1}

Add the numbers:

1 + 1 = 2

= 3 u^{2}

Simplify

0 \cdot u : 0

0 \cdot u

Apply rule

0 \cdot a = 0

= 0

- 2 u - 1 + 3 u^{2} = 0

- 2 u - 1 + 3 u^{2} = 0

- 2 u - 1 + 3 u^{2} = 0

Solve

- 2 u - 1 + 3 u^{2} = 0 : u = 1, u = - \frac{1}{3}

- 2 u - 1 + 3 u^{2} = 0

Write in the standard form

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

3 u^{2} - 2 u - 1 = 0

Solve with the quadratic formula

3 u^{2} - 2 u - 1 = 0

Quadratic Equation Formula:

For

a = 3, b = - 2, c = - 1

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

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

(- 2)^{2} - 4 \cdot 3 (- 1) = 4

(- 2)^{2} - 4 \cdot 3 (- 1)

Apply rule

- (- a) = a

= (- 2)^{2} + 4 \cdot 3 \cdot 1

Apply exponent rule:

(- a)^{n} = a^{n},

n

is even

(- 2)^{2} = 2^{2}

= 2^{2} + 4 \cdot 3 \cdot 1

Multiply the numbers:

4 \cdot 3 \cdot 1 = 12

= 2^{2} + 12

2^{2} = 4

= 4 + 12

Add the numbers:

4 + 12 = 16

= 16

Factor the number:

16 = 4^{2}

= 4^{2}

Apply radical rule:

4^{2} = 4

= 4

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

Separate the solutions

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

u = \frac{- ( - 2 ) + 4}{2 \cdot 3} : 1

\frac{- ( - 2 ) + 4}{2 \cdot 3}

Apply rule

- (- a) = a

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

Add the numbers:

2 + 4 = 6

= \frac{6}{2 \cdot 3}

Multiply the numbers:

2 \cdot 3 = 6

= \frac{6}{6}

Apply rule

\frac{a}{a} = 1

= 1

u = \frac{- ( - 2 ) - 4}{2 \cdot 3} : - \frac{1}{3}

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

Apply rule

- (- a) = a

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

Subtract the numbers:

2 - 4 = - 2

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

Multiply the numbers:

2 \cdot 3 = 6

= \frac{- 2}{6}

Apply the fraction rule:

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

= - \frac{2}{6}

Cancel the common factor:

2

= - \frac{1}{3}

The solutions to the quadratic equation are:

u = 1, u = - \frac{1}{3}

u = 1, u = - \frac{1}{3}

Verify Solutions

Find undefined (singularity) points:

u = 0

Take the denominator(s) of

- 2 - \frac{1}{u} + 3 u

and compare to zero

u = 0

The following points are undefined

u = 0

Combine undefined points with solutions:

u = 1, u = - \frac{1}{3}

Substitute back

u = sec (x)

sec (x) = 1, sec (x) = - \frac{1}{3}

sec (x) = 1, sec (x) = - \frac{1}{3}

sec (x) = 1 : x = 2 πn

sec (x) = 1

General solutions for

sec (x) = 1

sec (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} sec (x) 1 \frac{2 3}{3} 22 Undefined - 2 - 2 - \frac{2 3}{3} 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} sec (x) - 1 - \frac{2 3}{3} - 2 - 2 Undefined 22 \frac{2 3}{3}

x = 0 + 2 πn

x = 0 + 2 πn

Solve

x = 0 + 2 πn : x = 2 πn

x = 0 + 2 πn

0 + 2 πn = 2 πn

x = 2 πn

x = 2 πn

sec (x) = - \frac{1}{3} :

No Solution

sec (x) = - \frac{1}{3}

sec (x) \leq - 1 or sec (x) \geq 1

No Solution

Combine all the solutions

x = 2 πn

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Frequently Asked Questions (FAQ)

What is the general solution for 3sec(x)-cos(x)-2=0 ?
The general solution for 3sec(x)-cos(x)-2=0 is x=2pin