What is the general solution for 2sec^2(x)+3(1/(cos(x)))=2 ?

The general solution for 2sec^2(x)+3(1/(cos(x)))=2 is x=(2pi)/3+2pin,x=(4pi)/3+2pin

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Popular Trigonometry >

2sec^2(x)+3(1/(cos(x)))=2

Solution

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

Solution

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

Degrees

x = 12 0^{\circ} + 36 0^{\circ} n, x = 24 0^{\circ} + 36 0^{\circ} n

Solution steps

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

Subtract

2

from both sides

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

Simplify

2 sec^{2} (x) + \frac{3}{cos ( x )} - 2 : \frac{2 sec ^{2} ( x ) cos ( x ) + 3 - 2 cos ( x )}{cos ( x )}

2 sec^{2} (x) + \frac{3}{cos ( x )} - 2

Convert element to fraction:

2 sec^{2} (x) = \frac{2 sec ^{2} ( x ) cos ( x )}{cos ( x )}, 2 = \frac{2 cos ( x )}{cos ( x )}

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

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

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

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

Rewrite using trig identities

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

Use the basic trigonometric identity:

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

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

Simplify

3 - 2 \cdot \frac{1}{sec ( x )} + 2 \cdot \frac{1}{sec ( x )} sec^{2} (x) : 3 - \frac{2}{sec ( x )} + 2 sec (x)

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

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

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

Multiply fractions:

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

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

Multiply the numbers:

1 \cdot 2 = 2

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

2 \cdot \frac{1}{sec ( x )} sec^{2} (x) = 2 sec (x)

2 \cdot \frac{1}{sec ( x )} sec^{2} (x)

Multiply fractions:

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

= \frac{1 \cdot 2 sec ^{2} ( x )}{sec ( x )}

Multiply the numbers:

1 \cdot 2 = 2

= \frac{2 sec ^{2} ( x )}{sec ( x )}

Cancel the common factor:

sec (x)

= 2 sec (x)

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

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

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

Solve by substitution

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

Let:

sec (x) = u

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

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

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

Multiply both sides by

u

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

Multiply both sides by

u

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

Simplify

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

Simplify

- \frac{2}{u} u : - 2

- \frac{2}{u} u

Multiply fractions:

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

= - \frac{2 u}{u}

Cancel the common factor:

u

= - 2

Simplify

2 uu : 2 u^{2}

2 uu

Apply exponent rule:

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

uu = u^{1 + 1}

= 2 u^{1 + 1}

Add the numbers:

1 + 1 = 2

= 2 u^{2}

Simplify

0 \cdot u : 0

0 \cdot u

Apply rule

0 \cdot a = 0

= 0

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

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

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

Solve

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

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

Write in the standard form

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

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

Solve with the quadratic formula

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

Quadratic Equation Formula:

For

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

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

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

3^{2} - 4 \cdot 2 (- 2) = 5

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

Apply rule

- (- a) = a

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

Multiply the numbers:

4 \cdot 2 \cdot 2 = 16

= 3^{2} + 16

3^{2} = 9

= 9 + 16

Add the numbers:

9 + 16 = 25

= 25

Factor the number:

25 = 5^{2}

= 5^{2}

Apply radical rule:

5^{2} = 5

= 5

u_{1, 2} = \frac{- 3 \pm 5}{2 \cdot 2}

Separate the solutions

u_{1} = \frac{- 3 + 5}{2 \cdot 2}, u_{2} = \frac{- 3 - 5}{2 \cdot 2}

u = \frac{- 3 + 5}{2 \cdot 2} : \frac{1}{2}

\frac{- 3 + 5}{2 \cdot 2}

Add/Subtract the numbers:

- 3 + 5 = 2

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

Multiply the numbers:

2 \cdot 2 = 4

= \frac{2}{4}

Cancel the common factor:

2

= \frac{1}{2}

u = \frac{- 3 - 5}{2 \cdot 2} : - 2

\frac{- 3 - 5}{2 \cdot 2}

Subtract the numbers:

- 3 - 5 = - 8

= \frac{- 8}{2 \cdot 2}

Multiply the numbers:

2 \cdot 2 = 4

= \frac{- 8}{4}

Apply the fraction rule:

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

= - \frac{8}{4}

Divide the numbers:

\frac{8}{4} = 2

= - 2

The solutions to the quadratic equation are:

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

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

Verify Solutions

Find undefined (singularity) points:

u = 0

Take the denominator(s) of

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

and compare to zero

u = 0

The following points are undefined

u = 0

Combine undefined points with solutions:

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

Substitute back

u = sec (x)

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

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

sec (x) = \frac{1}{2} :

No Solution

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

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

No Solution

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

sec (x) = - 2

General solutions for

sec (x) = - 2

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

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

Combine all the solutions

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

Graph

Popular Examples

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Frequently Asked Questions (FAQ)

What is the general solution for 2sec^2(x)+3(1/(cos(x)))=2 ?
The general solution for 2sec^2(x)+3(1/(cos(x)))=2 is x=(2pi)/3+2pin,x=(4pi)/3+2pin