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Popular Trigonometry >

2+sqrt(3)sec(x)-4cos(x)=2sqrt(3)

Solution

2 + 3 sec (x) - 4 cos (x) = 23

Solution

x = \frac{π}{3} + 2 πn, x = \frac{5 π}{3} + 2 πn, x = \frac{5 π}{6} + 2 πn, x = \frac{7 π}{6} + 2 πn

Degrees

x = 6 0^{\circ} + 36 0^{\circ} n, x = 30 0^{\circ} + 36 0^{\circ} n, x = 15 0^{\circ} + 36 0^{\circ} n, x = 21 0^{\circ} + 36 0^{\circ} n

Solution steps

2 + 3 sec (x) - 4 cos (x) = 23

Subtract

23

from both sides

2 + 3 sec (x) - 4 cos (x) - 23 = 0

Rewrite using trig identities

2 - 23 - 4 cos (x) + sec (x) 3

Use the basic trigonometric identity:

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

= 2 - 23 - 4 \cdot \frac{1}{sec ( x )} + sec (x) 3

4 \cdot \frac{1}{sec ( x )} = \frac{4}{sec ( x )}

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

Multiply fractions:

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

= \frac{1 \cdot 4}{sec ( x )}

Multiply the numbers:

1 \cdot 4 = 4

= \frac{4}{sec ( x )}

= 2 - 23 - \frac{4}{sec ( x )} + 3 sec (x)

2 - \frac{4}{sec ( x )} - 23 + sec (x) 3 = 0

Solve by substitution

2 - \frac{4}{sec ( x )} - 23 + sec (x) 3 = 0

Let:

sec (x) = u

2 - \frac{4}{u} - 23 + u 3 = 0

2 - \frac{4}{u} - 23 + u 3 = 0 : u = 2, u = - \frac{2 3}{3}

2 - \frac{4}{u} - 23 + u 3 = 0

Multiply both sides by

u

2 - \frac{4}{u} - 23 + u 3 = 0

Multiply both sides by

u

2 u - \frac{4}{u} u - 23 u + u 3 u = 0 \cdot u

Simplify

2 u - \frac{4}{u} u - 23 u + u 3 u = 0 \cdot u

Simplify

- \frac{4}{u} u : - 4

- \frac{4}{u} u

Multiply fractions:

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

= - \frac{4 u}{u}

Cancel the common factor:

u

= - 4

Simplify

u 3 u : 3 u^{2}

u 3 u

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 - 4 - 23 u + 3 u^{2} = 0

2 u - 4 - 23 u + 3 u^{2} = 0

2 u - 4 - 23 u + 3 u^{2} = 0

Solve

2 u - 4 - 23 u + 3 u^{2} = 0 : u = 2, u = - \frac{2 3}{3}

2 u - 4 - 23 u + 3 u^{2} = 0

Write in the standard form

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

3 u^{2} + (2 - 23) u - 4 = 0

Solve with the quadratic formula

3 u^{2} + (2 - 23) u - 4 = 0

Quadratic Equation Formula:

For

a = 3, b = 2 - 23, c = - 4

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

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

(2 - 23)^{2} - 43 (- 4) = 23 + 2

(2 - 23)^{2} - 43 (- 4)

Apply rule

- (- a) = a

= (2 - 23)^{2} + 43 \cdot 4

Multiply the numbers:

4 \cdot 4 = 16

= (2 - 23)^{2} + 163

Expand

(2 - 23)^{2} + 163 : 16 + 83

(2 - 23)^{2} + 163

(2 - 23)^{2} : 16 - 83

Apply Perfect Square Formula:

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

a = 2, b = 23

= 2^{2} - 2 \cdot 2 \cdot 23 + (23)^{2}

Simplify

2^{2} - 2 \cdot 2 \cdot 23 + (23)^{2} : 16 - 83

2^{2} - 2 \cdot 2 \cdot 23 + (23)^{2}

2^{2} = 4

2^{2}

2^{2} = 4

= 4

2 \cdot 2 \cdot 23 = 83

2 \cdot 2 \cdot 23

Multiply the numbers:

2 \cdot 2 \cdot 2 = 8

= 83

(23)^{2} = 12

(23)^{2}

Apply exponent rule:

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

= 2^{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

= 2^{2} \cdot 3

2^{2} = 4

= 4 \cdot 3

Multiply the numbers:

4 \cdot 3 = 12

= 12

= 4 - 83 + 12

Add the numbers:

4 + 12 = 16

= 16 - 83

= 16 - 83

= 16 - 83 + 163

Add similar elements:

- 83 + 163 = 83

= 16 + 83

= 16 + 83

= 12 + 83 + 4

= 4 \cdot 3 + 83 + 4

= (4)^{2} (3)^{2} + 83 + (4)^{2}

4 = 2

4

Factor the number:

4 = 2^{2}

= 2^{2}

Apply radical rule:

2^{2} = 2

= 2

= 2^{2} (3)^{2} + 83 + 2^{2}

2 \cdot 23 \cdot 2 = 83

2 \cdot 23 \cdot 2

Multiply the numbers:

2 \cdot 2 \cdot 2 = 8

= 83

= (23)^{2} + 2 \cdot 23 \cdot 2 + 2^{2}

Apply Perfect Square Formula:

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

(23)^{2} + 2 \cdot 23 \cdot 2 + 2^{2} = (23 + 2)^{2}

= (23 + 2)^{2}

Apply radical rule:

(23 + 2)^{2} = 23 + 2

= 23 + 2

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

Separate the solutions

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

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

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

Expand

- (2 - 23) + 23 + 2 : 43

- (2 - 23) + 23 + 2

- (2 - 23) : - 2 + 23

- (2 - 23)

Distribute parentheses

= - (2) - (- 23)

Apply minus-plus rules

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

= - 2 + 23

= - 2 + 23 + 23 + 2

Simplify

- 2 + 23 + 23 + 2 : 43

- 2 + 23 + 23 + 2

Add similar elements:

23 + 23 = 43

= - 2 + 43 + 2

- 2 + 2 = 0

= 43

= 43

= \frac{4 3}{2 3}

Divide the numbers:

\frac{4}{2} = 2

= \frac{2 3}{3}

Cancel the common factor:

3

= 2

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

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

Expand

- (2 - 23) - (23 + 2) : - 4

- (2 - 23) - (23 + 2)

- (2 - 23) : - 2 + 23

- (2 - 23)

Distribute parentheses

= - (2) - (- 23)

Apply minus-plus rules

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

= - 2 + 23

= - 2 + 23 - (23 + 2)

- (23 + 2) : - 23 - 2

- (23 + 2)

Distribute parentheses

= - (23) - (2)

Apply minus-plus rules

+ (- a) = - a

= - 23 - 2

= - 2 + 23 - 23 - 2

Simplify

- 2 + 23 - 23 - 2 : - 4

- 2 + 23 - 23 - 2

Add similar elements:

23 - 23 = 0

= - 2 - 2

Subtract the numbers:

- 2 - 2 = - 4

= - 4

= - 4

= \frac{- 4}{2 3}

Apply the fraction rule:

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

= - \frac{4}{2 3}

Divide the numbers:

\frac{4}{2} = 2

= - \frac{2}{3}

Rationalize

- \frac{2}{3} : - \frac{2 3}{3}

- \frac{2}{3}

Multiply by the conjugate

\frac{3}{3}

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

33 = 3

33

Apply radical rule:

a a = a

33 = 3

= 3

= - \frac{2 3}{3}

= - \frac{2 3}{3}

The solutions to the quadratic equation are:

u = 2, u = - \frac{2 3}{3}

u = 2, u = - \frac{2 3}{3}

Verify Solutions

Find undefined (singularity) points:

u = 0

Take the denominator(s) of

2 - \frac{4}{u} - 23 + u 3

and compare to zero

u = 0

The following points are undefined

u = 0

Combine undefined points with solutions:

u = 2, u = - \frac{2 3}{3}

Substitute back

u = sec (x)

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

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

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

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

sec (x) = - \frac{2 3}{3} : x = \frac{5 π}{6} + 2 πn, x = \frac{7 π}{6} + 2 πn

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

General solutions for

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

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{5 π}{6} + 2 πn, x = \frac{7 π}{6} + 2 πn

x = \frac{5 π}{6} + 2 πn, x = \frac{7 π}{6} + 2 πn

Combine all the solutions

x = \frac{π}{3} + 2 πn, x = \frac{5 π}{3} + 2 πn, x = \frac{5 π}{6} + 2 πn, x = \frac{7 π}{6} + 2 πn

Graph

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