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Popular Trigonometry >

2sin(x)=(4cos(x)-1)/(tan(x))

Solution

2 sin (x) = \frac{4 cos ( x ) - 1}{tan ( x )}

Solution

x = \frac{2 π}{3} + 2 πn, x = \frac{4 π}{3} + 2 πn, x = 0.84106 \dots + 2 πn, x = 2 π - 0.84106 \dots + 2 πn

Degrees

x = 12 0^{\circ} + 36 0^{\circ} n, x = 24 0^{\circ} + 36 0^{\circ} n, x = 48.18968 \dots^{\circ} + 36 0^{\circ} n, x = 311.81031 \dots^{\circ} + 36 0^{\circ} n

Solution steps

2 sin (x) = \frac{4 cos ( x ) - 1}{tan ( x )}

Subtract

\frac{4 cos ( x ) - 1}{tan ( x )}

from both sides

2 sin (x) - \frac{4 cos ( x ) - 1}{tan ( x )} = 0

Simplify

2 sin (x) - \frac{4 cos ( x ) - 1}{tan ( x )} : \frac{2 sin ( x ) tan ( x ) - 4 cos ( x ) + 1}{tan ( x )}

2 sin (x) - \frac{4 cos ( x ) - 1}{tan ( x )}

Convert element to fraction:

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

= \frac{2 sin ( x ) tan ( x )}{tan ( x )} - \frac{4 cos ( x ) - 1}{tan ( x )}

Since the denominators are equal, combine the fractions:

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

= \frac{2 sin ( x ) tan ( x ) - ( 4 cos ( x ) - 1 )}{tan ( x )}

- (4 cos (x) - 1) : - 4 cos (x) + 1

- (4 cos (x) - 1)

Distribute parentheses

= - (4 cos (x)) - (- 1)

Apply minus-plus rules

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

= - 4 cos (x) + 1

= \frac{2 sin ( x ) tan ( x ) - 4 cos ( x ) + 1}{tan ( x )}

\frac{2 sin ( x ) tan ( x ) - 4 cos ( x ) + 1}{tan ( x )} = 0

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

2 sin (x) tan (x) - 4 cos (x) + 1 = 0

Rewrite using trig identities

1 - 4 cos (x) + 2 sin (x) tan (x)

Use the basic trigonometric identity:

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

= 1 - 4 cos (x) + 2 sin (x) \frac{sin ( x )}{cos ( x )}

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

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

Multiply fractions:

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

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

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

sin (x) \cdot 2 sin (x)

Apply exponent rule:

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

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

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

Add the numbers:

1 + 1 = 2

= 2 sin^{2} (x)

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

= 1 - 4 cos (x) + \frac{2 sin ^{2} ( x )}{cos ( x )}

Use the Pythagorean identity:

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

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

= 1 + \frac{2 ( 1 - cos ^{2} ( x ) )}{cos ( x )} - 4 cos (x)

Combine the fractions

\frac{2 ( - cos ^{2} ( x ) + 1 )}{cos ( x )} - 4 cos (x) : \frac{2 - 6 cos ^{2} ( x )}{cos ( x )}

\frac{2 ( - cos ^{2} ( x ) + 1 )}{cos ( x )} - 4 cos (x)

Convert element to fraction:

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

= \frac{2 ( 1 - cos ^{2} ( x ) )}{cos ( x )} - \frac{4 cos ( x ) 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 ( 1 - cos ^{2} ( x ) ) - 4 cos ( x ) cos ( x )}{cos ( x )}

2 (1 - cos^{2} (x)) - 4 cos (x) cos (x) = 2 (1 - cos^{2} (x)) - 4 cos^{2} (x)

2 (1 - cos^{2} (x)) - 4 cos (x) cos (x)

4 cos (x) cos (x) = 4 cos^{2} (x)

4 cos (x) cos (x)

Apply exponent rule:

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

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

= 4 cos^{1 + 1} (x)

Add the numbers:

1 + 1 = 2

= 4 cos^{2} (x)

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

= \frac{2 ( - cos ^{2} ( x ) + 1 ) - 4 cos ^{2} ( x )}{cos ( x )}

Expand

2 (1 - cos^{2} (x)) - 4 cos^{2} (x) : 2 - 6 cos^{2} (x)

2 (1 - cos^{2} (x)) - 4 cos^{2} (x)

Expand

2 (1 - cos^{2} (x)) : 2 - 2 cos^{2} (x)

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

Apply the distributive law:

a (b - c) = ab - a c

a = 2, b = 1, c = cos^{2} (x)

= 2 \cdot 1 - 2 cos^{2} (x)

Multiply the numbers:

2 \cdot 1 = 2

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

= 2 - 2 cos^{2} (x) - 4 cos^{2} (x)

Add similar elements:

- 2 cos^{2} (x) - 4 cos^{2} (x) = - 6 cos^{2} (x)

= 2 - 6 cos^{2} (x)

= \frac{2 - 6 cos ^{2} ( x )}{cos ( x )}

= \frac{2 - 6 cos ^{2} ( x )}{cos ( x )} + 1

1 + \frac{2 - 6 cos ^{2} ( x )}{cos ( x )} = 0

1 + \frac{2 - 6 cos ^{2} ( x )}{cos ( x )} = 0

Solve by substitution

1 + \frac{2 - 6 cos ^{2} ( x )}{cos ( x )} = 0

Let:

cos (x) = u

1 + \frac{2 - 6 u ^{2}}{u} = 0

1 + \frac{2 - 6 u ^{2}}{u} = 0 : u = - \frac{1}{2}, u = \frac{2}{3}

1 + \frac{2 - 6 u ^{2}}{u} = 0

Multiply both sides by

u

1 + \frac{2 - 6 u ^{2}}{u} = 0

Multiply both sides by

u

1 \cdot u + \frac{2 - 6 u ^{2}}{u} u = 0 \cdot u

Simplify

1 \cdot u + \frac{2 - 6 u ^{2}}{u} u = 0 \cdot u

Simplify

1 \cdot u : u

1 \cdot u

Multiply:

1 \cdot u = u

= u

Simplify

\frac{2 - 6 u ^{2}}{u} u : 2 - 6 u^{2}

\frac{2 - 6 u ^{2}}{u} u

Multiply fractions:

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

= \frac{( 2 - 6 u ^{2} ) u}{u}

Cancel the common factor:

u

= 2 - 6 u^{2}

Simplify

0 \cdot u : 0

0 \cdot u

Apply rule

0 \cdot a = 0

= 0

u + 2 - 6 u^{2} = 0

u + 2 - 6 u^{2} = 0

u + 2 - 6 u^{2} = 0

Solve

u + 2 - 6 u^{2} = 0 : u = - \frac{1}{2}, u = \frac{2}{3}

u + 2 - 6 u^{2} = 0

Write in the standard form

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

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

Solve with the quadratic formula

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

Quadratic Equation Formula:

For

a = - 6, b = 1, c = 2

u_{1, 2} = \frac{- 1 \pm 1 ^{2} - 4 ( - 6 ) \cdot 2}{2 ( - 6 )}

u_{1, 2} = \frac{- 1 \pm 1 ^{2} - 4 ( - 6 ) \cdot 2}{2 ( - 6 )}

1^{2} - 4 (- 6) \cdot 2 = 7

1^{2} - 4 (- 6) \cdot 2

Apply rule

1^{a} = 1

1^{2} = 1

= 1 - 4 (- 6) \cdot 2

Apply rule

- (- a) = a

= 1 + 4 \cdot 6 \cdot 2

Multiply the numbers:

4 \cdot 6 \cdot 2 = 48

= 1 + 48

Add the numbers:

1 + 48 = 49

= 49

Factor the number:

49 = 7^{2}

= 7^{2}

Apply radical rule:

7^{2} = 7

= 7

u_{1, 2} = \frac{- 1 \pm 7}{2 ( - 6 )}

Separate the solutions

u_{1} = \frac{- 1 + 7}{2 ( - 6 )}, u_{2} = \frac{- 1 - 7}{2 ( - 6 )}

u = \frac{- 1 + 7}{2 ( - 6 )} : - \frac{1}{2}

\frac{- 1 + 7}{2 ( - 6 )}

Remove parentheses:

(- a) = - a

= \frac{- 1 + 7}{- 2 \cdot 6}

Add/Subtract the numbers:

- 1 + 7 = 6

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

Multiply the numbers:

2 \cdot 6 = 12

= \frac{6}{- 12}

Apply the fraction rule:

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

= - \frac{6}{12}

Cancel the common factor:

6

= - \frac{1}{2}

u = \frac{- 1 - 7}{2 ( - 6 )} : \frac{2}{3}

\frac{- 1 - 7}{2 ( - 6 )}

Remove parentheses:

(- a) = - a

= \frac{- 1 - 7}{- 2 \cdot 6}

Subtract the numbers:

- 1 - 7 = - 8

= \frac{- 8}{- 2 \cdot 6}

Multiply the numbers:

2 \cdot 6 = 12

= \frac{- 8}{- 12}

Apply the fraction rule:

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

= \frac{8}{12}

Cancel the common factor:

4

= \frac{2}{3}

The solutions to the quadratic equation are:

u = - \frac{1}{2}, u = \frac{2}{3}

u = - \frac{1}{2}, u = \frac{2}{3}

Verify Solutions

Find undefined (singularity) points:

u = 0

Take the denominator(s) of

1 + \frac{2 - 6 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 = \frac{2}{3}

Substitute back

u = cos (x)

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

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

cos (x) = - \frac{1}{2} : x = \frac{2 π}{3} + 2 πn, x = \frac{4 π}{3} + 2 πn

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

General solutions for

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

cos (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} cos (x) 1 \frac{3}{2} \frac{2}{2} \frac{1}{2} 0 - \frac{1}{2} - \frac{2}{2} - \frac{3}{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} cos (x) - 1 - \frac{3}{2} - \frac{2}{2} - \frac{1}{2} 0 \frac{1}{2} \frac{2}{2} \frac{3}{2}

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

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

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

Apply trig inverse properties

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

General solutions for

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

cos (x) = a \Rightarrow x = arccos (a) + 2 πn, x = 2 π - arccos (a) + 2 πn

x = arccos (\frac{2}{3}) + 2 πn, x = 2 π - arccos (\frac{2}{3}) + 2 πn

x = arccos (\frac{2}{3}) + 2 πn, x = 2 π - arccos (\frac{2}{3}) + 2 πn

Combine all the solutions

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

Show solutions in decimal form

x = \frac{2 π}{3} + 2 πn, x = \frac{4 π}{3} + 2 πn, x = 0.84106 \dots + 2 πn, x = 2 π - 0.84106 \dots + 2 πn

Graph

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