What is the general solution for 6tan(x)-5csc(x)=0 ?

The general solution for 6tan(x)-5csc(x)=0 is x=0.84106…+2pin,x=2pi-0.84106…+2pin

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Popular >

6tan(x)-5csc(x)=0

Solution

6 tan (x) - 5 csc (x) = 0

Solution

x = 0.84106 \dots + 2 πn, x = 2 π - 0.84106 \dots + 2 πn

Degrees

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

Solution steps

6 tan (x) - 5 csc (x) = 0

Express with sin, cos

- 5 csc (x) + 6 tan (x)

Use the basic trigonometric identity:

csc (x) = \frac{1}{sin ( x )}

= - 5 \cdot \frac{1}{sin ( x )} + 6 tan (x)

Use the basic trigonometric identity:

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

= - 5 \cdot \frac{1}{sin ( x )} + 6 \cdot \frac{sin ( x )}{cos ( x )}

Simplify

- 5 \cdot \frac{1}{sin ( x )} + 6 \cdot \frac{sin ( x )}{cos ( x )} : \frac{- 5 cos ( x ) + 6 sin ^{2} ( x )}{sin ( x ) cos ( x )}

- 5 \cdot \frac{1}{sin ( x )} + 6 \cdot \frac{sin ( x )}{cos ( x )}

5 \cdot \frac{1}{sin ( x )} = \frac{5}{sin ( x )}

5 \cdot \frac{1}{sin ( x )}

Multiply fractions:

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

= \frac{1 \cdot 5}{sin ( x )}

Multiply the numbers:

1 \cdot 5 = 5

= \frac{5}{sin ( x )}

6 \cdot \frac{sin ( x )}{cos ( x )} = \frac{6 sin ( x )}{cos ( x )}

6 \cdot \frac{sin ( x )}{cos ( x )}

Multiply fractions:

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

= \frac{sin ( x ) \cdot 6}{cos ( x )}

= - \frac{5}{sin ( x )} + \frac{6 sin ( x )}{cos ( x )}

Least Common Multiplier of

sin (x), cos (x) : sin (x) cos (x)

sin (x), cos (x)

Lowest Common Multiplier (LCM)

Compute an expression comprised of factors that appear either in

sin (x)

cos (x)

= sin (x) cos (x)

Adjust Fractions based on the LCM

Multiply each numerator by the same amount needed to multiply its

corresponding denominator to turn it into the LCM

sin (x) cos (x)

For

\frac{5}{sin ( x )} :

multiply the denominator and numerator by

cos (x)

\frac{5}{sin ( x )} = \frac{5 cos ( x )}{sin ( x ) cos ( x )}

For

\frac{sin ( x ) \cdot 6}{cos ( x )} :

multiply the denominator and numerator by

sin (x)

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

= - \frac{5 cos ( x )}{sin ( x ) cos ( x )} + \frac{6 sin ^{2} ( x )}{sin ( x ) cos ( x )}

Since the denominators are equal, combine the fractions:

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

= \frac{- 5 cos ( x ) + 6 sin ^{2} ( x )}{sin ( x ) cos ( x )}

= \frac{- 5 cos ( x ) + 6 sin ^{2} ( x )}{sin ( x ) cos ( x )}

\frac{- 5 cos ( x ) + 6 sin ^{2} ( x )}{cos ( x ) sin ( x )} = 0

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

- 5 cos (x) + 6 sin^{2} (x) = 0

Rewrite using trig identities

- 5 cos (x) + 6 sin^{2} (x)

Use the Pythagorean identity:

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

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

= - 5 cos (x) + 6 (1 - cos^{2} (x))

(1 - cos^{2} (x)) \cdot 6 - 5 cos (x) = 0

Solve by substitution

(1 - cos^{2} (x)) \cdot 6 - 5 cos (x) = 0

Let:

cos (x) = u

(1 - u^{2}) \cdot 6 - 5 u = 0

(1 - u^{2}) \cdot 6 - 5 u = 0 : u = - \frac{3}{2}, u = \frac{2}{3}

(1 - u^{2}) \cdot 6 - 5 u = 0

Expand

(1 - u^{2}) \cdot 6 - 5 u : 6 - 6 u^{2} - 5 u

(1 - u^{2}) \cdot 6 - 5 u

= 6 (1 - u^{2}) - 5 u

Expand

6 (1 - u^{2}) : 6 - 6 u^{2}

6 (1 - u^{2})

Apply the distributive law:

a (b - c) = ab - a c

a = 6, b = 1, c = u^{2}

= 6 \cdot 1 - 6 u^{2}

Multiply the numbers:

6 \cdot 1 = 6

= 6 - 6 u^{2}

= 6 - 6 u^{2} - 5 u

6 - 6 u^{2} - 5 u = 0

Write in the standard form

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

- 6 u^{2} - 5 u + 6 = 0

Solve with the quadratic formula

- 6 u^{2} - 5 u + 6 = 0

Quadratic Equation Formula:

For

a = - 6, b = - 5, c = 6

u_{1, 2} = \frac{- ( - 5 ) \pm ( - 5 ) ^{2} - 4 ( - 6 ) \cdot 6}{2 ( - 6 )}

u_{1, 2} = \frac{- ( - 5 ) \pm ( - 5 ) ^{2} - 4 ( - 6 ) \cdot 6}{2 ( - 6 )}

(- 5)^{2} - 4 (- 6) \cdot 6 = 13

(- 5)^{2} - 4 (- 6) \cdot 6

Apply rule

- (- a) = a

= (- 5)^{2} + 4 \cdot 6 \cdot 6

Apply exponent rule:

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

n

is even

(- 5)^{2} = 5^{2}

= 5^{2} + 4 \cdot 6 \cdot 6

Multiply the numbers:

4 \cdot 6 \cdot 6 = 144

= 5^{2} + 144

5^{2} = 25

= 25 + 144

Add the numbers:

25 + 144 = 169

= 169

Factor the number:

169 = 1 3^{2}

= 1 3^{2}

Apply radical rule:

1 3^{2} = 13

= 13

u_{1, 2} = \frac{- ( - 5 ) \pm 13}{2 ( - 6 )}

Separate the solutions

u_{1} = \frac{- ( - 5 ) + 13}{2 ( - 6 )}, u_{2} = \frac{- ( - 5 ) - 13}{2 ( - 6 )}

u = \frac{- ( - 5 ) + 13}{2 ( - 6 )} : - \frac{3}{2}

\frac{- ( - 5 ) + 13}{2 ( - 6 )}

Remove parentheses:

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

= \frac{5 + 13}{- 2 \cdot 6}

Add the numbers:

5 + 13 = 18

= \frac{18}{- 2 \cdot 6}

Multiply the numbers:

2 \cdot 6 = 12

= \frac{18}{- 12}

Apply the fraction rule:

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

= - \frac{18}{12}

Cancel the common factor:

6

= - \frac{3}{2}

u = \frac{- ( - 5 ) - 13}{2 ( - 6 )} : \frac{2}{3}

\frac{- ( - 5 ) - 13}{2 ( - 6 )}

Remove parentheses:

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

= \frac{5 - 13}{- 2 \cdot 6}

Subtract the numbers:

5 - 13 = - 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{3}{2}, u = \frac{2}{3}

Substitute back

u = cos (x)

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

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

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

No Solution

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

- 1 \leq cos (x) \leq 1

No Solution

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

Show solutions in decimal form

x = 0.84106 \dots + 2 πn, x = 2 π - 0.84106 \dots + 2 πn

Popular Examples

2(19)tan(θ/2)=6.3

2 (19) tan (\frac{θ}{2}) = 6.3

3csc(x)-6=0,0<= x<= 360

3 csc (x) - 6 = 0, 0^{\circ} \leq x \leq 36 0^{\circ}

tan(x/2+pi/3)=-1

tan (\frac{x}{2} + \frac{π}{3}) = - 1

-sec^2(x)+1=-5sec(x)-5

- sec^{2} (x) + 1 = - 5 sec (x) - 5

sec(t)= 1/(-(sqrt(2))/2)

sec (t) = \frac{1}{- \frac{2}{2}}

Frequently Asked Questions (FAQ)

What is the general solution for 6tan(x)-5csc(x)=0 ?
The general solution for 6tan(x)-5csc(x)=0 is x=0.84106…+2pin,x=2pi-0.84106…+2pin