What is the general solution for tan(x)*sin(x)=5219 ?

The general solution for tan(x)*sin(x)=5219 is x=1.57060…+2pin,x=2pi-1.57060…+2pin

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tan(x)*sin(x)=5219

Solution

tan (x) \cdot sin (x) = 5219

Solution

x = 1.57060 \dots + 2 πn, x = 2 π - 1.57060 \dots + 2 πn

Degrees

x = 89.98902 \dots^{\circ} + 36 0^{\circ} n, x = 270.01097 \dots^{\circ} + 36 0^{\circ} n

Solution steps

tan (x) sin (x) = 5219

Subtract

5219

from both sides

tan (x) sin (x) - 5219 = 0

Express with sin, cos

\frac{sin ( x )}{cos ( x )} sin (x) - 5219 = 0

Simplify

\frac{sin ( x )}{cos ( x )} sin (x) - 5219 : \frac{sin ^{2} ( x ) - 5219 cos ( x )}{cos ( x )}

\frac{sin ( x )}{cos ( x )} sin (x) - 5219

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

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

Multiply fractions:

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

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

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

sin (x) sin (x)

Apply exponent rule:

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

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

= sin^{1 + 1} (x)

Add the numbers:

1 + 1 = 2

= sin^{2} (x)

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

= \frac{sin ^{2} ( x )}{cos ( x )} - 5219

Convert element to fraction:

5219 = \frac{5219 cos ( x )}{cos ( x )}

= \frac{sin ^{2} ( x )}{cos ( x )} - \frac{5219 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{sin ^{2} ( x ) - 5219 cos ( x )}{cos ( x )}

\frac{sin ^{2} ( x ) - 5219 cos ( x )}{cos ( x )} = 0

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

sin^{2} (x) - 5219 cos (x) = 0

Add

5219 cos (x)

to both sides

sin^{2} (x) = 5219 cos (x)

Square both sides

(sin^{2} (x))^{2} = (5219 cos (x))^{2}

Subtract

(5219 cos (x))^{2}

from both sides

sin^{4} (x) - 27237961 cos^{2} (x) = 0

Factor

sin^{4} (x) - 27237961 cos^{2} (x) : (sin^{2} (x) + 5219 cos (x)) (sin^{2} (x) - 5219 cos (x))

sin^{4} (x) - 27237961 cos^{2} (x)

Rewrite

sin^{4} (x) - 27237961 cos^{2} (x)

(sin^{2} (x))^{2} - (5219 cos (x))^{2}

sin^{4} (x) - 27237961 cos^{2} (x)

Rewrite

27237961

521 9^{2}

= sin^{4} (x) - 521 9^{2} cos^{2} (x)

Apply exponent rule:

a^{b c} = (a^{b})^{c}

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

= (sin^{2} (x))^{2} - 521 9^{2} cos^{2} (x)

Apply exponent rule:

a^{m} b^{m} = (ab)^{m}

521 9^{2} cos^{2} (x) = (5219 cos (x))^{2}

= (sin^{2} (x))^{2} - (5219 cos (x))^{2}

= (sin^{2} (x))^{2} - (5219 cos (x))^{2}

Apply Difference of Two Squares Formula:

x^{2} - y^{2} = (x + y) (x - y)

(sin^{2} (x))^{2} - (5219 cos (x))^{2} = (sin^{2} (x) + 5219 cos (x)) (sin^{2} (x) - 5219 cos (x))

= (sin^{2} (x) + 5219 cos (x)) (sin^{2} (x) - 5219 cos (x))

(sin^{2} (x) + 5219 cos (x)) (sin^{2} (x) - 5219 cos (x)) = 0

Solving each part separately

sin^{2} (x) + 5219 cos (x) = 0 or sin^{2} (x) - 5219 cos (x) = 0

sin^{2} (x) + 5219 cos (x) = 0 : x = arccos (- \frac{- 5219 + 27237965}{2}) + 2 πn, x = - arccos (- \frac{- 5219 + 27237965}{2}) + 2 πn

sin^{2} (x) + 5219 cos (x) = 0

Rewrite using trig identities

sin^{2} (x) + 5219 cos (x)

Use the Pythagorean identity:

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

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

= 1 - cos^{2} (x) + 5219 cos (x)

1 - cos^{2} (x) + 5219 cos (x) = 0

Solve by substitution

1 - cos^{2} (x) + 5219 cos (x) = 0

Let:

cos (x) = u

1 - u^{2} + 5219 u = 0

1 - u^{2} + 5219 u = 0 : u = - \frac{- 5219 + 27237965}{2}, u = \frac{5219 + 27237965}{2}

1 - u^{2} + 5219 u = 0

Write in the standard form

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

- u^{2} + 5219 u + 1 = 0

Solve with the quadratic formula

- u^{2} + 5219 u + 1 = 0

Quadratic Equation Formula:

For

a = - 1, b = 5219, c = 1

u_{1, 2} = \frac{- 5219 \pm 521 9 ^{2} - 4 ( - 1 ) \cdot 1}{2 ( - 1 )}

u_{1, 2} = \frac{- 5219 \pm 521 9 ^{2} - 4 ( - 1 ) \cdot 1}{2 ( - 1 )}

521 9^{2} - 4 (- 1) \cdot 1 = 27237965

521 9^{2} - 4 (- 1) \cdot 1

Apply rule

- (- a) = a

= 521 9^{2} + 4 \cdot 1 \cdot 1

Multiply the numbers:

4 \cdot 1 \cdot 1 = 4

= 521 9^{2} + 4

521 9^{2} = 27237961

= 27237961 + 4

Add the numbers:

27237961 + 4 = 27237965

= 27237965

u_{1, 2} = \frac{- 5219 \pm 27237965}{2 ( - 1 )}

Separate the solutions

u_{1} = \frac{- 5219 + 27237965}{2 ( - 1 )}, u_{2} = \frac{- 5219 - 27237965}{2 ( - 1 )}

u = \frac{- 5219 + 27237965}{2 ( - 1 )} : - \frac{- 5219 + 27237965}{2}

\frac{- 5219 + 27237965}{2 ( - 1 )}

Remove parentheses:

(- a) = - a

= \frac{- 5219 + 27237965}{- 2 \cdot 1}

Multiply the numbers:

2 \cdot 1 = 2

= \frac{- 5219 + 27237965}{- 2}

Apply the fraction rule:

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

= - \frac{- 5219 + 27237965}{2}

u = \frac{- 5219 - 27237965}{2 ( - 1 )} : \frac{5219 + 27237965}{2}

\frac{- 5219 - 27237965}{2 ( - 1 )}

Remove parentheses:

(- a) = - a

= \frac{- 5219 - 27237965}{- 2 \cdot 1}

Multiply the numbers:

2 \cdot 1 = 2

= \frac{- 5219 - 27237965}{- 2}

Apply the fraction rule:

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

- 5219 - 27237965 = - (5219 + 27237965)

= \frac{5219 + 27237965}{2}

The solutions to the quadratic equation are:

u = - \frac{- 5219 + 27237965}{2}, u = \frac{5219 + 27237965}{2}

Substitute back

u = cos (x)

cos (x) = - \frac{- 5219 + 27237965}{2}, cos (x) = \frac{5219 + 27237965}{2}

cos (x) = - \frac{- 5219 + 27237965}{2}, cos (x) = \frac{5219 + 27237965}{2}

cos (x) = - \frac{- 5219 + 27237965}{2} : x = arccos (- \frac{- 5219 + 27237965}{2}) + 2 πn, x = - arccos (- \frac{- 5219 + 27237965}{2}) + 2 πn

cos (x) = - \frac{- 5219 + 27237965}{2}

Apply trig inverse properties

cos (x) = - \frac{- 5219 + 27237965}{2}

General solutions for

cos (x) = - \frac{- 5219 + 27237965}{2}

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

x = arccos (- \frac{- 5219 + 27237965}{2}) + 2 πn, x = - arccos (- \frac{- 5219 + 27237965}{2}) + 2 πn

x = arccos (- \frac{- 5219 + 27237965}{2}) + 2 πn, x = - arccos (- \frac{- 5219 + 27237965}{2}) + 2 πn

cos (x) = \frac{5219 + 27237965}{2} :

No Solution

cos (x) = \frac{5219 + 27237965}{2}

- 1 \leq cos (x) \leq 1

No Solution

Combine all the solutions

x = arccos (- \frac{- 5219 + 27237965}{2}) + 2 πn, x = - arccos (- \frac{- 5219 + 27237965}{2}) + 2 πn

sin^{2} (x) - 5219 cos (x) = 0 : x = arccos (\frac{27237965 - 5219}{2}) + 2 πn, x = 2 π - arccos (\frac{27237965 - 5219}{2}) + 2 πn

sin^{2} (x) - 5219 cos (x) = 0

Rewrite using trig identities

sin^{2} (x) - 5219 cos (x)

Use the Pythagorean identity:

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

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

= 1 - cos^{2} (x) - 5219 cos (x)

1 - cos^{2} (x) - 5219 cos (x) = 0

Solve by substitution

1 - cos^{2} (x) - 5219 cos (x) = 0

Let:

cos (x) = u

1 - u^{2} - 5219 u = 0

1 - u^{2} - 5219 u = 0 : u = - \frac{5219 + 27237965}{2}, u = \frac{27237965 - 5219}{2}

1 - u^{2} - 5219 u = 0

Write in the standard form

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

- u^{2} - 5219 u + 1 = 0

Solve with the quadratic formula

- u^{2} - 5219 u + 1 = 0

Quadratic Equation Formula:

For

a = - 1, b = - 5219, c = 1

u_{1, 2} = \frac{- ( - 5219 ) \pm ( - 5219 ) ^{2} - 4 ( - 1 ) \cdot 1}{2 ( - 1 )}

u_{1, 2} = \frac{- ( - 5219 ) \pm ( - 5219 ) ^{2} - 4 ( - 1 ) \cdot 1}{2 ( - 1 )}

(- 5219)^{2} - 4 (- 1) \cdot 1 = 27237965

(- 5219)^{2} - 4 (- 1) \cdot 1

Apply rule

- (- a) = a

= (- 5219)^{2} + 4 \cdot 1 \cdot 1

Apply exponent rule:

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

n

is even

(- 5219)^{2} = 521 9^{2}

= 521 9^{2} + 4 \cdot 1 \cdot 1

Multiply the numbers:

4 \cdot 1 \cdot 1 = 4

= 521 9^{2} + 4

521 9^{2} = 27237961

= 27237961 + 4

Add the numbers:

27237961 + 4 = 27237965

= 27237965

u_{1, 2} = \frac{- ( - 5219 ) \pm 27237965}{2 ( - 1 )}

Separate the solutions

u_{1} = \frac{- ( - 5219 ) + 27237965}{2 ( - 1 )}, u_{2} = \frac{- ( - 5219 ) - 27237965}{2 ( - 1 )}

u = \frac{- ( - 5219 ) + 27237965}{2 ( - 1 )} : - \frac{5219 + 27237965}{2}

\frac{- ( - 5219 ) + 27237965}{2 ( - 1 )}

Remove parentheses:

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

= \frac{5219 + 27237965}{- 2 \cdot 1}

Multiply the numbers:

2 \cdot 1 = 2

= \frac{5219 + 27237965}{- 2}

Apply the fraction rule:

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

= - \frac{5219 + 27237965}{2}

u = \frac{- ( - 5219 ) - 27237965}{2 ( - 1 )} : \frac{27237965 - 5219}{2}

\frac{- ( - 5219 ) - 27237965}{2 ( - 1 )}

Remove parentheses:

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

= \frac{5219 - 27237965}{- 2 \cdot 1}

Multiply the numbers:

2 \cdot 1 = 2

= \frac{5219 - 27237965}{- 2}

Apply the fraction rule:

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

5219 - 27237965 = - (27237965 - 5219)

= \frac{27237965 - 5219}{2}

The solutions to the quadratic equation are:

u = - \frac{5219 + 27237965}{2}, u = \frac{27237965 - 5219}{2}

Substitute back

u = cos (x)

cos (x) = - \frac{5219 + 27237965}{2}, cos (x) = \frac{27237965 - 5219}{2}

cos (x) = - \frac{5219 + 27237965}{2}, cos (x) = \frac{27237965 - 5219}{2}

cos (x) = - \frac{5219 + 27237965}{2} :

No Solution

cos (x) = - \frac{5219 + 27237965}{2}

- 1 \leq cos (x) \leq 1

No Solution

cos (x) = \frac{27237965 - 5219}{2} : x = arccos (\frac{27237965 - 5219}{2}) + 2 πn, x = 2 π - arccos (\frac{27237965 - 5219}{2}) + 2 πn

cos (x) = \frac{27237965 - 5219}{2}

Apply trig inverse properties

cos (x) = \frac{27237965 - 5219}{2}

General solutions for

cos (x) = \frac{27237965 - 5219}{2}

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

x = arccos (\frac{27237965 - 5219}{2}) + 2 πn, x = 2 π - arccos (\frac{27237965 - 5219}{2}) + 2 πn

x = arccos (\frac{27237965 - 5219}{2}) + 2 πn, x = 2 π - arccos (\frac{27237965 - 5219}{2}) + 2 πn

Combine all the solutions

x = arccos (\frac{27237965 - 5219}{2}) + 2 πn, x = 2 π - arccos (\frac{27237965 - 5219}{2}) + 2 πn

Combine all the solutions

x = arccos (- \frac{- 5219 + 27237965}{2}) + 2 πn, x = - arccos (- \frac{- 5219 + 27237965}{2}) + 2 πn, x = arccos (\frac{27237965 - 5219}{2}) + 2 πn, x = 2 π - arccos (\frac{27237965 - 5219}{2}) + 2 πn

Verify solutions by plugging them into the original equation

Check the solutions by plugging them into

tan (x) sin (x) = 5219

Remove the ones that don't agree with the equation.

Check the solution

arccos (- \frac{- 5219 + 27237965}{2}) + 2 πn :

False

arccos (- \frac{- 5219 + 27237965}{2}) + 2 πn

Plug in

n = 1

arccos (- \frac{- 5219 + 27237965}{2}) + 2 π 1

For

tan (x) sin (x) = 5219

plug in

x = arccos (- \frac{- 5219 + 27237965}{2}) + 2 π 1

tan (arccos (- \frac{- 5219 + 27237965}{2}) + 2 π 1) sin (arccos (- \frac{- 5219 + 27237965}{2}) + 2 π 1) = 5219

Refine

- 5218.99999 \dots = 5219

\Rightarrow False

Check the solution

- arccos (- \frac{- 5219 + 27237965}{2}) + 2 πn :

False

- arccos (- \frac{- 5219 + 27237965}{2}) + 2 πn

Plug in

n = 1

- arccos (- \frac{- 5219 + 27237965}{2}) + 2 π 1

For

tan (x) sin (x) = 5219

plug in

x = - arccos (- \frac{- 5219 + 27237965}{2}) + 2 π 1

tan (- arccos (- \frac{- 5219 + 27237965}{2}) + 2 π 1) sin (- arccos (- \frac{- 5219 + 27237965}{2}) + 2 π 1) = 5219

Refine

- 5218.99999 \dots = 5219

\Rightarrow False

Check the solution

arccos (\frac{27237965 - 5219}{2}) + 2 πn :

True

arccos (\frac{27237965 - 5219}{2}) + 2 πn

Plug in

n = 1

arccos (\frac{27237965 - 5219}{2}) + 2 π 1

For

tan (x) sin (x) = 5219

plug in

x = arccos (\frac{27237965 - 5219}{2}) + 2 π 1

tan (arccos (\frac{27237965 - 5219}{2}) + 2 π 1) sin (arccos (\frac{27237965 - 5219}{2}) + 2 π 1) = 5219

Refine

5218.99999 \dots = 5219

\Rightarrow True

Check the solution

2 π - arccos (\frac{27237965 - 5219}{2}) + 2 πn :

True

2 π - arccos (\frac{27237965 - 5219}{2}) + 2 πn

Plug in

n = 1

2 π - arccos (\frac{27237965 - 5219}{2}) + 2 π 1

For

tan (x) sin (x) = 5219

plug in

x = 2 π - arccos (\frac{27237965 - 5219}{2}) + 2 π 1

tan (2 π - arccos (\frac{27237965 - 5219}{2}) + 2 π 1) sin (2 π - arccos (\frac{27237965 - 5219}{2}) + 2 π 1) = 5219

Refine

5218.99999 \dots = 5219

\Rightarrow True

x = arccos (\frac{27237965 - 5219}{2}) + 2 πn, x = 2 π - arccos (\frac{27237965 - 5219}{2}) + 2 πn

Show solutions in decimal form

x = 1.57060 \dots + 2 πn, x = 2 π - 1.57060 \dots + 2 πn

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Frequently Asked Questions (FAQ)

What is the general solution for tan(x)*sin(x)=5219 ?
The general solution for tan(x)*sin(x)=5219 is x=1.57060…+2pin,x=2pi-1.57060…+2pin