What is the general solution for 2sin(x)*tan(x)+5sin(x)=-2cos(x) ?

The general solution for 2sin(x)*tan(x)+5sin(x)=-2cos(x) is x=-0.46364…+pin,x=-1.10714…+pin

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2sin(x)*tan(x)+5sin(x)=-2cos(x)

Solution

2 sin (x) \cdot tan (x) + 5 sin (x) = - 2 cos (x)

Solution

x = - 0.46364 \dots + πn, x = - 1.10714 \dots + πn

Degrees

x = - 26.56505 \dots^{\circ} + 18 0^{\circ} n, x = - 63.43494 \dots^{\circ} + 18 0^{\circ} n

Solution steps

2 sin (x) tan (x) + 5 sin (x) = - 2 cos (x)

Subtract

- 2 cos (x)

from both sides

2 sin (x) tan (x) + 5 sin (x) + 2 cos (x) = 0

Express with sin, cos

2 cos (x) + 5 sin (x) + 2 sin (x) tan (x)

Use the basic trigonometric identity:

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

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

Simplify

2 cos (x) + 5 sin (x) + 2 sin (x) \frac{sin ( x )}{cos ( x )} : \frac{2 cos ^{2} ( x ) + 5 sin ( x ) cos ( x ) + 2 sin ^{2} ( x )}{cos ( x )}

2 cos (x) + 5 sin (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 )}

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

Convert element to fraction:

2 cos (x) = \frac{2 cos ( x ) cos ( x )}{cos ( x )}, 5 sin (x) = \frac{5 sin ( x ) cos ( x )}{cos ( x )}

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

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

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

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

2 cos (x) cos (x)

Apply exponent rule:

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

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

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

Add the numbers:

1 + 1 = 2

= 2 cos^{2} (x)

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

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

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

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

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

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

Factor

2 cos^{2} (x) + 2 sin^{2} (x) + 5 cos (x) sin (x) : (2 sin (x) + cos (x)) (sin (x) + 2 cos (x))

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

Break the expression into groups

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

Definition

Factors of

4 : 1, 2, 4

4

Divisors (Factors)

Find the Prime factors of

4 : 2, 2

4

4

divides by

24 = 2 \cdot 2

= 2 \cdot 2

2

is a prime number, therefore no further factorization is possible

= 2 \cdot 2

Add the prime factors:

2

Add 1 and the number

4

itself

1, 4

The factors of

4

1, 2, 4

For every two factors such that

u * v = 4,

check if

u + v = 5

Check

u = 1, v = 4 : u * v = 4, u + v = 5 \Rightarrow

TrueCheck

u = 2, v = 2 : u * v = 4, u + v = 4 \Rightarrow

False

u = 1, v = 4

Group into

(a x^{2} + ux y) + (vx y + c y^{2})

(2 sin^{2} (x) + sin (x) cos (x)) + (4 sin (x) cos (x) + 2 cos^{2} (x))

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

Factor out

sin (x)

from

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

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

Apply exponent rule:

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

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

= 2 sin (x) sin (x) + sin (x) cos (x)

Factor out common term

sin (x)

= sin (x) (2 sin (x) + cos (x))

Factor out

2 cos (x)

from

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

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

Apply exponent rule:

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

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

= 4 sin (x) cos (x) + 2 cos (x) cos (x)

Rewrite

4

2 \cdot 2

= 2 \cdot 2 sin (x) cos (x) + 2 cos (x) cos (x)

Factor out common term

2 cos (x)

= 2 cos (x) (2 sin (x) + cos (x))

= sin (x) (2 sin (x) + cos (x)) + 2 cos (x) (2 sin (x) + cos (x))

Factor out common term

2 sin (x) + cos (x)

= (2 sin (x) + cos (x)) (sin (x) + 2 cos (x))

(2 sin (x) + cos (x)) (sin (x) + 2 cos (x)) = 0

Solving each part separately

2 sin (x) + cos (x) = 0 or sin (x) + 2 cos (x) = 0

2 sin (x) + cos (x) = 0 : x = arctan (- \frac{1}{2}) + πn

2 sin (x) + cos (x) = 0

Rewrite using trig identities

2 sin (x) + cos (x) = 0

Divide both sides by

cos (x), cos (x) \neq = 0

\frac{2 sin ( x ) + cos ( x )}{cos ( x )} = \frac{0}{cos ( x )}

Simplify

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

Use the basic trigonometric identity:

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

2 tan (x) + 1 = 0

2 tan (x) + 1 = 0

Move

1

to the right side

2 tan (x) + 1 = 0

Subtract

1

from both sides

2 tan (x) + 1 - 1 = 0 - 1

Simplify

2 tan (x) = - 1

2 tan (x) = - 1

Divide both sides by

2

2 tan (x) = - 1

Divide both sides by

2

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

Simplify

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

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

Apply trig inverse properties

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

General solutions for

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

tan (x) = - a \Rightarrow x = arctan (- a) + πn

x = arctan (- \frac{1}{2}) + πn

x = arctan (- \frac{1}{2}) + πn

sin (x) + 2 cos (x) = 0 : x = arctan (- 2) + πn

sin (x) + 2 cos (x) = 0

Rewrite using trig identities

sin (x) + 2 cos (x) = 0

Divide both sides by

cos (x), cos (x) \neq = 0

\frac{sin ( x ) + 2 cos ( x )}{cos ( x )} = \frac{0}{cos ( x )}

Simplify

\frac{sin ( x )}{cos ( x )} + 2 = 0

Use the basic trigonometric identity:

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

tan (x) + 2 = 0

tan (x) + 2 = 0

Move

2

to the right side

tan (x) + 2 = 0

Subtract

2

from both sides

tan (x) + 2 - 2 = 0 - 2

Simplify

tan (x) = - 2

tan (x) = - 2

Apply trig inverse properties

tan (x) = - 2

General solutions for

tan (x) = - 2

tan (x) = - a \Rightarrow x = arctan (- a) + πn

x = arctan (- 2) + πn

x = arctan (- 2) + πn

Combine all the solutions

x = arctan (- \frac{1}{2}) + πn, x = arctan (- 2) + πn

Show solutions in decimal form

x = - 0.46364 \dots + πn, x = - 1.10714 \dots + πn

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Frequently Asked Questions (FAQ)

What is the general solution for 2sin(x)*tan(x)+5sin(x)=-2cos(x) ?
The general solution for 2sin(x)*tan(x)+5sin(x)=-2cos(x) is x=-0.46364…+pin,x=-1.10714…+pin