What is the general solution for 4cos(x)+4sin(x)tan(x)=8 ?

The general solution for 4cos(x)+4sin(x)tan(x)=8 is x= pi/3+2pin,x=(5pi)/3+2pin

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4cos(x)+4sin(x)tan(x)=8

Solution

4 cos (x) + 4 sin (x) tan (x) = 8

Solution

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

Degrees

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

Solution steps

4 cos (x) + 4 sin (x) tan (x) = 8

Subtract

8

from both sides

4 cos (x) + 4 sin (x) tan (x) - 8 = 0

Rewrite using trig identities

- 8 + 4 cos (x) + 4 sin (x) tan (x)

Use the basic trigonometric identity:

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

= - 8 + 4 cos (x) + 4 sin (x) \frac{sin ( x )}{cos ( x )}

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

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

Multiply fractions:

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

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

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

sin (x) \cdot 4 sin (x)

Apply exponent rule:

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

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

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

Add the numbers:

1 + 1 = 2

= 4 sin^{2} (x)

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

= - 8 + 4 cos (x) + \frac{4 sin ^{2} ( x )}{cos ( x )}

Use the Pythagorean identity:

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

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

= - 8 + \frac{4 ( 1 - cos ^{2} ( x ) )}{cos ( x )} + 4 cos (x)

Combine the fractions

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

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

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

4 (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)

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

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

Expand

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

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

Expand

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

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

Apply the distributive law:

a (b - c) = ab - a c

a = 4, b = 1, c = cos^{2} (x)

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

Multiply the numbers:

4 \cdot 1 = 4

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

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

Add similar elements:

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

= 4

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

= \frac{4}{cos ( x )} - 8

- 8 + \frac{4}{cos ( x )} = 0

- 8 + \frac{4}{cos ( x )} = 0

Multiply both sides by

cos (x)

- 8 + \frac{4}{cos ( x )} = 0

Multiply both sides by

cos (x)

- 8 cos (x) + \frac{4}{cos ( x )} cos (x) = 0 \cdot cos (x)

Simplify

- 8 cos (x) + \frac{4}{cos ( x )} cos (x) = 0 \cdot cos (x)

Simplify

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

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

Multiply fractions:

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

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

Cancel the common factor:

cos (x)

= 4

Simplify

0 \cdot cos (x) : 0

0 \cdot cos (x)

Apply rule

0 \cdot a = 0

= 0

- 8 cos (x) + 4 = 0

- 8 cos (x) + 4 = 0

- 8 cos (x) + 4 = 0

Move

4

to the right side

- 8 cos (x) + 4 = 0

Subtract

4

from both sides

- 8 cos (x) + 4 - 4 = 0 - 4

Simplify

- 8 cos (x) = - 4

- 8 cos (x) = - 4

Divide both sides by

- 8

- 8 cos (x) = - 4

Divide both sides by

- 8

\frac{- 8 cos ( x )}{- 8} = \frac{- 4}{- 8}

Simplify

\frac{- 8 cos ( x )}{- 8} = \frac{- 4}{- 8}

Simplify

\frac{- 8 cos ( x )}{- 8} : cos (x)

\frac{- 8 cos ( x )}{- 8}

Apply the fraction rule:

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

= \frac{8 cos ( x )}{8}

Divide the numbers:

\frac{8}{8} = 1

= cos (x)

Simplify

\frac{- 4}{- 8} : \frac{1}{2}

\frac{- 4}{- 8}

Apply the fraction rule:

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

= \frac{4}{8}

Cancel the common factor:

4

= \frac{1}{2}

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

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

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

Verify Solutions

Find undefined (singularity) points:

cos (x) = 0

Take the denominator(s) of

- 8 + \frac{4}{cos ( x )}

and compare to zero

cos (x) = 0

The following points are undefined

cos (x) = 0

Combine undefined points with solutions:

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

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

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Frequently Asked Questions (FAQ)

What is the general solution for 4cos(x)+4sin(x)tan(x)=8 ?
The general solution for 4cos(x)+4sin(x)tan(x)=8 is x= pi/3+2pin,x=(5pi)/3+2pin