What is the general solution for 8tan(x/2)+8cos(x)tan(x/2)=1,0<x<360 ?

The general solution for 8tan(x/2)+8cos(x)tan(x/2)=1,0<x<360 is x=0.12532…,x=180-0.12532…

English

Español

Português

Français

Deutsch

Italiano

Русский

中文(简体)

한국어

日本語

Tiếng Việt

עברית

العربية

Popular >

8tan(x/2)+8cos(x)tan(x/2)=1,0<x<360

Solution

8 tan (\frac{x}{2}) + 8 cos (x) tan (\frac{x}{2}) = 1, 0^{\circ} < x < 36 0^{\circ}

Solution

x = 0.12532 \dots, x = 18 0^{\circ} - 0.12532 \dots

Radians

x = 0.12532 \dots, x = π - 0.12532 \dots

Solution steps

8 tan (\frac{x}{2}) + 8 cos (x) tan (\frac{x}{2}) = 1, 0^{\circ} < x < 36 0^{\circ}

Subtract

1

from both sides

8 tan (\frac{x}{2}) + 8 cos (x) tan (\frac{x}{2}) - 1 = 0

Let:

u = \frac{x}{2}

8 tan (u) + 8 cos (2 u) tan (u) - 1 = 0, 0 < u < 18 0^{\circ}

Rewrite using trig identities

- 1 + 8 tan (u) + 8 cos (2 u) tan (u)

Use the Double Angle identity:

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

= - 1 + 8 tan (u) + 8 (2 cos^{2} (u) - 1) tan (u)

Simplify

- 1 + 8 tan (u) + 8 (2 cos^{2} (u) - 1) tan (u) : 16 cos^{2} (u) tan (u) - 1

- 1 + 8 tan (u) + 8 (2 cos^{2} (u) - 1) tan (u)

= - 1 + 8 tan (u) + 8 tan (u) (2 cos^{2} (u) - 1)

Expand

8 tan (u) (2 cos^{2} (u) - 1) : 16 cos^{2} (u) tan (u) - 8 tan (u)

8 tan (u) (2 cos^{2} (u) - 1)

Apply the distributive law:

a (b - c) = ab - a c

a = 8 tan (u), b = 2 cos^{2} (u), c = 1

= 8 tan (u) \cdot 2 cos^{2} (u) - 8 tan (u) \cdot 1

= 8 \cdot 2 cos^{2} (u) tan (u) - 8 \cdot 1 \cdot tan (u)

Simplify

8 \cdot 2 cos^{2} (u) tan (u) - 8 \cdot 1 \cdot tan (u) : 16 cos^{2} (u) tan (u) - 8 tan (u)

8 \cdot 2 cos^{2} (u) tan (u) - 8 \cdot 1 \cdot tan (u)

Multiply the numbers:

8 \cdot 2 = 16

= 16 cos^{2} (u) tan (u) - 8 \cdot 1 \cdot tan (u)

Multiply the numbers:

8 \cdot 1 = 8

= 16 cos^{2} (u) tan (u) - 8 tan (u)

= 16 cos^{2} (u) tan (u) - 8 tan (u)

= - 1 + 8 tan (u) + 16 cos^{2} (u) tan (u) - 8 tan (u)

Simplify

- 1 + 8 tan (u) + 16 cos^{2} (u) tan (u) - 8 tan (u) : 16 cos^{2} (u) tan (u) - 1

- 1 + 8 tan (u) + 16 cos^{2} (u) tan (u) - 8 tan (u)

Group like terms

= 8 tan (u) + 16 cos^{2} (u) tan (u) - 8 tan (u) - 1

Add similar elements:

8 tan (u) - 8 tan (u) = 0

= 16 cos^{2} (u) tan (u) - 1

= 16 cos^{2} (u) tan (u) - 1

= 16 cos^{2} (u) tan (u) - 1

Use the basic trigonometric identity:

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

= - 1 + 16 cos^{2} (u) \frac{sin ( u )}{cos ( u )}

16 cos^{2} (u) \frac{sin ( u )}{cos ( u )} = 16 sin (u) cos (u)

16 cos^{2} (u) \frac{sin ( u )}{cos ( u )}

Multiply fractions:

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

= \frac{sin ( u ) \cdot 16 cos ^{2} ( u )}{cos ( u )}

Cancel the common factor:

cos (u)

= 16 sin (u) cos (u)

= - 1 + 16 sin (u) cos (u)

Use the Double Angle identity:

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

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

= - 1 + 16 \cdot \frac{sin ( 2 u )}{2}

- 1 + 16 \cdot \frac{sin ( 2 u )}{2} = 0

16 \cdot \frac{sin ( 2 u )}{2} = 8 sin (2 u)

16 \cdot \frac{sin ( 2 u )}{2}

Multiply fractions:

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

= \frac{sin ( 2 u ) \cdot 16}{2}

Divide the numbers:

\frac{16}{2} = 8

= 8 sin (2 u)

- 1 + 8 sin (2 u) = 0

Move

1

to the right side

- 1 + 8 sin (2 u) = 0

Add

1

to both sides

- 1 + 8 sin (2 u) + 1 = 0 + 1

Simplify

8 sin (2 u) = 1

8 sin (2 u) = 1

Divide both sides by

8

8 sin (2 u) = 1

Divide both sides by

8

\frac{8 sin ( 2 u )}{8} = \frac{1}{8}

Simplify

sin (2 u) = \frac{1}{8}

sin (2 u) = \frac{1}{8}

Apply trig inverse properties

sin (2 u) = \frac{1}{8}

General solutions for

sin (2 u) = \frac{1}{8}

sin (x) = a \Rightarrow x = arcsin (a) + 36 0^{\circ} n, x = 18 0^{\circ} - arcsin (a) + 36 0^{\circ} n

2 u = arcsin (\frac{1}{8}) + 36 0^{\circ} n, 2 u = 18 0^{\circ} - arcsin (\frac{1}{8}) + 36 0^{\circ} n

2 u = arcsin (\frac{1}{8}) + 36 0^{\circ} n, 2 u = 18 0^{\circ} - arcsin (\frac{1}{8}) + 36 0^{\circ} n

Solve

2 u = arcsin (\frac{1}{8}) + 36 0^{\circ} n : u = \frac{arcsin ( \frac{1}{8} )}{2} + 18 0^{\circ} n

2 u = arcsin (\frac{1}{8}) + 36 0^{\circ} n

Divide both sides by

2

2 u = arcsin (\frac{1}{8}) + 36 0^{\circ} n

Divide both sides by

2

\frac{2 u}{2} = \frac{arcsin ( \frac{1}{8} )}{2} + \frac{36 0 ^{\circ} n}{2}

Simplify

u = \frac{arcsin ( \frac{1}{8} )}{2} + 18 0^{\circ} n

u = \frac{arcsin ( \frac{1}{8} )}{2} + 18 0^{\circ} n

Solve

2 u = 18 0^{\circ} - arcsin (\frac{1}{8}) + 36 0^{\circ} n : u = 9 0^{\circ} - \frac{arcsin ( \frac{1}{8} )}{2} + 18 0^{\circ} n

2 u = 18 0^{\circ} - arcsin (\frac{1}{8}) + 36 0^{\circ} n

Divide both sides by

2

2 u = 18 0^{\circ} - arcsin (\frac{1}{8}) + 36 0^{\circ} n

Divide both sides by

2

\frac{2 u}{2} = 9 0^{\circ} - \frac{arcsin ( \frac{1}{8} )}{2} + \frac{36 0 ^{\circ} n}{2}

Simplify

u = 9 0^{\circ} - \frac{arcsin ( \frac{1}{8} )}{2} + 18 0^{\circ} n

u = 9 0^{\circ} - \frac{arcsin ( \frac{1}{8} )}{2} + 18 0^{\circ} n

u = \frac{arcsin ( \frac{1}{8} )}{2} + 18 0^{\circ} n, u = 9 0^{\circ} - \frac{arcsin ( \frac{1}{8} )}{2} + 18 0^{\circ} n

Solutions for the range

0 < u < 18 0^{\circ}

u = \frac{arcsin ( \frac{1}{8} )}{2}, u = \frac{18 0 ^{\circ} - arcsin ( \frac{1}{8} )}{2}

Substitute back

u = \frac{x}{2}

\frac{x}{2} = \frac{arcsin ( \frac{1}{8} )}{2} : x = arcsin (\frac{1}{8})

\frac{x}{2} = \frac{arcsin ( \frac{1}{8} )}{2}

Multiply both sides by

2

\frac{x}{2} = \frac{arcsin ( \frac{1}{8} )}{2}

Multiply both sides by

2

\frac{2 x}{2} = \frac{2 arcsin ( \frac{1}{8} )}{2}

Simplify

x = arcsin (\frac{1}{8})

x = arcsin (\frac{1}{8})

\frac{x}{2} = \frac{18 0 ^{\circ} - arcsin ( \frac{1}{8} )}{2} : x = 18 0^{\circ} - arcsin (\frac{1}{8})

\frac{x}{2} = \frac{18 0 ^{\circ} - arcsin ( \frac{1}{8} )}{2}

Multiply both sides by

2

\frac{x}{2} = \frac{18 0 ^{\circ} - arcsin ( \frac{1}{8} )}{2}

Multiply both sides by

2

\frac{2 x}{2} = \frac{2 ( 18 0 ^{\circ} - arcsin ( \frac{1}{8} ) )}{2}

Simplify

x = 18 0^{\circ} - arcsin (\frac{1}{8})

x = 18 0^{\circ} - arcsin (\frac{1}{8})

x = arcsin (\frac{1}{8}), x = 18 0^{\circ} - arcsin (\frac{1}{8})

Show solutions in decimal form

x = 0.12532 \dots, x = 18 0^{\circ} - 0.12532 \dots

Popular Examples

cos(θ)=-4/5 ,tan(θ)<0

cos (θ) = - \frac{4}{5}, tan (θ) < 0

16sin^2(x)=8

16 sin^{2} (x) = 8

-2cot(x)=0

- 2 cot (x) = 0

cos(θ/2)=0

cos (\frac{θ}{2}) = 0

cos(x)=((sqrt(2)))/2

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

Frequently Asked Questions (FAQ)

What is the general solution for 8tan(x/2)+8cos(x)tan(x/2)=1,0<x<360 ?
The general solution for 8tan(x/2)+8cos(x)tan(x/2)=1,0<x<360 is x=0.12532…,x=180-0.12532…