What is the general solution for sec^2(x)cot(x)=8,0<x<pi ?

Q: What is the general solution for sec^2(x)cot(x)=8,0<x<pi ?

The general solution for sec^2(x)cot(x)=8,0<x<pi is x=(0.25268…)/2 ,x=(pi-0.25268…)/2

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sec^2(x)cot(x)=8,0<x<pi

Solution

sec^{2} (x) cot (x) = 8, 0 < x < π

Solution

x = \frac{0.25268 \dots}{2}, x = \frac{π - 0.25268 \dots}{2}

Degrees

x = 7.23875 \dots^{\circ}, x = 82.76124 \dots^{\circ}

Solution steps

sec^{2} (x) cot (x) = 8, 0 < x < π

Subtract

8

from both sides

sec^{2} (x) cot (x) - 8 = 0

Express with sin, cos

- 8 + cot (x) sec^{2} (x)

Use the basic trigonometric identity:

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

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

Use the basic trigonometric identity:

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

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

Simplify

- 8 + \frac{cos ( x )}{sin ( x )} (\frac{1}{cos ( x )})^{2} : \frac{- 8 cos ( x ) sin ( x ) + 1}{cos ( x ) sin ( x )}

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

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

\frac{cos ( x )}{sin ( x )} (\frac{1}{cos ( x )})^{2}

Multiply fractions:

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

= \frac{cos ( x ) ( \frac{1}{c o s ( x )} ) ^{2}}{sin ( x )}

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

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

Apply exponent rule:

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

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

Apply rule

1^{a} = 1

1^{2} = 1

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

= \frac{\frac{1}{c o s ^{2} ( x )} cos ( x )}{sin ( x )}

Multiply

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

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

Multiply fractions:

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

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

Multiply:

1 \cdot cos (x) = cos (x)

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

Cancel the common factor:

cos (x)

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

= \frac{\frac{1}{c o s ( x )}}{sin ( x )}

Apply the fraction rule:

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

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

= - 8 + \frac{1}{cos ( x ) sin ( x )}

Convert element to fraction:

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

= - \frac{8 cos ( x ) sin ( x )}{cos ( x ) sin ( x )} + \frac{1}{cos ( x ) sin ( x )}

Since the denominators are equal, combine the fractions:

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

= \frac{- 8 cos ( x ) sin ( x ) + 1}{cos ( x ) sin ( x )}

= \frac{- 8 cos ( x ) sin ( x ) + 1}{cos ( x ) sin ( x )}

\frac{1 - 8 cos ( x ) sin ( x )}{cos ( x ) sin ( x )} = 0

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

1 - 8 cos (x) sin (x) = 0

Rewrite using trig identities

1 - 8 cos (x) sin (x)

Use the Double Angle identity:

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

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

= 1 - 8 \cdot \frac{sin ( 2 x )}{2}

1 - 8 \cdot \frac{sin ( 2 x )}{2} = 0

8 \cdot \frac{sin ( 2 x )}{2} = 4 sin (2 x)

8 \cdot \frac{sin ( 2 x )}{2}

Multiply fractions:

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

= \frac{sin ( 2 x ) \cdot 8}{2}

Divide the numbers:

\frac{8}{2} = 4

= 4 sin (2 x)

1 - 4 sin (2 x) = 0

Move

1

to the right side

1 - 4 sin (2 x) = 0

Subtract

1

from both sides

1 - 4 sin (2 x) - 1 = 0 - 1

Simplify

- 4 sin (2 x) = - 1

- 4 sin (2 x) = - 1

Divide both sides by

- 4

- 4 sin (2 x) = - 1

Divide both sides by

- 4

\frac{- 4 sin ( 2 x )}{- 4} = \frac{- 1}{- 4}

Simplify

sin (2 x) = \frac{1}{4}

sin (2 x) = \frac{1}{4}

Apply trig inverse properties

sin (2 x) = \frac{1}{4}

General solutions for

sin (2 x) = \frac{1}{4}

sin (x) = a \Rightarrow x = arcsin (a) + 2 πn, x = π - arcsin (a) + 2 πn

2 x = arcsin (\frac{1}{4}) + 2 πn, 2 x = π - arcsin (\frac{1}{4}) + 2 πn

2 x = arcsin (\frac{1}{4}) + 2 πn, 2 x = π - arcsin (\frac{1}{4}) + 2 πn

Solve

2 x = arcsin (\frac{1}{4}) + 2 πn : x = \frac{arcsin ( \frac{1}{4} )}{2} + πn

2 x = arcsin (\frac{1}{4}) + 2 πn

Divide both sides by

2

2 x = arcsin (\frac{1}{4}) + 2 πn

Divide both sides by

2

\frac{2 x}{2} = \frac{arcsin ( \frac{1}{4} )}{2} + \frac{2 πn}{2}

Simplify

x = \frac{arcsin ( \frac{1}{4} )}{2} + πn

x = \frac{arcsin ( \frac{1}{4} )}{2} + πn

Solve

2 x = π - arcsin (\frac{1}{4}) + 2 πn : x = \frac{π}{2} - \frac{arcsin ( \frac{1}{4} )}{2} + πn

2 x = π - arcsin (\frac{1}{4}) + 2 πn

Divide both sides by

2

2 x = π - arcsin (\frac{1}{4}) + 2 πn

Divide both sides by

2

\frac{2 x}{2} = \frac{π}{2} - \frac{arcsin ( \frac{1}{4} )}{2} + \frac{2 πn}{2}

Simplify

x = \frac{π}{2} - \frac{arcsin ( \frac{1}{4} )}{2} + πn

x = \frac{π}{2} - \frac{arcsin ( \frac{1}{4} )}{2} + πn

x = \frac{arcsin ( \frac{1}{4} )}{2} + πn, x = \frac{π}{2} - \frac{arcsin ( \frac{1}{4} )}{2} + πn

Solutions for the range

0 < x < π

x = \frac{arcsin ( \frac{1}{4} )}{2}, x = \frac{π - arcsin ( \frac{1}{4} )}{2}

Show solutions in decimal form

x = \frac{0.25268 \dots}{2}, x = \frac{π - 0.25268 \dots}{2}

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Frequently Asked Questions (FAQ)

What is the general solution for sec^2(x)cot(x)=8,0<x<pi ?
The general solution for sec^2(x)cot(x)=8,0<x<pi is x=(0.25268…)/2 ,x=(pi-0.25268…)/2