What is the general solution for csc(x)sec(x)=2sqrt(2) ?

The general solution for csc(x)sec(x)=2sqrt(2) is x= pi/8+pin,x=(3pi)/8+pin

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csc(x)sec(x)=2sqrt(2)

Solution

csc (x) sec (x) = 22

Solution

x = \frac{π}{8} + πn, x = \frac{3 π}{8} + πn

Degrees

x = 22. 5^{\circ} + 18 0^{\circ} n, x = 67. 5^{\circ} + 18 0^{\circ} n

Solution steps

csc (x) sec (x) = 22

Subtract

22

from both sides

csc (x) sec (x) - 22 = 0

Express with sin, cos

- 22 + csc (x) sec (x)

Use the basic trigonometric identity:

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

= - 22 + \frac{1}{sin ( x )} sec (x)

Use the basic trigonometric identity:

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

= - 22 + \frac{1}{sin ( x )} \cdot \frac{1}{cos ( x )}

Simplify

- 22 + \frac{1}{sin ( x )} \cdot \frac{1}{cos ( x )} : \frac{- 2 2 sin ( x ) cos ( x ) + 1}{sin ( x ) cos ( x )}

- 22 + \frac{1}{sin ( x )} \cdot \frac{1}{cos ( x )}

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

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

Multiply fractions:

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

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

Multiply the numbers:

1 \cdot 1 = 1

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

= - 22 + \frac{1}{sin ( x ) cos ( x )}

Convert element to fraction:

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

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

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

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

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

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

Rewrite using trig identities

1 - 2 cos (x) sin (x) 2

Use the Double Angle identity:

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

= 1 - sin (2 x) 2

1 - sin (2 x) 2 = 0

Move

1

to the right side

1 - sin (2 x) 2 = 0

Subtract

1

from both sides

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

Simplify

- sin (2 x) 2 = - 1

- sin (2 x) 2 = - 1

Divide both sides by

- 2

- sin (2 x) 2 = - 1

Divide both sides by

- 2

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

Simplify

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

Simplify

\frac{- sin ( 2 x ) 2}{- 2} : sin (2 x)

\frac{- sin ( 2 x ) 2}{- 2}

Apply the fraction rule:

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

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

Cancel the common factor:

2

= sin (2 x)

Simplify

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

\frac{- 1}{- 2}

Apply the fraction rule:

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

= \frac{1}{2}

Rationalize

\frac{1}{2} : \frac{2}{2}

\frac{1}{2}

Multiply by the conjugate

\frac{2}{2}

= \frac{1 \cdot 2}{2 2}

1 \cdot 2 = 2

22 = 2

22

Apply radical rule:

a a = a

22 = 2

= 2

= \frac{2}{2}

= \frac{2}{2}

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

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

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

General solutions for

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

sin (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} sin (x) 0 \frac{1}{2} \frac{2}{2} \frac{3}{2} 1 \frac{3}{2} \frac{2}{2} \frac{1}{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} sin (x) 0 - \frac{1}{2} - \frac{2}{2} - \frac{3}{2} - 1 - \frac{3}{2} - \frac{2}{2} - \frac{1}{2}

2 x = \frac{π}{4} + 2 πn, 2 x = \frac{3 π}{4} + 2 πn

2 x = \frac{π}{4} + 2 πn, 2 x = \frac{3 π}{4} + 2 πn

Solve

2 x = \frac{π}{4} + 2 πn : x = \frac{π}{8} + πn

2 x = \frac{π}{4} + 2 πn

Divide both sides by

2

2 x = \frac{π}{4} + 2 πn

Divide both sides by

2

\frac{2 x}{2} = \frac{\frac{π}{4}}{2} + \frac{2 πn}{2}

Simplify

\frac{2 x}{2} = \frac{\frac{π}{4}}{2} + \frac{2 πn}{2}

Simplify

\frac{2 x}{2} : x

\frac{2 x}{2}

Divide the numbers:

\frac{2}{2} = 1

= x

Simplify

\frac{\frac{π}{4}}{2} + \frac{2 πn}{2} : \frac{π}{8} + πn

\frac{\frac{π}{4}}{2} + \frac{2 πn}{2}

\frac{\frac{π}{4}}{2} = \frac{π}{8}

\frac{\frac{π}{4}}{2}

Apply the fraction rule:

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

= \frac{π}{4 \cdot 2}

Multiply the numbers:

4 \cdot 2 = 8

= \frac{π}{8}

\frac{2 πn}{2} = πn

\frac{2 πn}{2}

Divide the numbers:

\frac{2}{2} = 1

= πn

= \frac{π}{8} + πn

x = \frac{π}{8} + πn

x = \frac{π}{8} + πn

x = \frac{π}{8} + πn

Solve

2 x = \frac{3 π}{4} + 2 πn : x = \frac{3 π}{8} + πn

2 x = \frac{3 π}{4} + 2 πn

Divide both sides by

2

2 x = \frac{3 π}{4} + 2 πn

Divide both sides by

2

\frac{2 x}{2} = \frac{\frac{3 π}{4}}{2} + \frac{2 πn}{2}

Simplify

\frac{2 x}{2} = \frac{\frac{3 π}{4}}{2} + \frac{2 πn}{2}

Simplify

\frac{2 x}{2} : x

\frac{2 x}{2}

Divide the numbers:

\frac{2}{2} = 1

= x

Simplify

\frac{\frac{3 π}{4}}{2} + \frac{2 πn}{2} : \frac{3 π}{8} + πn

\frac{\frac{3 π}{4}}{2} + \frac{2 πn}{2}

\frac{\frac{3 π}{4}}{2} = \frac{3 π}{8}

\frac{\frac{3 π}{4}}{2}

Apply the fraction rule:

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

= \frac{3 π}{4 \cdot 2}

Multiply the numbers:

4 \cdot 2 = 8

= \frac{3 π}{8}

\frac{2 πn}{2} = πn

\frac{2 πn}{2}

Divide the numbers:

\frac{2}{2} = 1

= πn

= \frac{3 π}{8} + πn

x = \frac{3 π}{8} + πn

x = \frac{3 π}{8} + πn

x = \frac{3 π}{8} + πn

x = \frac{π}{8} + πn, x = \frac{3 π}{8} + πn

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Frequently Asked Questions (FAQ)

What is the general solution for csc(x)sec(x)=2sqrt(2) ?
The general solution for csc(x)sec(x)=2sqrt(2) is x= pi/8+pin,x=(3pi)/8+pin