What is the general solution for sec(2x-10)=csc(50) ?

The general solution for sec(2x-10)=csc(50) is x=180n+5+(0.69813…)/2 ,x=180+180n+5-(0.69813…)/2

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sec(2x-10)=csc(50)

Solution

sec (2 x - 1 0^{\circ}) = csc (5 0^{\circ})

Solution

x = 18 0^{\circ} n + 5^{\circ} + \frac{0.69813 \dots}{2}, x = 18 0^{\circ} + 18 0^{\circ} n + 5^{\circ} - \frac{0.69813 \dots}{2}

Radians

x = \frac{π}{36} + \frac{0.69813 \dots}{2} + πn, x = π + \frac{π}{36} - \frac{0.69813 \dots}{2} + πn

Solution steps

sec (2 x - 1 0^{\circ}) = csc (5 0^{\circ})

Apply trig inverse properties

sec (2 x - 1 0^{\circ}) = csc (5 0^{\circ})

General solutions for

sec (2 x - 1 0^{\circ}) = csc (5 0^{\circ})

sec (x) = a \Rightarrow x = arcsec (a) + 36 0^{\circ} n, x = 36 0^{\circ} - arcsec (a) + 36 0^{\circ} n

2 x - 1 0^{\circ} = arcsec (csc (5 0^{\circ})) + 36 0^{\circ} n, 2 x - 1 0^{\circ} = 36 0^{\circ} - arcsec (csc (5 0^{\circ})) + 36 0^{\circ} n

2 x - 1 0^{\circ} = arcsec (csc (5 0^{\circ})) + 36 0^{\circ} n, 2 x - 1 0^{\circ} = 36 0^{\circ} - arcsec (csc (5 0^{\circ})) + 36 0^{\circ} n

Solve

2 x - 1 0^{\circ} = arcsec (csc (5 0^{\circ})) + 36 0^{\circ} n : x = 18 0^{\circ} n + 5^{\circ} + \frac{arcsec ( csc ( 5 0 ^{\circ} ) )}{2}

2 x - 1 0^{\circ} = arcsec (csc (5 0^{\circ})) + 36 0^{\circ} n

Move

1 0^{\circ}

to the right side

2 x - 1 0^{\circ} = arcsec (csc (5 0^{\circ})) + 36 0^{\circ} n

Add

1 0^{\circ}

to both sides

2 x - 1 0^{\circ} + 1 0^{\circ} = arcsec (csc (5 0^{\circ})) + 36 0^{\circ} n + 1 0^{\circ}

Simplify

2 x = arcsec (csc (5 0^{\circ})) + 36 0^{\circ} n + 1 0^{\circ}

2 x = arcsec (csc (5 0^{\circ})) + 36 0^{\circ} n + 1 0^{\circ}

Divide both sides by

2

2 x = arcsec (csc (5 0^{\circ})) + 36 0^{\circ} n + 1 0^{\circ}

Divide both sides by

2

\frac{2 x}{2} = \frac{arcsec ( csc ( 5 0 ^{\circ} ) )}{2} + \frac{36 0 ^{\circ} n}{2} + \frac{1 0 ^{\circ}}{2}

Simplify

\frac{2 x}{2} = \frac{arcsec ( csc ( 5 0 ^{\circ} ) )}{2} + \frac{36 0 ^{\circ} n}{2} + \frac{1 0 ^{\circ}}{2}

Simplify

\frac{2 x}{2} : x

\frac{2 x}{2}

Divide the numbers:

\frac{2}{2} = 1

= x

Simplify

\frac{arcsec ( csc ( 5 0 ^{\circ} ) )}{2} + \frac{36 0 ^{\circ} n}{2} + \frac{1 0 ^{\circ}}{2} : 18 0^{\circ} n + 5^{\circ} + \frac{arcsec ( csc ( 5 0 ^{\circ} ) )}{2}

\frac{arcsec ( csc ( 5 0 ^{\circ} ) )}{2} + \frac{36 0 ^{\circ} n}{2} + \frac{1 0 ^{\circ}}{2}

Group like terms

= \frac{36 0 ^{\circ} n}{2} + \frac{1 0 ^{\circ}}{2} + \frac{arcsec ( csc ( 5 0 ^{\circ} ) )}{2}

\frac{36 0 ^{\circ} n}{2} = 18 0^{\circ} n

\frac{36 0 ^{\circ} n}{2}

Divide the numbers:

\frac{2}{2} = 1

= 18 0^{\circ} n

\frac{1 0 ^{\circ}}{2} = 5^{\circ}

\frac{1 0 ^{\circ}}{2}

Apply the fraction rule:

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

= \frac{18 0 ^{\circ}}{18 \cdot 2}

Multiply the numbers:

18 \cdot 2 = 36

= 5^{\circ}

= 18 0^{\circ} n + 5^{\circ} + \frac{arcsec ( csc ( 5 0 ^{\circ} ) )}{2}

x = 18 0^{\circ} n + 5^{\circ} + \frac{arcsec ( csc ( 5 0 ^{\circ} ) )}{2}

x = 18 0^{\circ} n + 5^{\circ} + \frac{arcsec ( csc ( 5 0 ^{\circ} ) )}{2}

x = 18 0^{\circ} n + 5^{\circ} + \frac{arcsec ( csc ( 5 0 ^{\circ} ) )}{2}

Solve

2 x - 1 0^{\circ} = 36 0^{\circ} - arcsec (csc (5 0^{\circ})) + 36 0^{\circ} n : x = 18 0^{\circ} + 18 0^{\circ} n + 5^{\circ} - \frac{arcsec ( csc ( 5 0 ^{\circ} ) )}{2}

2 x - 1 0^{\circ} = 36 0^{\circ} - arcsec (csc (5 0^{\circ})) + 36 0^{\circ} n

Move

1 0^{\circ}

to the right side

2 x - 1 0^{\circ} = 36 0^{\circ} - arcsec (csc (5 0^{\circ})) + 36 0^{\circ} n

Add

1 0^{\circ}

to both sides

2 x - 1 0^{\circ} + 1 0^{\circ} = 36 0^{\circ} - arcsec (csc (5 0^{\circ})) + 36 0^{\circ} n + 1 0^{\circ}

Simplify

2 x = 36 0^{\circ} - arcsec (csc (5 0^{\circ})) + 36 0^{\circ} n + 1 0^{\circ}

2 x = 36 0^{\circ} - arcsec (csc (5 0^{\circ})) + 36 0^{\circ} n + 1 0^{\circ}

Divide both sides by

2

2 x = 36 0^{\circ} - arcsec (csc (5 0^{\circ})) + 36 0^{\circ} n + 1 0^{\circ}

Divide both sides by

2

\frac{2 x}{2} = 18 0^{\circ} - \frac{arcsec ( csc ( 5 0 ^{\circ} ) )}{2} + \frac{36 0 ^{\circ} n}{2} + \frac{1 0 ^{\circ}}{2}

Simplify

\frac{2 x}{2} = 18 0^{\circ} - \frac{arcsec ( csc ( 5 0 ^{\circ} ) )}{2} + \frac{36 0 ^{\circ} n}{2} + \frac{1 0 ^{\circ}}{2}

Simplify

\frac{2 x}{2} : x

\frac{2 x}{2}

Divide the numbers:

\frac{2}{2} = 1

= x

Simplify

18 0^{\circ} - \frac{arcsec ( csc ( 5 0 ^{\circ} ) )}{2} + \frac{36 0 ^{\circ} n}{2} + \frac{1 0 ^{\circ}}{2} : 18 0^{\circ} + 18 0^{\circ} n + 5^{\circ} - \frac{arcsec ( csc ( 5 0 ^{\circ} ) )}{2}

18 0^{\circ} - \frac{arcsec ( csc ( 5 0 ^{\circ} ) )}{2} + \frac{36 0 ^{\circ} n}{2} + \frac{1 0 ^{\circ}}{2}

Group like terms

= 18 0^{\circ} + \frac{36 0 ^{\circ} n}{2} + \frac{1 0 ^{\circ}}{2} - \frac{arcsec ( csc ( 5 0 ^{\circ} ) )}{2}

18 0^{\circ} = 18 0^{\circ}

18 0^{\circ}

Divide the numbers:

\frac{2}{2} = 1

= 18 0^{\circ}

\frac{36 0 ^{\circ} n}{2} = 18 0^{\circ} n

\frac{36 0 ^{\circ} n}{2}

Divide the numbers:

\frac{2}{2} = 1

= 18 0^{\circ} n

\frac{1 0 ^{\circ}}{2} = 5^{\circ}

\frac{1 0 ^{\circ}}{2}

Apply the fraction rule:

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

= \frac{18 0 ^{\circ}}{18 \cdot 2}

Multiply the numbers:

18 \cdot 2 = 36

= 5^{\circ}

= 18 0^{\circ} + 18 0^{\circ} n + 5^{\circ} - \frac{arcsec ( csc ( 5 0 ^{\circ} ) )}{2}

x = 18 0^{\circ} + 18 0^{\circ} n + 5^{\circ} - \frac{arcsec ( csc ( 5 0 ^{\circ} ) )}{2}

x = 18 0^{\circ} + 18 0^{\circ} n + 5^{\circ} - \frac{arcsec ( csc ( 5 0 ^{\circ} ) )}{2}

x = 18 0^{\circ} + 18 0^{\circ} n + 5^{\circ} - \frac{arcsec ( csc ( 5 0 ^{\circ} ) )}{2}

x = 18 0^{\circ} n + 5^{\circ} + \frac{arcsec ( csc ( 5 0 ^{\circ} ) )}{2}, x = 18 0^{\circ} + 18 0^{\circ} n + 5^{\circ} - \frac{arcsec ( csc ( 5 0 ^{\circ} ) )}{2}

Show solutions in decimal form

x = 18 0^{\circ} n + 5^{\circ} + \frac{0.69813 \dots}{2}, x = 18 0^{\circ} + 18 0^{\circ} n + 5^{\circ} - \frac{0.69813 \dots}{2}

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Frequently Asked Questions (FAQ)

What is the general solution for sec(2x-10)=csc(50) ?
The general solution for sec(2x-10)=csc(50) is x=180n+5+(0.69813…)/2 ,x=180+180n+5-(0.69813…)/2