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cos(76)sec(8x+4)=1

Solution

cos (7 6^{\circ}) sec (8 x + 4) = 1

Solution

x = - \frac{1}{2} + \frac{18 0 ^{\circ} n}{4} + \frac{1.32645 \dots}{8}, x = - \frac{1}{2} + 4 5^{\circ} + \frac{18 0 ^{\circ} n}{4} - \frac{1.32645 \dots}{8}

Radians

x = - \frac{1}{2} + \frac{1.32645 \dots}{8} + \frac{π}{4} n, x = - \frac{1}{2} + \frac{π}{4} - \frac{1.32645 \dots}{8} + \frac{π}{4} n

Solution steps

cos (7 6^{\circ}) sec (8 x + 4) = 1

Divide both sides by

cos (7 6^{\circ})

cos (7 6^{\circ}) sec (8 x + 4) = 1

Divide both sides by

cos (7 6^{\circ})

\frac{cos ( 7 6 ^{\circ} ) sec ( 8 x + 4 )}{cos ( 7 6 ^{\circ} )} = \frac{1}{cos ( 7 6 ^{\circ} )}

Simplify

sec (8 x + 4) = \frac{1}{cos ( 7 6 ^{\circ} )}

sec (8 x + 4) = \frac{1}{cos ( 7 6 ^{\circ} )}

Apply trig inverse properties

sec (8 x + 4) = \frac{1}{cos ( 7 6 ^{\circ} )}

General solutions for

sec (8 x + 4) = \frac{1}{cos ( 7 6 ^{\circ} )}

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

8 x + 4 = arcsec (\frac{1}{cos ( 7 6 ^{\circ} )}) + 36 0^{\circ} n, 8 x + 4 = 36 0^{\circ} - arcsec (\frac{1}{cos ( 7 6 ^{\circ} )}) + 36 0^{\circ} n

8 x + 4 = arcsec (\frac{1}{cos ( 7 6 ^{\circ} )}) + 36 0^{\circ} n, 8 x + 4 = 36 0^{\circ} - arcsec (\frac{1}{cos ( 7 6 ^{\circ} )}) + 36 0^{\circ} n

Solve

8 x + 4 = arcsec (\frac{1}{cos ( 7 6 ^{\circ} )}) + 36 0^{\circ} n : x = - \frac{1}{2} + \frac{18 0 ^{\circ} n}{4} + \frac{arcsec ( \frac{1}{c o s ( 7 6 ^{\circ} )} )}{8}

8 x + 4 = arcsec (\frac{1}{cos ( 7 6 ^{\circ} )}) + 36 0^{\circ} n

Move

4

to the right side

8 x + 4 = arcsec (\frac{1}{cos ( 7 6 ^{\circ} )}) + 36 0^{\circ} n

Subtract

4

from both sides

8 x + 4 - 4 = arcsec (\frac{1}{cos ( 7 6 ^{\circ} )}) + 36 0^{\circ} n - 4

Simplify

8 x = arcsec (\frac{1}{cos ( 7 6 ^{\circ} )}) + 36 0^{\circ} n - 4

8 x = arcsec (\frac{1}{cos ( 7 6 ^{\circ} )}) + 36 0^{\circ} n - 4

Divide both sides by

8

8 x = arcsec (\frac{1}{cos ( 7 6 ^{\circ} )}) + 36 0^{\circ} n - 4

Divide both sides by

8

\frac{8 x}{8} = \frac{arcsec ( \frac{1}{c o s ( 7 6 ^{\circ} )} )}{8} + \frac{36 0 ^{\circ} n}{8} - \frac{4}{8}

Simplify

\frac{8 x}{8} = \frac{arcsec ( \frac{1}{c o s ( 7 6 ^{\circ} )} )}{8} + \frac{36 0 ^{\circ} n}{8} - \frac{4}{8}

Simplify

\frac{8 x}{8} : x

\frac{8 x}{8}

Divide the numbers:

\frac{8}{8} = 1

= x

Simplify

\frac{arcsec ( \frac{1}{c o s ( 7 6 ^{\circ} )} )}{8} + \frac{36 0 ^{\circ} n}{8} - \frac{4}{8} : - \frac{1}{2} + \frac{18 0 ^{\circ} n}{4} + \frac{arcsec ( \frac{1}{c o s ( 7 6 ^{\circ} )} )}{8}

\frac{arcsec ( \frac{1}{c o s ( 7 6 ^{\circ} )} )}{8} + \frac{36 0 ^{\circ} n}{8} - \frac{4}{8}

Group like terms

= - \frac{4}{8} + \frac{36 0 ^{\circ} n}{8} + \frac{arcsec ( \frac{1}{c o s ( 7 6 ^{\circ} )} )}{8}

Cancel

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

\frac{4}{8}

Cancel the common factor:

4

= \frac{1}{2}

= - \frac{1}{2} + \frac{36 0 ^{\circ} n}{8} + \frac{arcsec ( \frac{1}{c o s ( 7 6 ^{\circ} )} )}{8}

Cancel

\frac{36 0 ^{\circ} n}{8} : \frac{18 0 ^{\circ} n}{4}

\frac{36 0 ^{\circ} n}{8}

Cancel the common factor:

2

= \frac{18 0 ^{\circ} n}{4}

= - \frac{1}{2} + \frac{18 0 ^{\circ} n}{4} + \frac{arcsec ( \frac{1}{c o s ( 7 6 ^{\circ} )} )}{8}

x = - \frac{1}{2} + \frac{18 0 ^{\circ} n}{4} + \frac{arcsec ( \frac{1}{c o s ( 7 6 ^{\circ} )} )}{8}

x = - \frac{1}{2} + \frac{18 0 ^{\circ} n}{4} + \frac{arcsec ( \frac{1}{c o s ( 7 6 ^{\circ} )} )}{8}

x = - \frac{1}{2} + \frac{18 0 ^{\circ} n}{4} + \frac{arcsec ( \frac{1}{c o s ( 7 6 ^{\circ} )} )}{8}

Solve

8 x + 4 = 36 0^{\circ} - arcsec (\frac{1}{cos ( 7 6 ^{\circ} )}) + 36 0^{\circ} n : x = - \frac{1}{2} + 4 5^{\circ} + \frac{18 0 ^{\circ} n}{4} - \frac{arcsec ( \frac{1}{c o s ( 7 6 ^{\circ} )} )}{8}

8 x + 4 = 36 0^{\circ} - arcsec (\frac{1}{cos ( 7 6 ^{\circ} )}) + 36 0^{\circ} n

Move

4

to the right side

8 x + 4 = 36 0^{\circ} - arcsec (\frac{1}{cos ( 7 6 ^{\circ} )}) + 36 0^{\circ} n

Subtract

4

from both sides

8 x + 4 - 4 = 36 0^{\circ} - arcsec (\frac{1}{cos ( 7 6 ^{\circ} )}) + 36 0^{\circ} n - 4

Simplify

8 x = 36 0^{\circ} - arcsec (\frac{1}{cos ( 7 6 ^{\circ} )}) + 36 0^{\circ} n - 4

8 x = 36 0^{\circ} - arcsec (\frac{1}{cos ( 7 6 ^{\circ} )}) + 36 0^{\circ} n - 4

Divide both sides by

8

8 x = 36 0^{\circ} - arcsec (\frac{1}{cos ( 7 6 ^{\circ} )}) + 36 0^{\circ} n - 4

Divide both sides by

8

\frac{8 x}{8} = 4 5^{\circ} - \frac{arcsec ( \frac{1}{c o s ( 7 6 ^{\circ} )} )}{8} + \frac{36 0 ^{\circ} n}{8} - \frac{4}{8}

Simplify

\frac{8 x}{8} = 4 5^{\circ} - \frac{arcsec ( \frac{1}{c o s ( 7 6 ^{\circ} )} )}{8} + \frac{36 0 ^{\circ} n}{8} - \frac{4}{8}

Simplify

\frac{8 x}{8} : x

\frac{8 x}{8}

Divide the numbers:

\frac{8}{8} = 1

= x

Simplify

4 5^{\circ} - \frac{arcsec ( \frac{1}{c o s ( 7 6 ^{\circ} )} )}{8} + \frac{36 0 ^{\circ} n}{8} - \frac{4}{8} : - \frac{1}{2} + 4 5^{\circ} + \frac{18 0 ^{\circ} n}{4} - \frac{arcsec ( \frac{1}{c o s ( 7 6 ^{\circ} )} )}{8}

4 5^{\circ} - \frac{arcsec ( \frac{1}{c o s ( 7 6 ^{\circ} )} )}{8} + \frac{36 0 ^{\circ} n}{8} - \frac{4}{8}

Group like terms

= - \frac{4}{8} + 4 5^{\circ} + \frac{36 0 ^{\circ} n}{8} - \frac{arcsec ( \frac{1}{c o s ( 7 6 ^{\circ} )} )}{8}

Cancel

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

\frac{4}{8}

Cancel the common factor:

4

= \frac{1}{2}

= - \frac{1}{2} + 4 5^{\circ} + \frac{36 0 ^{\circ} n}{8} - \frac{arcsec ( \frac{1}{c o s ( 7 6 ^{\circ} )} )}{8}

Cancel

4 5^{\circ} : 4 5^{\circ}

4 5^{\circ}

Cancel the common factor:

2

= 4 5^{\circ}

= - \frac{1}{2} + 4 5^{\circ} + \frac{36 0 ^{\circ} n}{8} - \frac{arcsec ( \frac{1}{c o s ( 7 6 ^{\circ} )} )}{8}

Cancel

\frac{36 0 ^{\circ} n}{8} : \frac{18 0 ^{\circ} n}{4}

\frac{36 0 ^{\circ} n}{8}

Cancel the common factor:

2

= \frac{18 0 ^{\circ} n}{4}

= - \frac{1}{2} + 4 5^{\circ} + \frac{18 0 ^{\circ} n}{4} - \frac{arcsec ( \frac{1}{c o s ( 7 6 ^{\circ} )} )}{8}

x = - \frac{1}{2} + 4 5^{\circ} + \frac{18 0 ^{\circ} n}{4} - \frac{arcsec ( \frac{1}{c o s ( 7 6 ^{\circ} )} )}{8}

x = - \frac{1}{2} + 4 5^{\circ} + \frac{18 0 ^{\circ} n}{4} - \frac{arcsec ( \frac{1}{c o s ( 7 6 ^{\circ} )} )}{8}

x = - \frac{1}{2} + 4 5^{\circ} + \frac{18 0 ^{\circ} n}{4} - \frac{arcsec ( \frac{1}{c o s ( 7 6 ^{\circ} )} )}{8}

x = - \frac{1}{2} + \frac{18 0 ^{\circ} n}{4} + \frac{arcsec ( \frac{1}{c o s ( 7 6 ^{\circ} )} )}{8}, x = - \frac{1}{2} + 4 5^{\circ} + \frac{18 0 ^{\circ} n}{4} - \frac{arcsec ( \frac{1}{c o s ( 7 6 ^{\circ} )} )}{8}

Show solutions in decimal form

x = - \frac{1}{2} + \frac{18 0 ^{\circ} n}{4} + \frac{1.32645 \dots}{8}, x = - \frac{1}{2} + 4 5^{\circ} + \frac{18 0 ^{\circ} n}{4} - \frac{1.32645 \dots}{8}

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