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cos(7x-30)sec(x)=1

Solution

cos (7 x - 30) sec (x) = 1

Solution

x = \frac{2 πn}{3} + 5, x = \frac{π + 2 πn}{3} + 5, x = \frac{πn}{2} + \frac{15}{4}, x = \frac{π}{4} + \frac{15}{4} + \frac{πn}{2}

Degrees

x = 286.47889 \dots^{\circ} + 12 0^{\circ} n, x = 346.47889 \dots^{\circ} + 12 0^{\circ} n, x = 214.85917 \dots^{\circ} + 9 0^{\circ} n, x = 259.85917 \dots^{\circ} + 9 0^{\circ} n

Solution steps

cos (7 x - 30) sec (x) = 1

Subtract

1

from both sides

cos (7 x - 30) sec (x) - 1 = 0

Express with sin, cos

- 1 + cos (- 30 + 7 x) sec (x)

Use the basic trigonometric identity:

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

= - 1 + cos (- 30 + 7 x) \frac{1}{cos ( x )}

Simplify

- 1 + cos (- 30 + 7 x) \frac{1}{cos ( x )} : \frac{- cos ( x ) + cos ( - 30 + 7 x )}{cos ( x )}

- 1 + cos (- 30 + 7 x) \frac{1}{cos ( x )}

cos (- 30 + 7 x) \frac{1}{cos ( x )} = \frac{cos ( - 30 + 7 x )}{cos ( x )}

cos (- 30 + 7 x) \frac{1}{cos ( x )}

Multiply fractions:

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

= \frac{1 \cdot cos ( - 30 + 7 x )}{cos ( x )}

Multiply:

1 \cdot cos (- 30 + 7 x) = cos (- 30 + 7 x)

= \frac{cos ( - 30 + 7 x )}{cos ( x )}

= - 1 + \frac{cos ( 7 x - 30 )}{cos ( x )}

Convert element to fraction:

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

= - \frac{1 \cdot cos ( x )}{cos ( x )} + \frac{cos ( - 30 + 7 x )}{cos ( x )}

Since the denominators are equal, combine the fractions:

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

= \frac{- 1 \cdot cos ( x ) + cos ( - 30 + 7 x )}{cos ( x )}

Multiply:

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

= \frac{- cos ( x ) + cos ( 7 x - 30 )}{cos ( x )}

= \frac{- cos ( x ) + cos ( - 30 + 7 x )}{cos ( x )}

\frac{cos ( - 30 + 7 x ) - cos ( x )}{cos ( x )} = 0

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

cos (- 30 + 7 x) - cos (x) = 0

Rewrite using trig identities

cos (- 30 + 7 x) - cos (x)

Use the Sum to Product identity:

cos (s) - cos (t) = - 2 sin (\frac{s + t}{2}) sin (\frac{s - t}{2})

= - 2 sin (\frac{- 30 + 7 x + x}{2}) sin (\frac{- 30 + 7 x - x}{2})

Simplify

- 2 sin (\frac{- 30 + 7 x + x}{2}) sin (\frac{- 30 + 7 x - x}{2}) : - 2 sin (- 15 + 4 x) sin (3 (x - 5))

- 2 sin (\frac{- 30 + 7 x + x}{2}) sin (\frac{- 30 + 7 x - x}{2})

\frac{- 30 + 7 x + x}{2} = - 15 + 4 x

\frac{- 30 + 7 x + x}{2}

Add similar elements:

7 x + x = 8 x

= \frac{- 30 + 8 x}{2}

Factor

- 30 + 8 x : 2 (- 15 + 4 x)

- 30 + 8 x

Rewrite as

= - 2 \cdot 15 + 2 \cdot 4 x

Factor out common term

2

= 2 (- 15 + 4 x)

= \frac{2 ( - 15 + 4 x )}{2}

Divide the numbers:

\frac{2}{2} = 1

= - 15 + 4 x

= - 2 sin ((4 x - 15)) sin (\frac{7 x - x - 30}{2})

\frac{- 30 + 7 x - x}{2} = 3 (x - 5)

\frac{- 30 + 7 x - x}{2}

Add similar elements:

7 x - x = 6 x

= \frac{- 30 + 6 x}{2}

Factor

- 30 + 6 x : 6 (- 5 + x)

- 30 + 6 x

Rewrite as

= - 6 \cdot 5 + 6 x

Factor out common term

6

= 6 (- 5 + x)

= \frac{6 ( - 5 + x )}{2}

Divide the numbers:

\frac{6}{2} = 3

= 3 (x - 5)

= - 2 sin ((4 x - 15)) sin (3 (x - 5))

Remove parentheses:

(- a) = - a

= - 2 sin (- 15 + 4 x) sin (3 (x - 5))

= - 2 sin (- 15 + 4 x) sin (3 (x - 5))

- 2 sin ((- 5 + x) \cdot 3) sin (- 15 + 4 x) = 0

Solving each part separately

sin ((- 5 + x) \cdot 3) = 0 or sin (- 15 + 4 x) = 0

sin ((- 5 + x) \cdot 3) = 0 : x = \frac{2 πn}{3} + 5, x = \frac{π + 2 πn}{3} + 5

sin ((- 5 + x) \cdot 3) = 0

General solutions for

sin ((- 5 + x) 3) = 0

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}

(- 5 + x) \cdot 3 = 0 + 2 πn, (- 5 + x) \cdot 3 = π + 2 πn

(- 5 + x) \cdot 3 = 0 + 2 πn, (- 5 + x) \cdot 3 = π + 2 πn

Solve

(- 5 + x) 3 = 0 + 2 πn : x = \frac{2 πn}{3} + 5

(- 5 + x) \cdot 3 = 0 + 2 πn

0 + 2 πn = 2 πn

(- 5 + x) \cdot 3 = 2 πn

Divide both sides by

3

(- 5 + x) \cdot 3 = 2 πn

Divide both sides by

3

\frac{( - 5 + x ) \cdot 3}{3} = \frac{2 πn}{3}

Simplify

- 5 + x = \frac{2 πn}{3}

- 5 + x = \frac{2 πn}{3}

Move

5

to the right side

- 5 + x = \frac{2 πn}{3}

Add

5

to both sides

- 5 + x + 5 = \frac{2 πn}{3} + 5

Simplify

x = \frac{2 πn}{3} + 5

x = \frac{2 πn}{3} + 5

Solve

(- 5 + x) 3 = π + 2 πn : x = \frac{π + 2 πn}{3} + 5

(- 5 + x) \cdot 3 = π + 2 πn

Divide both sides by

3

(- 5 + x) \cdot 3 = π + 2 πn

Divide both sides by

3

\frac{( - 5 + x ) \cdot 3}{3} = \frac{π}{3} + \frac{2 πn}{3}

Simplify

- 5 + x = \frac{π}{3} + \frac{2 πn}{3}

- 5 + x = \frac{π}{3} + \frac{2 πn}{3}

Move

5

to the right side

- 5 + x = \frac{π}{3} + \frac{2 πn}{3}

Add

5

to both sides

- 5 + x + 5 = \frac{π}{3} + \frac{2 πn}{3} + 5

Simplify

- 5 + x + 5 = \frac{π}{3} + \frac{2 πn}{3} + 5

Simplify

- 5 + x + 5 : x

- 5 + x + 5

Add similar elements:

- 5 + 5 = 0

= x

Simplify

\frac{π}{3} + \frac{2 πn}{3} + 5 : \frac{π + 2 πn}{3} + 5

\frac{π}{3} + \frac{2 πn}{3} + 5

Combine the fractions

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

Apply rule

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

= \frac{π + 2 πn}{3}

= \frac{π + 2 πn}{3} + 5

x = \frac{π + 2 πn}{3} + 5

x = \frac{π + 2 πn}{3} + 5

x = \frac{π + 2 πn}{3} + 5

x = \frac{2 πn}{3} + 5, x = \frac{π + 2 πn}{3} + 5

sin (- 15 + 4 x) = 0 : x = \frac{πn}{2} + \frac{15}{4}, x = \frac{π}{4} + \frac{15}{4} + \frac{πn}{2}

sin (- 15 + 4 x) = 0

General solutions for

sin (- 15 + 4 x) = 0

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}

- 15 + 4 x = 0 + 2 πn, - 15 + 4 x = π + 2 πn

- 15 + 4 x = 0 + 2 πn, - 15 + 4 x = π + 2 πn

Solve

- 15 + 4 x = 0 + 2 πn : x = \frac{πn}{2} + \frac{15}{4}

- 15 + 4 x = 0 + 2 πn

0 + 2 πn = 2 πn

- 15 + 4 x = 2 πn

Move

15

to the right side

- 15 + 4 x = 2 πn

Add

15

to both sides

- 15 + 4 x + 15 = 2 πn + 15

Simplify

4 x = 2 πn + 15

4 x = 2 πn + 15

Divide both sides by

4

4 x = 2 πn + 15

Divide both sides by

4

\frac{4 x}{4} = \frac{2 πn}{4} + \frac{15}{4}

Simplify

x = \frac{πn}{2} + \frac{15}{4}

x = \frac{πn}{2} + \frac{15}{4}

Solve

- 15 + 4 x = π + 2 πn : x = \frac{π}{4} + \frac{15}{4} + \frac{πn}{2}

- 15 + 4 x = π + 2 πn

Move

15

to the right side

- 15 + 4 x = π + 2 πn

Add

15

to both sides

- 15 + 4 x + 15 = π + 2 πn + 15

Simplify

4 x = π + 2 πn + 15

4 x = π + 2 πn + 15

Divide both sides by

4

4 x = π + 2 πn + 15

Divide both sides by

4

\frac{4 x}{4} = \frac{π}{4} + \frac{2 πn}{4} + \frac{15}{4}

Simplify

\frac{4 x}{4} = \frac{π}{4} + \frac{2 πn}{4} + \frac{15}{4}

Simplify

\frac{4 x}{4} : x

\frac{4 x}{4}

Divide the numbers:

\frac{4}{4} = 1

= x

Simplify

\frac{π}{4} + \frac{2 πn}{4} + \frac{15}{4} : \frac{π}{4} + \frac{15}{4} + \frac{πn}{2}

\frac{π}{4} + \frac{2 πn}{4} + \frac{15}{4}

Group like terms

= \frac{π}{4} + \frac{15}{4} + \frac{2 πn}{4}

Cancel

\frac{2 πn}{4} : \frac{πn}{2}

\frac{2 πn}{4}

Cancel the common factor:

2

= \frac{πn}{2}

= \frac{π}{4} + \frac{15}{4} + \frac{πn}{2}

x = \frac{π}{4} + \frac{15}{4} + \frac{πn}{2}

x = \frac{π}{4} + \frac{15}{4} + \frac{πn}{2}

x = \frac{π}{4} + \frac{15}{4} + \frac{πn}{2}

x = \frac{πn}{2} + \frac{15}{4}, x = \frac{π}{4} + \frac{15}{4} + \frac{πn}{2}

Combine all the solutions

x = \frac{2 πn}{3} + 5, x = \frac{π + 2 πn}{3} + 5, x = \frac{πn}{2} + \frac{15}{4}, x = \frac{π}{4} + \frac{15}{4} + \frac{πn}{2}

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