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6sec^2(9x)-5=7

Solution

6 sec^{2} (9 x) - 5 = 7

Solution

x = \frac{π}{36} + \frac{2 πn}{9}, x = \frac{7 π}{36} + \frac{2 πn}{9}, x = \frac{π}{12} + \frac{2 πn}{9}, x = \frac{5 π}{36} + \frac{2 πn}{9}

Degrees

x = 5^{\circ} + 4 0^{\circ} n, x = 3 5^{\circ} + 4 0^{\circ} n, x = 1 5^{\circ} + 4 0^{\circ} n, x = 2 5^{\circ} + 4 0^{\circ} n

Solution steps

6 sec^{2} (9 x) - 5 = 7

Solve by substitution

6 sec^{2} (9 x) - 5 = 7

Let:

sec (9 x) = u

6 u^{2} - 5 = 7

6 u^{2} - 5 = 7 : u = 2, u = - 2

6 u^{2} - 5 = 7

Move

5

to the right side

6 u^{2} - 5 = 7

Add

5

to both sides

6 u^{2} - 5 + 5 = 7 + 5

Simplify

6 u^{2} = 12

6 u^{2} = 12

Divide both sides by

6

6 u^{2} = 12

Divide both sides by

6

\frac{6 u ^{2}}{6} = \frac{12}{6}

Simplify

u^{2} = 2

u^{2} = 2

For

x^{2} = f (a)

the solutions are

x = f (a), - f (a)

u = 2, u = - 2

Substitute back

u = sec (9 x)

sec (9 x) = 2, sec (9 x) = - 2

sec (9 x) = 2, sec (9 x) = - 2

sec (9 x) = 2 : x = \frac{π}{36} + \frac{2 πn}{9}, x = \frac{7 π}{36} + \frac{2 πn}{9}

sec (9 x) = 2

General solutions for

sec (9 x) = 2

sec (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} sec (x) 1 \frac{2 3}{3} 22 Undefined - 2 - 2 - \frac{2 3}{3} 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} sec (x) - 1 - \frac{2 3}{3} - 2 - 2 Undefined 22 \frac{2 3}{3}

9 x = \frac{π}{4} + 2 πn, 9 x = \frac{7 π}{4} + 2 πn

9 x = \frac{π}{4} + 2 πn, 9 x = \frac{7 π}{4} + 2 πn

Solve

9 x = \frac{π}{4} + 2 πn : x = \frac{π}{36} + \frac{2 πn}{9}

9 x = \frac{π}{4} + 2 πn

Divide both sides by

9

9 x = \frac{π}{4} + 2 πn

Divide both sides by

9

\frac{9 x}{9} = \frac{\frac{π}{4}}{9} + \frac{2 πn}{9}

Simplify

\frac{9 x}{9} = \frac{\frac{π}{4}}{9} + \frac{2 πn}{9}

Simplify

\frac{9 x}{9} : x

\frac{9 x}{9}

Divide the numbers:

\frac{9}{9} = 1

= x

Simplify

\frac{\frac{π}{4}}{9} + \frac{2 πn}{9} : \frac{π}{36} + \frac{2 πn}{9}

\frac{\frac{π}{4}}{9} + \frac{2 πn}{9}

\frac{\frac{π}{4}}{9} = \frac{π}{36}

\frac{\frac{π}{4}}{9}

Apply the fraction rule:

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

= \frac{π}{4 \cdot 9}

Multiply the numbers:

4 \cdot 9 = 36

= \frac{π}{36}

= \frac{π}{36} + \frac{2 πn}{9}

x = \frac{π}{36} + \frac{2 πn}{9}

x = \frac{π}{36} + \frac{2 πn}{9}

x = \frac{π}{36} + \frac{2 πn}{9}

Solve

9 x = \frac{7 π}{4} + 2 πn : x = \frac{7 π}{36} + \frac{2 πn}{9}

9 x = \frac{7 π}{4} + 2 πn

Divide both sides by

9

9 x = \frac{7 π}{4} + 2 πn

Divide both sides by

9

\frac{9 x}{9} = \frac{\frac{7 π}{4}}{9} + \frac{2 πn}{9}

Simplify

\frac{9 x}{9} = \frac{\frac{7 π}{4}}{9} + \frac{2 πn}{9}

Simplify

\frac{9 x}{9} : x

\frac{9 x}{9}

Divide the numbers:

\frac{9}{9} = 1

= x

Simplify

\frac{\frac{7 π}{4}}{9} + \frac{2 πn}{9} : \frac{7 π}{36} + \frac{2 πn}{9}

\frac{\frac{7 π}{4}}{9} + \frac{2 πn}{9}

\frac{\frac{7 π}{4}}{9} = \frac{7 π}{36}

\frac{\frac{7 π}{4}}{9}

Apply the fraction rule:

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

= \frac{7 π}{4 \cdot 9}

Multiply the numbers:

4 \cdot 9 = 36

= \frac{7 π}{36}

= \frac{7 π}{36} + \frac{2 πn}{9}

x = \frac{7 π}{36} + \frac{2 πn}{9}

x = \frac{7 π}{36} + \frac{2 πn}{9}

x = \frac{7 π}{36} + \frac{2 πn}{9}

x = \frac{π}{36} + \frac{2 πn}{9}, x = \frac{7 π}{36} + \frac{2 πn}{9}

sec (9 x) = - 2 : x = \frac{π}{12} + \frac{2 πn}{9}, x = \frac{5 π}{36} + \frac{2 πn}{9}

sec (9 x) = - 2

General solutions for

sec (9 x) = - 2

sec (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} sec (x) 1 \frac{2 3}{3} 22 Undefined - 2 - 2 - \frac{2 3}{3} 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} sec (x) - 1 - \frac{2 3}{3} - 2 - 2 Undefined 22 \frac{2 3}{3}

9 x = \frac{3 π}{4} + 2 πn, 9 x = \frac{5 π}{4} + 2 πn

9 x = \frac{3 π}{4} + 2 πn, 9 x = \frac{5 π}{4} + 2 πn

Solve

9 x = \frac{3 π}{4} + 2 πn : x = \frac{π}{12} + \frac{2 πn}{9}

9 x = \frac{3 π}{4} + 2 πn

Divide both sides by

9

9 x = \frac{3 π}{4} + 2 πn

Divide both sides by

9

\frac{9 x}{9} = \frac{\frac{3 π}{4}}{9} + \frac{2 πn}{9}

Simplify

\frac{9 x}{9} = \frac{\frac{3 π}{4}}{9} + \frac{2 πn}{9}

Simplify

\frac{9 x}{9} : x

\frac{9 x}{9}

Divide the numbers:

\frac{9}{9} = 1

= x

Simplify

\frac{\frac{3 π}{4}}{9} + \frac{2 πn}{9} : \frac{π}{12} + \frac{2 πn}{9}

\frac{\frac{3 π}{4}}{9} + \frac{2 πn}{9}

\frac{\frac{3 π}{4}}{9} = \frac{π}{12}

\frac{\frac{3 π}{4}}{9}

Apply the fraction rule:

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

= \frac{3 π}{4 \cdot 9}

Multiply the numbers:

4 \cdot 9 = 36

= \frac{3 π}{36}

Cancel the common factor:

3

= \frac{π}{12}

= \frac{π}{12} + \frac{2 πn}{9}

x = \frac{π}{12} + \frac{2 πn}{9}

x = \frac{π}{12} + \frac{2 πn}{9}

x = \frac{π}{12} + \frac{2 πn}{9}

Solve

9 x = \frac{5 π}{4} + 2 πn : x = \frac{5 π}{36} + \frac{2 πn}{9}

9 x = \frac{5 π}{4} + 2 πn

Divide both sides by

9

9 x = \frac{5 π}{4} + 2 πn

Divide both sides by

9

\frac{9 x}{9} = \frac{\frac{5 π}{4}}{9} + \frac{2 πn}{9}

Simplify

\frac{9 x}{9} = \frac{\frac{5 π}{4}}{9} + \frac{2 πn}{9}

Simplify

\frac{9 x}{9} : x

\frac{9 x}{9}

Divide the numbers:

\frac{9}{9} = 1

= x

Simplify

\frac{\frac{5 π}{4}}{9} + \frac{2 πn}{9} : \frac{5 π}{36} + \frac{2 πn}{9}

\frac{\frac{5 π}{4}}{9} + \frac{2 πn}{9}

\frac{\frac{5 π}{4}}{9} = \frac{5 π}{36}

\frac{\frac{5 π}{4}}{9}

Apply the fraction rule:

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

= \frac{5 π}{4 \cdot 9}

Multiply the numbers:

4 \cdot 9 = 36

= \frac{5 π}{36}

= \frac{5 π}{36} + \frac{2 πn}{9}

x = \frac{5 π}{36} + \frac{2 πn}{9}

x = \frac{5 π}{36} + \frac{2 πn}{9}

x = \frac{5 π}{36} + \frac{2 πn}{9}

x = \frac{π}{12} + \frac{2 πn}{9}, x = \frac{5 π}{36} + \frac{2 πn}{9}

Combine all the solutions

x = \frac{π}{36} + \frac{2 πn}{9}, x = \frac{7 π}{36} + \frac{2 πn}{9}, x = \frac{π}{12} + \frac{2 πn}{9}, x = \frac{5 π}{36} + \frac{2 πn}{9}

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