What is the general solution for 8cos(x)=sec^2(x) ?

The general solution for 8cos(x)=sec^2(x) is x= pi/3+2pin,x=(5pi)/3+2pin

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8cos(x)=sec^2(x)

Solution

8 cos (x) = sec^{2} (x)

Solution

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

Degrees

x = 6 0^{\circ} + 36 0^{\circ} n, x = 30 0^{\circ} + 36 0^{\circ} n

Solution steps

8 cos (x) = sec^{2} (x)

Subtract

sec^{2} (x)

from both sides

8 cos (x) - sec^{2} (x) = 0

Rewrite using trig identities

- sec^{2} (x) + 8 cos (x)

Use the basic trigonometric identity:

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

= - sec^{2} (x) + 8 \cdot \frac{1}{sec ( x )}

8 \cdot \frac{1}{sec ( x )} = \frac{8}{sec ( x )}

8 \cdot \frac{1}{sec ( x )}

Multiply fractions:

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

= \frac{1 \cdot 8}{sec ( x )}

Multiply the numbers:

1 \cdot 8 = 8

= \frac{8}{sec ( x )}

= - sec^{2} (x) + \frac{8}{sec ( x )}

\frac{8}{sec ( x )} - sec^{2} (x) = 0

Solve by substitution

\frac{8}{sec ( x )} - sec^{2} (x) = 0

Let:

sec (x) = u

\frac{8}{u} - u^{2} = 0

\frac{8}{u} - u^{2} = 0 : u = 2, u = - 1 + 3 i, u = - 1 - 3 i

\frac{8}{u} - u^{2} = 0

Multiply both sides by

u

\frac{8}{u} - u^{2} = 0

Multiply both sides by

u

\frac{8}{u} u - u^{2} u = 0 \cdot u

Simplify

\frac{8}{u} u - u^{2} u = 0 \cdot u

Simplify

\frac{8}{u} u : 8

\frac{8}{u} u

Multiply fractions:

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

= \frac{8 u}{u}

Cancel the common factor:

u

= 8

Simplify

- u^{2} u : - u^{3}

- u^{2} u

Apply exponent rule:

a^{b} \cdot a^{c} = a^{b + c}

u^{2} u = u^{2 + 1}

= - u^{2 + 1}

Add the numbers:

2 + 1 = 3

= - u^{3}

Simplify

0 \cdot u : 0

0 \cdot u

Apply rule

0 \cdot a = 0

= 0

8 - u^{3} = 0

8 - u^{3} = 0

8 - u^{3} = 0

Solve

8 - u^{3} = 0 : u = 2, u = - 1 + 3 i, u = - 1 - 3 i

8 - u^{3} = 0

Move

8

to the right side

8 - u^{3} = 0

Subtract

8

from both sides

8 - u^{3} - 8 = 0 - 8

Simplify

- u^{3} = - 8

- u^{3} = - 8

Divide both sides by

- 1

- u^{3} = - 8

Divide both sides by

- 1

\frac{- u ^{3}}{- 1} = \frac{- 8}{- 1}

Simplify

u^{3} = 8

u^{3} = 8

For

x^{3} = f (a)

the solutions are

Factor the number:

8 = 2^{3}

Apply radical rule:

= 2

Simplify

Factor the number:

8 = 2^{3}

Apply radical rule:

= 2

= 2 \cdot \frac{- 1 + 3 i}{2}

Multiply fractions:

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

= \frac{( - 1 + 3 i ) \cdot 2}{2}

Cancel the common factor:

2

= - - 1 + 3 i

Simplify

Factor the number:

8 = 2^{3}

Apply radical rule:

= 2

= 2 \cdot \frac{- 1 - 3 i}{2}

Multiply fractions:

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

= \frac{( - 1 - 3 i ) \cdot 2}{2}

Cancel the common factor:

2

= - - 1 - 3 i

u = 2, u = - 1 + 3 i, u = - 1 - 3 i

u = 2, u = - 1 + 3 i, u = - 1 - 3 i

Verify Solutions

Find undefined (singularity) points:

u = 0

Take the denominator(s) of

\frac{8}{u} - u^{2}

and compare to zero

u = 0

The following points are undefined

u = 0

Combine undefined points with solutions:

u = 2, u = - 1 + 3 i, u = - 1 - 3 i

Substitute back

u = sec (x)

sec (x) = 2, sec (x) = - 1 + 3 i, sec (x) = - 1 - 3 i

sec (x) = 2, sec (x) = - 1 + 3 i, sec (x) = - 1 - 3 i

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

sec (x) = 2

General solutions for

sec (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}

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

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

sec (x) = - 1 + 3 i :

No Solution

sec (x) = - 1 + 3 i

No Solution

sec (x) = - 1 - 3 i :

No Solution

sec (x) = - 1 - 3 i

No Solution

Combine all the solutions

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

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Frequently Asked Questions (FAQ)

What is the general solution for 8cos(x)=sec^2(x) ?
The general solution for 8cos(x)=sec^2(x) is x= pi/3+2pin,x=(5pi)/3+2pin