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cos^2(x)+sec^2(x)=11

Solution

cos^{2} (x) + sec^{2} (x) = 11

Solution

x = 1.26319 \dots + 2 πn, x = 2 π - 1.26319 \dots + 2 πn, x = 1.87839 \dots + 2 πn, x = - 1.87839 \dots + 2 πn

Degrees

x = 72.37560 \dots^{\circ} + 36 0^{\circ} n, x = 287.62439 \dots^{\circ} + 36 0^{\circ} n, x = 107.62439 \dots^{\circ} + 36 0^{\circ} n, x = - 107.62439 \dots^{\circ} + 36 0^{\circ} n

Solution steps

cos^{2} (x) + sec^{2} (x) = 11

Subtract

11

from both sides

cos^{2} (x) + sec^{2} (x) - 11 = 0

Rewrite using trig identities

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

Use the basic trigonometric identity:

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

= - 11 + (\frac{1}{sec ( x )})^{2} + sec^{2} (x)

(\frac{1}{sec ( x )})^{2} = \frac{1}{sec ^{2} ( x )}

(\frac{1}{sec ( x )})^{2}

Apply exponent rule:

(\frac{a}{b})^{c} = \frac{a ^{c}}{b ^{c}}

= \frac{1 ^{2}}{sec ^{2} ( x )}

Apply rule

1^{a} = 1

1^{2} = 1

= \frac{1}{sec ^{2} ( x )}

= - 11 + \frac{1}{sec ^{2} ( x )} + sec^{2} (x)

- 11 + \frac{1}{sec ^{2} ( x )} + sec^{2} (x) = 0

Solve by substitution

- 11 + \frac{1}{sec ^{2} ( x )} + sec^{2} (x) = 0

Let:

sec (x) = u

- 11 + \frac{1}{u ^{2}} + u^{2} = 0

- 11 + \frac{1}{u ^{2}} + u^{2} = 0 : u = \frac{11 + 3 13}{2}, u = - \frac{11 + 3 13}{2}, u = \frac{11 - 3 13}{2}, u = - \frac{11 - 3 13}{2}

- 11 + \frac{1}{u ^{2}} + u^{2} = 0

Multiply both sides by

u^{2}

- 11 + \frac{1}{u ^{2}} + u^{2} = 0

Multiply both sides by

u^{2}

- 11 u^{2} + \frac{1}{u ^{2}} u^{2} + u^{2} u^{2} = 0 \cdot u^{2}

Simplify

- 11 u^{2} + \frac{1}{u ^{2}} u^{2} + u^{2} u^{2} = 0 \cdot u^{2}

Simplify

\frac{1}{u ^{2}} u^{2} : 1

\frac{1}{u ^{2}} u^{2}

Multiply fractions:

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

= \frac{1 \cdot u ^{2}}{u ^{2}}

Cancel the common factor:

u^{2}

= 1

Simplify

u^{2} u^{2} : u^{4}

u^{2} u^{2}

Apply exponent rule:

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

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

= u^{2 + 2}

Add the numbers:

2 + 2 = 4

= u^{4}

Simplify

0 \cdot u^{2} : 0

0 \cdot u^{2}

Apply rule

0 \cdot a = 0

= 0

- 11 u^{2} + 1 + u^{4} = 0

- 11 u^{2} + 1 + u^{4} = 0

- 11 u^{2} + 1 + u^{4} = 0

Solve

- 11 u^{2} + 1 + u^{4} = 0 : u = \frac{11 + 3 13}{2}, u = - \frac{11 + 3 13}{2}, u = \frac{11 - 3 13}{2}, u = - \frac{11 - 3 13}{2}

- 11 u^{2} + 1 + u^{4} = 0

Write in the standard form

a_{n} x^{n} + \dots + a_{1} x + a_{0} = 0

u^{4} - 11 u^{2} + 1 = 0

Rewrite the equation with

v = u^{2}

and

v^{2} = u^{4}

v^{2} - 11 v + 1 = 0

Solve

v^{2} - 11 v + 1 = 0 : v = \frac{11 + 3 13}{2}, v = \frac{11 - 3 13}{2}

v^{2} - 11 v + 1 = 0

Solve with the quadratic formula

v^{2} - 11 v + 1 = 0

Quadratic Equation Formula:

For

a = 1, b = - 11, c = 1

v_{1, 2} = \frac{- ( - 11 ) \pm ( - 11 ) ^{2} - 4 \cdot 1 \cdot 1}{2 \cdot 1}

v_{1, 2} = \frac{- ( - 11 ) \pm ( - 11 ) ^{2} - 4 \cdot 1 \cdot 1}{2 \cdot 1}

(- 11)^{2} - 4 \cdot 1 \cdot 1 = 313

(- 11)^{2} - 4 \cdot 1 \cdot 1

Apply exponent rule:

(- a)^{n} = a^{n},

n

is even

(- 11)^{2} = 1 1^{2}

= 1 1^{2} - 4 \cdot 1 \cdot 1

Multiply the numbers:

4 \cdot 1 \cdot 1 = 4

= 1 1^{2} - 4

1 1^{2} = 121

= 121 - 4

Subtract the numbers:

121 - 4 = 117

= 117

Prime factorization of

117 : 3^{2} \cdot 13

117

117

divides by

3117 = 39 \cdot 3

= 3 \cdot 39

39

divides by

339 = 13 \cdot 3

= 3 \cdot 3 \cdot 13

3, 13

are all prime numbers, therefore no further factorization is possible

= 3 \cdot 3 \cdot 13

= 3^{2} \cdot 13

= 3^{2} \cdot 13

Apply radical rule:

= 13 3^{2}

Apply radical rule:

3^{2} = 3

= 313

v_{1, 2} = \frac{- ( - 11 ) \pm 3 13}{2 \cdot 1}

Separate the solutions

v_{1} = \frac{- ( - 11 ) + 3 13}{2 \cdot 1}, v_{2} = \frac{- ( - 11 ) - 3 13}{2 \cdot 1}

v = \frac{- ( - 11 ) + 3 13}{2 \cdot 1} : \frac{11 + 3 13}{2}

\frac{- ( - 11 ) + 3 13}{2 \cdot 1}

Apply rule

- (- a) = a

= \frac{11 + 3 13}{2 \cdot 1}

Multiply the numbers:

2 \cdot 1 = 2

= \frac{11 + 3 13}{2}

v = \frac{- ( - 11 ) - 3 13}{2 \cdot 1} : \frac{11 - 3 13}{2}

\frac{- ( - 11 ) - 3 13}{2 \cdot 1}

Apply rule

- (- a) = a

= \frac{11 - 3 13}{2 \cdot 1}

Multiply the numbers:

2 \cdot 1 = 2

= \frac{11 - 3 13}{2}

The solutions to the quadratic equation are:

v = \frac{11 + 3 13}{2}, v = \frac{11 - 3 13}{2}

v = \frac{11 + 3 13}{2}, v = \frac{11 - 3 13}{2}

Substitute back

v = u^{2},

solve for

u

Solve

u^{2} = \frac{11 + 3 13}{2} : u = \frac{11 + 3 13}{2}, u = - \frac{11 + 3 13}{2}

u^{2} = \frac{11 + 3 13}{2}

For

x^{2} = f (a)

the solutions are

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

u = \frac{11 + 3 13}{2}, u = - \frac{11 + 3 13}{2}

Solve

u^{2} = \frac{11 - 3 13}{2} : u = \frac{11 - 3 13}{2}, u = - \frac{11 - 3 13}{2}

u^{2} = \frac{11 - 3 13}{2}

For

x^{2} = f (a)

the solutions are

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

u = \frac{11 - 3 13}{2}, u = - \frac{11 - 3 13}{2}

The solutions are

u = \frac{11 + 3 13}{2}, u = - \frac{11 + 3 13}{2}, u = \frac{11 - 3 13}{2}, u = - \frac{11 - 3 13}{2}

u = \frac{11 + 3 13}{2}, u = - \frac{11 + 3 13}{2}, u = \frac{11 - 3 13}{2}, u = - \frac{11 - 3 13}{2}

Verify Solutions

Find undefined (singularity) points:

u = 0

Take the denominator(s) of

- 11 + \frac{1}{u ^{2}} + u^{2}

and compare to zero

Solve

u^{2} = 0 : u = 0

u^{2} = 0

Apply rule

x^{n} = 0 \Rightarrow x = 0

u = 0

The following points are undefined

u = 0

Combine undefined points with solutions:

u = \frac{11 + 3 13}{2}, u = - \frac{11 + 3 13}{2}, u = \frac{11 - 3 13}{2}, u = - \frac{11 - 3 13}{2}

Substitute back

u = sec (x)

sec (x) = \frac{11 + 3 13}{2}, sec (x) = - \frac{11 + 3 13}{2}, sec (x) = \frac{11 - 3 13}{2}, sec (x) = - \frac{11 - 3 13}{2}

sec (x) = \frac{11 + 3 13}{2}, sec (x) = - \frac{11 + 3 13}{2}, sec (x) = \frac{11 - 3 13}{2}, sec (x) = - \frac{11 - 3 13}{2}

sec (x) = \frac{11 + 3 13}{2} : x = arcsec \frac{11 + 3 13}{2} + 2 πn, x = 2 π - arcsec \frac{11 + 3 13}{2} + 2 πn

sec (x) = \frac{11 + 3 13}{2}

Apply trig inverse properties

sec (x) = \frac{11 + 3 13}{2}

General solutions for

sec (x) = \frac{11 + 3 13}{2}

sec (x) = a \Rightarrow x = arcsec (a) + 2 πn, x = 2 π - arcsec (a) + 2 πn

x = arcsec \frac{11 + 3 13}{2} + 2 πn, x = 2 π - arcsec \frac{11 + 3 13}{2} + 2 πn

x = arcsec \frac{11 + 3 13}{2} + 2 πn, x = 2 π - arcsec \frac{11 + 3 13}{2} + 2 πn

sec (x) = - \frac{11 + 3 13}{2} : x = arcsec - \frac{11 + 3 13}{2} + 2 πn, x = - arcsec - \frac{11 + 3 13}{2} + 2 πn

sec (x) = - \frac{11 + 3 13}{2}

Apply trig inverse properties

sec (x) = - \frac{11 + 3 13}{2}

General solutions for

sec (x) = - \frac{11 + 3 13}{2}

sec (x) = - a \Rightarrow x = arcsec (- a) + 2 πn, x = - arcsec (- a) + 2 πn

x = arcsec - \frac{11 + 3 13}{2} + 2 πn, x = - arcsec - \frac{11 + 3 13}{2} + 2 πn

x = arcsec - \frac{11 + 3 13}{2} + 2 πn, x = - arcsec - \frac{11 + 3 13}{2} + 2 πn

sec (x) = \frac{11 - 3 13}{2} :

No Solution

sec (x) = \frac{11 - 3 13}{2}

sec (x) \leq - 1 or sec (x) \geq 1

No Solution

sec (x) = - \frac{11 - 3 13}{2} :

No Solution

sec (x) = - \frac{11 - 3 13}{2}

sec (x) \leq - 1 or sec (x) \geq 1

No Solution

Combine all the solutions

x = arcsec \frac{11 + 3 13}{2} + 2 πn, x = 2 π - arcsec \frac{11 + 3 13}{2} + 2 πn, x = arcsec - \frac{11 + 3 13}{2} + 2 πn, x = - arcsec - \frac{11 + 3 13}{2} + 2 πn

Show solutions in decimal form

x = 1.26319 \dots + 2 πn, x = 2 π - 1.26319 \dots + 2 πn, x = 1.87839 \dots + 2 πn, x = - 1.87839 \dots + 2 πn

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