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Popular >

2sec(x^2)+tan(x^2)-3=0

Solution

2 sec (x^{2}) + tan (x^{2}) - 3 = 0

Solution

x = 0.56432 \dots + 2 πn, x = - 0.56432 \dots + 2 πn, x = - 1.20782 \dots + 2 πn, x = - - 1.20782 \dots + 2 πn

Degrees

x = 0^{\circ} + 149.93016 \dots^{\circ} n, x = 0^{\circ} - 149.93016 \dots^{\circ} n, x = 0^{\circ} + 129.07910 \dots^{\circ} n, x = 0^{\circ} - 129.07910 \dots^{\circ} n

Solution steps

2 sec (x^{2}) + tan (x^{2}) - 3 = 0

Express with sin, cos

2 \cdot \frac{1}{cos ( x ^{2} )} + \frac{sin ( x ^{2} )}{cos ( x ^{2} )} - 3 = 0

Simplify

2 \cdot \frac{1}{cos ( x ^{2} )} + \frac{sin ( x ^{2} )}{cos ( x ^{2} )} - 3 : \frac{2 + sin ( x ^{2} ) - 3 cos ( x ^{2} )}{cos ( x ^{2} )}

2 \cdot \frac{1}{cos ( x ^{2} )} + \frac{sin ( x ^{2} )}{cos ( x ^{2} )} - 3

2 \cdot \frac{1}{cos ( x ^{2} )} = \frac{2}{cos ( x ^{2} )}

2 \cdot \frac{1}{cos ( x ^{2} )}

Multiply fractions:

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

= \frac{1 \cdot 2}{cos ( x ^{2} )}

Multiply the numbers:

1 \cdot 2 = 2

= \frac{2}{cos ( x ^{2} )}

= \frac{2}{cos ( x ^{2} )} + \frac{sin ( x ^{2} )}{cos ( x ^{2} )} - 3

Combine the fractions

\frac{2}{cos ( x ^{2} )} + \frac{sin ( x ^{2} )}{cos ( x ^{2} )} : \frac{2 + sin ( x ^{2} )}{cos ( x ^{2} )}

Apply rule

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

= \frac{2 + sin ( x ^{2} )}{cos ( x ^{2} )}

= \frac{sin ( x ^{2} ) + 2}{cos ( x ^{2} )} - 3

Convert element to fraction:

3 = \frac{3 cos ( x ^{2} )}{cos ( x ^{2} )}

= \frac{2 + sin ( x ^{2} )}{cos ( x ^{2} )} - \frac{3 cos ( x ^{2} )}{cos ( x ^{2} )}

Since the denominators are equal, combine the fractions:

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

= \frac{2 + sin ( x ^{2} ) - 3 cos ( x ^{2} )}{cos ( x ^{2} )}

\frac{2 + sin ( x ^{2} ) - 3 cos ( x ^{2} )}{cos ( x ^{2} )} = 0

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

2 + sin (x^{2}) - 3 cos (x^{2}) = 0

Add

3 cos (x^{2})

to both sides

2 + sin (x^{2}) = 3 cos (x^{2})

Square both sides

(2 + sin (x^{2}))^{2} = (3 cos (x^{2}))^{2}

Subtract

(3 cos (x^{2}))^{2}

from both sides

(2 + sin (x^{2}))^{2} - 9 cos^{2} (x^{2}) = 0

Rewrite using trig identities

(2 + sin (x^{2}))^{2} - 9 cos^{2} (x^{2})

Use the Pythagorean identity:

cos^{2} (x) + sin^{2} (x) = 1

cos^{2} (x) = 1 - sin^{2} (x)

= (2 + sin (x^{2}))^{2} - 9 (1 - sin^{2} (x^{2}))

Simplify

(2 + sin (x^{2}))^{2} - 9 (1 - sin^{2} (x^{2})) : 10 sin^{2} (x^{2}) + 4 sin (x^{2}) - 5

(2 + sin (x^{2}))^{2} - 9 (1 - sin^{2} (x^{2}))

(2 + sin (x^{2}))^{2} : 4 + 4 sin (x^{2}) + sin^{2} (x^{2})

Apply Perfect Square Formula:

(a + b)^{2} = a^{2} + 2 ab + b^{2}

a = 2, b = sin (x^{2})

= 2^{2} + 2 \cdot 2 sin (x^{2}) + sin^{2} (x^{2})

Simplify

2^{2} + 2 \cdot 2 sin (x^{2}) + sin^{2} (x^{2}) : 4 + 4 sin (x^{2}) + sin^{2} (x^{2})

2^{2} + 2 \cdot 2 sin (x^{2}) + sin^{2} (x^{2})

2^{2} = 4

= 4 + 2 \cdot 2 sin (x^{2}) + sin^{2} (x^{2})

Multiply the numbers:

2 \cdot 2 = 4

= 4 + 4 sin (x^{2}) + sin^{2} (x^{2})

= 4 + 4 sin (x^{2}) + sin^{2} (x^{2})

= 4 + 4 sin (x^{2}) + sin^{2} (x^{2}) - 9 (1 - sin^{2} (x^{2}))

Expand

- 9 (1 - sin^{2} (x^{2})) : - 9 + 9 sin^{2} (x^{2})

- 9 (1 - sin^{2} (x^{2}))

Apply the distributive law:

a (b - c) = ab - a c

a = - 9, b = 1, c = sin^{2} (x^{2})

= - 9 \cdot 1 - (- 9) sin^{2} (x^{2})

Apply minus-plus rules

- (- a) = a

= - 9 \cdot 1 + 9 sin^{2} (x^{2})

Multiply the numbers:

9 \cdot 1 = 9

= - 9 + 9 sin^{2} (x^{2})

= 4 + 4 sin (x^{2}) + sin^{2} (x^{2}) - 9 + 9 sin^{2} (x^{2})

Simplify

4 + 4 sin (x^{2}) + sin^{2} (x^{2}) - 9 + 9 sin^{2} (x^{2}) : 10 sin^{2} (x^{2}) + 4 sin (x^{2}) - 5

4 + 4 sin (x^{2}) + sin^{2} (x^{2}) - 9 + 9 sin^{2} (x^{2})

Group like terms

= 4 sin (x^{2}) + sin^{2} (x^{2}) + 9 sin^{2} (x^{2}) + 4 - 9

Add similar elements:

sin^{2} (x^{2}) + 9 sin^{2} (x^{2}) = 10 sin^{2} (x^{2})

= 4 sin (x^{2}) + 10 sin^{2} (x^{2}) + 4 - 9

Add/Subtract the numbers:

4 - 9 = - 5

= 10 sin^{2} (x^{2}) + 4 sin (x^{2}) - 5

= 10 sin^{2} (x^{2}) + 4 sin (x^{2}) - 5

= 10 sin^{2} (x^{2}) + 4 sin (x^{2}) - 5

- 5 + 10 sin^{2} (x^{2}) + 4 sin (x^{2}) = 0

Solve by substitution

- 5 + 10 sin^{2} (x^{2}) + 4 sin (x^{2}) = 0

Let:

sin (x^{2}) = u

- 5 + 10 u^{2} + 4 u = 0

- 5 + 10 u^{2} + 4 u = 0 : u = \frac{- 2 + 3 6}{10}, u = - \frac{2 + 3 6}{10}

- 5 + 10 u^{2} + 4 u = 0

Write in the standard form

a x^{2} + b x + c = 0

10 u^{2} + 4 u - 5 = 0

Solve with the quadratic formula

10 u^{2} + 4 u - 5 = 0

Quadratic Equation Formula:

For

a = 10, b = 4, c = - 5

u_{1, 2} = \frac{- 4 \pm 4 ^{2} - 4 \cdot 10 ( - 5 )}{2 \cdot 10}

u_{1, 2} = \frac{- 4 \pm 4 ^{2} - 4 \cdot 10 ( - 5 )}{2 \cdot 10}

4^{2} - 4 \cdot 10 (- 5) = 66

4^{2} - 4 \cdot 10 (- 5)

Apply rule

- (- a) = a

= 4^{2} + 4 \cdot 10 \cdot 5

Multiply the numbers:

4 \cdot 10 \cdot 5 = 200

= 4^{2} + 200

4^{2} = 16

= 16 + 200

Add the numbers:

16 + 200 = 216

= 216

Prime factorization of

216 : 2^{3} \cdot 3^{3}

216

216

divides by

2216 = 108 \cdot 2

= 2 \cdot 108

108

divides by

2108 = 54 \cdot 2

= 2 \cdot 2 \cdot 54

54

divides by

254 = 27 \cdot 2

= 2 \cdot 2 \cdot 2 \cdot 27

27

divides by

327 = 9 \cdot 3

= 2 \cdot 2 \cdot 2 \cdot 3 \cdot 9

9

divides by

39 = 3 \cdot 3

= 2 \cdot 2 \cdot 2 \cdot 3 \cdot 3 \cdot 3

2, 3

are all prime numbers, therefore no further factorization is possible

= 2 \cdot 2 \cdot 2 \cdot 3 \cdot 3 \cdot 3

= 2^{3} \cdot 3^{3}

= 2^{3} \cdot 3^{3}

Apply exponent rule:

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

= 2^{2} \cdot 3^{2} \cdot 2 \cdot 3

Apply radical rule:

= 2^{2} 3^{2} 2 \cdot 3

Apply radical rule:

2^{2} = 2

= 2 3^{2} 2 \cdot 3

Apply radical rule:

3^{2} = 3

= 2 \cdot 3 2 \cdot 3

Refine

= 66

u_{1, 2} = \frac{- 4 \pm 6 6}{2 \cdot 10}

Separate the solutions

u_{1} = \frac{- 4 + 6 6}{2 \cdot 10}, u_{2} = \frac{- 4 - 6 6}{2 \cdot 10}

u = \frac{- 4 + 6 6}{2 \cdot 10} : \frac{- 2 + 3 6}{10}

\frac{- 4 + 6 6}{2 \cdot 10}

Multiply the numbers:

2 \cdot 10 = 20

= \frac{- 4 + 6 6}{20}

Factor

- 4 + 66 : 2 (- 2 + 36)

- 4 + 66

Rewrite as

= - 2 \cdot 2 + 2 \cdot 36

Factor out common term

2

= 2 (- 2 + 36)

= \frac{2 ( - 2 + 3 6 )}{20}

Cancel the common factor:

2

= \frac{- 2 + 3 6}{10}

u = \frac{- 4 - 6 6}{2 \cdot 10} : - \frac{2 + 3 6}{10}

\frac{- 4 - 6 6}{2 \cdot 10}

Multiply the numbers:

2 \cdot 10 = 20

= \frac{- 4 - 6 6}{20}

Factor

- 4 - 66 : - 2 (2 + 36)

- 4 - 66

Rewrite as

= - 2 \cdot 2 - 2 \cdot 36

Factor out common term

2

= - 2 (2 + 36)

= - \frac{2 ( 2 + 3 6 )}{20}

Cancel the common factor:

2

= - \frac{2 + 3 6}{10}

The solutions to the quadratic equation are:

u = \frac{- 2 + 3 6}{10}, u = - \frac{2 + 3 6}{10}

Substitute back

u = sin (x^{2})

sin (x^{2}) = \frac{- 2 + 3 6}{10}, sin (x^{2}) = - \frac{2 + 3 6}{10}

sin (x^{2}) = \frac{- 2 + 3 6}{10}, sin (x^{2}) = - \frac{2 + 3 6}{10}

sin (x^{2}) = \frac{- 2 + 3 6}{10} : x = arcsin (\frac{- 2 + 3 6}{10}) + 2 πn, x = - arcsin (\frac{- 2 + 3 6}{10}) + 2 πn, x = π - arcsin (\frac{- 2 + 3 6}{10}) + 2 πn, x = - π - arcsin (\frac{- 2 + 3 6}{10}) + 2 πn

sin (x^{2}) = \frac{- 2 + 3 6}{10}

Apply trig inverse properties

sin (x^{2}) = \frac{- 2 + 3 6}{10}

General solutions for

sin (x^{2}) = \frac{- 2 + 3 6}{10}

sin (x) = a \Rightarrow x = arcsin (a) + 2 πn, x = π - arcsin (a) + 2 πn

x^{2} = arcsin (\frac{- 2 + 3 6}{10}) + 2 πn, x^{2} = π - arcsin (\frac{- 2 + 3 6}{10}) + 2 πn

x^{2} = arcsin (\frac{- 2 + 3 6}{10}) + 2 πn, x^{2} = π - arcsin (\frac{- 2 + 3 6}{10}) + 2 πn

Solve

x^{2} = arcsin (\frac{- 2 + 3 6}{10}) + 2 πn : x = arcsin (\frac{- 2 + 3 6}{10}) + 2 πn, x = - arcsin (\frac{- 2 + 3 6}{10}) + 2 πn

x^{2} = arcsin (\frac{- 2 + 3 6}{10}) + 2 πn

For

x^{2} = f (a)

the solutions are

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

x = arcsin (\frac{- 2 + 3 6}{10}) + 2 πn, x = - arcsin (\frac{- 2 + 3 6}{10}) + 2 πn

Solve

x^{2} = π - arcsin (\frac{- 2 + 3 6}{10}) + 2 πn : x = π - arcsin (\frac{- 2 + 3 6}{10}) + 2 πn, x = - π - arcsin (\frac{- 2 + 3 6}{10}) + 2 πn

x^{2} = π - arcsin (\frac{- 2 + 3 6}{10}) + 2 πn

For

x^{2} = f (a)

the solutions are

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

x = π - arcsin (\frac{- 2 + 3 6}{10}) + 2 πn, x = - π - arcsin (\frac{- 2 + 3 6}{10}) + 2 πn

x = arcsin (\frac{- 2 + 3 6}{10}) + 2 πn, x = - arcsin (\frac{- 2 + 3 6}{10}) + 2 πn, x = π - arcsin (\frac{- 2 + 3 6}{10}) + 2 πn, x = - π - arcsin (\frac{- 2 + 3 6}{10}) + 2 πn

sin (x^{2}) = - \frac{2 + 3 6}{10} : x = - arcsin (\frac{2 + 3 6}{10}) + 2 πn, x = - - arcsin (\frac{2 + 3 6}{10}) + 2 πn, x = π + arcsin (\frac{2 + 3 6}{10}) + 2 πn, x = - π + arcsin (\frac{2 + 3 6}{10}) + 2 πn

sin (x^{2}) = - \frac{2 + 3 6}{10}

Apply trig inverse properties

sin (x^{2}) = - \frac{2 + 3 6}{10}

General solutions for

sin (x^{2}) = - \frac{2 + 3 6}{10}

sin (x) = - a \Rightarrow x = arcsin (- a) + 2 πn, x = π + arcsin (a) + 2 πn

x^{2} = arcsin (- \frac{2 + 3 6}{10}) + 2 πn, x^{2} = π + arcsin (\frac{2 + 3 6}{10}) + 2 πn

x^{2} = arcsin (- \frac{2 + 3 6}{10}) + 2 πn, x^{2} = π + arcsin (\frac{2 + 3 6}{10}) + 2 πn

Solve

x^{2} = arcsin (- \frac{2 + 3 6}{10}) + 2 πn : x = - arcsin (\frac{2 + 3 6}{10}) + 2 πn, x = - - arcsin (\frac{2 + 3 6}{10}) + 2 πn

x^{2} = arcsin (- \frac{2 + 3 6}{10}) + 2 πn

Use the following property:

arcsin (- x) = - arcsin (x)

arcsin (- \frac{2 + 3 6}{10}) = - arcsin (\frac{2 + 3 6}{10})

x^{2} = - arcsin (\frac{2 + 3 6}{10}) + 2 πn

For

x^{2} = f (a)

the solutions are

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

x = - arcsin (\frac{2 + 3 6}{10}) + 2 πn, x = - - arcsin (\frac{2 + 3 6}{10}) + 2 πn

Solve

x^{2} = π + arcsin (\frac{2 + 3 6}{10}) + 2 πn : x = π + arcsin (\frac{2 + 3 6}{10}) + 2 πn, x = - π + arcsin (\frac{2 + 3 6}{10}) + 2 πn

x^{2} = π + arcsin (\frac{2 + 3 6}{10}) + 2 πn

For

x^{2} = f (a)

the solutions are

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

x = π + arcsin (\frac{2 + 3 6}{10}) + 2 πn, x = - π + arcsin (\frac{2 + 3 6}{10}) + 2 πn

x = - arcsin (\frac{2 + 3 6}{10}) + 2 πn, x = - - arcsin (\frac{2 + 3 6}{10}) + 2 πn, x = π + arcsin (\frac{2 + 3 6}{10}) + 2 πn, x = - π + arcsin (\frac{2 + 3 6}{10}) + 2 πn

Combine all the solutions

x = arcsin (\frac{- 2 + 3 6}{10}) + 2 πn, x = - arcsin (\frac{- 2 + 3 6}{10}) + 2 πn, x = π - arcsin (\frac{- 2 + 3 6}{10}) + 2 πn, x = - π - arcsin (\frac{- 2 + 3 6}{10}) + 2 πn, x = - arcsin (\frac{2 + 3 6}{10}) + 2 πn, x = - - arcsin (\frac{2 + 3 6}{10}) + 2 πn, x = π + arcsin (\frac{2 + 3 6}{10}) + 2 πn, x = - π + arcsin (\frac{2 + 3 6}{10}) + 2 πn

Verify solutions by plugging them into the original equation

Check the solutions by plugging them into

2 sec (x^{2}) + tan (x^{2}) - 3 = 0

Remove the ones that don't agree with the equation.

Check the solution

arcsin (\frac{- 2 + 3 6}{10}) + 2 πn :

True

arcsin (\frac{- 2 + 3 6}{10}) + 2 πn

Plug in

n = 1

arcsin (\frac{- 2 + 3 6}{10}) + 2 π 1

For

2 sec (x^{2}) + tan (x^{2}) - 3 = 0

plug in

x = arcsin (\frac{- 2 + 3 6}{10}) + 2 π 1

2 sec arcsin (\frac{- 2 + 3 6}{10}) + 2 π 1^{2} + tan arcsin (\frac{- 2 + 3 6}{10}) + 2 π 1^{2} - 3 = 0

Refine

0 = 0

\Rightarrow True

Check the solution

- arcsin (\frac{- 2 + 3 6}{10}) + 2 πn :

True

- arcsin (\frac{- 2 + 3 6}{10}) + 2 πn

Plug in

n = 1

- arcsin (\frac{- 2 + 3 6}{10}) + 2 π 1

For

2 sec (x^{2}) + tan (x^{2}) - 3 = 0

plug in

x = - arcsin (\frac{- 2 + 3 6}{10}) + 2 π 1

2 sec - arcsin (\frac{- 2 + 3 6}{10}) + 2 π 1^{2} + tan - arcsin (\frac{- 2 + 3 6}{10}) + 2 π 1^{2} - 3 = 0

Refine

0 = 0

\Rightarrow True

Check the solution

π - arcsin (\frac{- 2 + 3 6}{10}) + 2 πn :

False

π - arcsin (\frac{- 2 + 3 6}{10}) + 2 πn

Plug in

n = 1

π - arcsin (\frac{- 2 + 3 6}{10}) + 2 π 1

For

2 sec (x^{2}) + tan (x^{2}) - 3 = 0

plug in

x = π - arcsin (\frac{- 2 + 3 6}{10}) + 2 π 1

2 sec π - arcsin (\frac{- 2 + 3 6}{10}) + 2 π 1^{2} + tan π - arcsin (\frac{- 2 + 3 6}{10}) + 2 π 1^{2} - 3 = 0

Refine

- 6 = 0

\Rightarrow False

Check the solution

- π - arcsin (\frac{- 2 + 3 6}{10}) + 2 πn :

False

- π - arcsin (\frac{- 2 + 3 6}{10}) + 2 πn

Plug in

n = 1

- π - arcsin (\frac{- 2 + 3 6}{10}) + 2 π 1

For

2 sec (x^{2}) + tan (x^{2}) - 3 = 0

plug in

x = - π - arcsin (\frac{- 2 + 3 6}{10}) + 2 π 1

2 sec - π - arcsin (\frac{- 2 + 3 6}{10}) + 2 π 1^{2} + tan - π - arcsin (\frac{- 2 + 3 6}{10}) + 2 π 1^{2} - 3 = 0

Refine

- 6 = 0

\Rightarrow False

Check the solution

- arcsin (\frac{2 + 3 6}{10}) + 2 πn :

True

- arcsin (\frac{2 + 3 6}{10}) + 2 πn

Plug in

n = 1

- arcsin (\frac{2 + 3 6}{10}) + 2 π 1

For

2 sec (x^{2}) + tan (x^{2}) - 3 = 0

plug in

x = - arcsin (\frac{2 + 3 6}{10}) + 2 π 1

2 sec - arcsin (\frac{2 + 3 6}{10}) + 2 π 1^{2} + tan - arcsin (\frac{2 + 3 6}{10}) + 2 π 1^{2} - 3 = 0

Refine

0 = 0

\Rightarrow True

Check the solution

- - arcsin (\frac{2 + 3 6}{10}) + 2 πn :

True

- - arcsin (\frac{2 + 3 6}{10}) + 2 πn

Plug in

n = 1

- - arcsin (\frac{2 + 3 6}{10}) + 2 π 1

For

2 sec (x^{2}) + tan (x^{2}) - 3 = 0

plug in

x = - - arcsin (\frac{2 + 3 6}{10}) + 2 π 1

2 sec - - arcsin (\frac{2 + 3 6}{10}) + 2 π 1^{2} + tan - - arcsin (\frac{2 + 3 6}{10}) + 2 π 1^{2} - 3 = 0

Refine

0 = 0

\Rightarrow True

Check the solution

π + arcsin (\frac{2 + 3 6}{10}) + 2 πn :

False

π + arcsin (\frac{2 + 3 6}{10}) + 2 πn

Plug in

n = 1

π + arcsin (\frac{2 + 3 6}{10}) + 2 π 1

For

2 sec (x^{2}) + tan (x^{2}) - 3 = 0

plug in

x = π + arcsin (\frac{2 + 3 6}{10}) + 2 π 1

2 sec π + arcsin (\frac{2 + 3 6}{10}) + 2 π 1^{2} + tan π + arcsin (\frac{2 + 3 6}{10}) + 2 π 1^{2} - 3 = 0

Refine

- 6 = 0

\Rightarrow False

Check the solution

- π + arcsin (\frac{2 + 3 6}{10}) + 2 πn :

False

- π + arcsin (\frac{2 + 3 6}{10}) + 2 πn

Plug in

n = 1

- π + arcsin (\frac{2 + 3 6}{10}) + 2 π 1

For

2 sec (x^{2}) + tan (x^{2}) - 3 = 0

plug in

x = - π + arcsin (\frac{2 + 3 6}{10}) + 2 π 1

2 sec - π + arcsin (\frac{2 + 3 6}{10}) + 2 π 1^{2} + tan - π + arcsin (\frac{2 + 3 6}{10}) + 2 π 1^{2} - 3 = 0

Refine

- 6 = 0

\Rightarrow False

x = arcsin (\frac{- 2 + 3 6}{10}) + 2 πn, x = - arcsin (\frac{- 2 + 3 6}{10}) + 2 πn, x = - arcsin (\frac{2 + 3 6}{10}) + 2 πn, x = - - arcsin (\frac{2 + 3 6}{10}) + 2 πn

Show solutions in decimal form

x = 0.56432 \dots + 2 πn, x = - 0.56432 \dots + 2 πn, x = - 1.20782 \dots + 2 πn, x = - - 1.20782 \dots + 2 πn

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