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Popular >

sqrt(3)sec^2(x)+2tan^2(x)=2

Solution

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

Solution

x = 0.26179 \dots + 2 πn, x = 2 π - 0.26179 \dots + 2 πn, x = 2.87979 \dots + 2 πn, x = - 2.87979 \dots + 2 πn

Degrees

x = 1 5^{\circ} + 36 0^{\circ} n, x = 34 5^{\circ} + 36 0^{\circ} n, x = 16 5^{\circ} + 36 0^{\circ} n, x = - 16 5^{\circ} + 36 0^{\circ} n

Solution steps

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

Subtract

2

from both sides

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

Rewrite using trig identities

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

Use the Pythagorean identity:

tan^{2} (x) + 1 = sec^{2} (x)

tan^{2} (x) = sec^{2} (x) - 1

= - 2 + 2 (sec^{2} (x) - 1) + sec^{2} (x) 3

Simplify

- 2 + 2 (sec^{2} (x) - 1) + sec^{2} (x) 3 : 2 sec^{2} (x) + 3 sec^{2} (x) - 4

- 2 + 2 (sec^{2} (x) - 1) + sec^{2} (x) 3

= - 2 + 2 (sec^{2} (x) - 1) + 3 sec^{2} (x)

Expand

2 (sec^{2} (x) - 1) : 2 sec^{2} (x) - 2

2 (sec^{2} (x) - 1)

Apply the distributive law:

a (b - c) = ab - a c

a = 2, b = sec^{2} (x), c = 1

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

Multiply the numbers:

2 \cdot 1 = 2

= 2 sec^{2} (x) - 2

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

Simplify

- 2 + 2 sec^{2} (x) - 2 + sec^{2} (x) 3 : 2 sec^{2} (x) + 3 sec^{2} (x) - 4

- 2 + 2 sec^{2} (x) - 2 + sec^{2} (x) 3

Group like terms

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

Subtract the numbers:

- 2 - 2 = - 4

= 2 sec^{2} (x) + 3 sec^{2} (x) - 4

= 2 sec^{2} (x) + 3 sec^{2} (x) - 4

= 2 sec^{2} (x) + 3 sec^{2} (x) - 4

- 4 + 2 sec^{2} (x) + sec^{2} (x) 3 = 0

Solve by substitution

- 4 + 2 sec^{2} (x) + sec^{2} (x) 3 = 0

Let:

sec (x) = u

- 4 + 2 u^{2} + u^{2} 3 = 0

- 4 + 2 u^{2} + u^{2} 3 = 0 : u = 2 2 - 3, u = - 2 2 - 3

- 4 + 2 u^{2} + u^{2} 3 = 0

Move

4

to the right side

- 4 + 2 u^{2} + u^{2} 3 = 0

Add

4

to both sides

- 4 + 2 u^{2} + u^{2} 3 + 4 = 0 + 4

Simplify

2 u^{2} + u^{2} 3 = 4

2 u^{2} + u^{2} 3 = 4

Factor

2 u^{2} + u^{2} 3 : (2 + 3) u^{2}

2 u^{2} + u^{2} 3

Factor out common term

u^{2}

= u^{2} (2 + 3)

(2 + 3) u^{2} = 4

Divide both sides by

2 + 3

(2 + 3) u^{2} = 4

Divide both sides by

2 + 3

\frac{( 2 + 3 ) u ^{2}}{2 + 3} = \frac{4}{2 + 3}

Simplify

\frac{( 2 + 3 ) u ^{2}}{2 + 3} = \frac{4}{2 + 3}

Simplify

\frac{( 2 + 3 ) u ^{2}}{2 + 3} : u^{2}

\frac{( 2 + 3 ) u ^{2}}{2 + 3}

Cancel the common factor:

2 + 3

= u^{2}

Simplify

\frac{4}{2 + 3} : 4 (2 - 3)

\frac{4}{2 + 3}

Multiply by the conjugate

\frac{2 - 3}{2 - 3}

= \frac{4 ( 2 - 3 )}{( 2 + 3 ) ( 2 - 3 )}

(2 + 3) (2 - 3) = 1

(2 + 3) (2 - 3)

Apply Difference of Two Squares Formula:

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

a = 2, b = 3

= 2^{2} - (3)^{2}

Simplify

2^{2} - (3)^{2} : 1

2^{2} - (3)^{2}

2^{2} = 4

2^{2}

2^{2} = 4

= 4

(3)^{2} = 3

(3)^{2}

Apply radical rule:

a = a^{\frac{1}{2}}

= (3^{\frac{1}{2}})^{2}

Apply exponent rule:

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

= 3^{\frac{1}{2} \cdot 2}

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

\frac{1}{2} \cdot 2

Multiply fractions:

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

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

Cancel the common factor:

2

= 1

= 3

= 4 - 3

Subtract the numbers:

4 - 3 = 1

= 1

= 1

= \frac{4 ( 2 - 3 )}{1}

Apply rule

\frac{a}{1} = a

= 4 (2 - 3)

u^{2} = 4 (2 - 3)

u^{2} = 4 (2 - 3)

u^{2} = 4 (2 - 3)

For

x^{2} = f (a)

the solutions are

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

u = 4 (2 - 3), u = - 4 (2 - 3)

4 (2 - 3) = 2 2 - 3

4 (2 - 3)

Apply radical rule:

n ab = n a n b,

assuming

a \geq 0, b \geq 0

= 4 2 - 3

4 = 2

4

Factor the number:

4 = 2^{2}

= 2^{2}

Apply radical rule:

n a^{n} = a

2^{2} = 2

= 2

= 2 2 - 3

- 4 (2 - 3) = - 2 2 - 3

- 4 (2 - 3)

Simplify

4 (2 - 3) : 2 2 - 3

4 (2 - 3)

Apply radical rule:

n ab = n a n b,

assuming

a \geq 0, b \geq 0

= 4 2 - 3

4 = 2

4

Factor the number:

4 = 2^{2}

= 2^{2}

Apply radical rule:

n a^{n} = a

2^{2} = 2

= 2

= 2 2 - 3

= - 2 2 - 3

u = 2 2 - 3, u = - 2 2 - 3

Substitute back

u = sec (x)

sec (x) = 2 2 - 3, sec (x) = - 2 2 - 3

sec (x) = 2 2 - 3, sec (x) = - 2 2 - 3

sec (x) = 2 2 - 3 : x = arcsec (2 2 - 3) + 2 πn, x = 2 π - arcsec (2 2 - 3) + 2 πn

sec (x) = 2 2 - 3

Apply trig inverse properties

sec (x) = 2 2 - 3

General solutions for

sec (x) = 2 2 - 3

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

x = arcsec (2 2 - 3) + 2 πn, x = 2 π - arcsec (2 2 - 3) + 2 πn

x = arcsec (2 2 - 3) + 2 πn, x = 2 π - arcsec (2 2 - 3) + 2 πn

sec (x) = - 2 2 - 3 : x = arcsec (- 2 2 - 3) + 2 πn, x = - arcsec (- 2 2 - 3) + 2 πn

sec (x) = - 2 2 - 3

Apply trig inverse properties

sec (x) = - 2 2 - 3

General solutions for

sec (x) = - 2 2 - 3

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

x = arcsec (- 2 2 - 3) + 2 πn, x = - arcsec (- 2 2 - 3) + 2 πn

x = arcsec (- 2 2 - 3) + 2 πn, x = - arcsec (- 2 2 - 3) + 2 πn

Combine all the solutions

x = arcsec (2 2 - 3) + 2 πn, x = 2 π - arcsec (2 2 - 3) + 2 πn, x = arcsec (- 2 2 - 3) + 2 πn, x = - arcsec (- 2 2 - 3) + 2 πn

Show solutions in decimal form

x = 0.26179 \dots + 2 πn, x = 2 π - 0.26179 \dots + 2 πn, x = 2.87979 \dots + 2 πn, x = - 2.87979 \dots + 2 πn

Graph

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