English

Español

Português

Français

Deutsch

Italiano

Русский

中文(简体)

한국어

日本語

Tiếng Việt

עברית

العربية

Popular >

sec^2(x)=3tan^2(x)-1

Solution

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

Solution

x = \frac{π}{4} + 2 πn, x = \frac{7 π}{4} + 2 πn, x = \frac{3 π}{4} + 2 πn, x = \frac{5 π}{4} + 2 πn

Degrees

x = 4 5^{\circ} + 36 0^{\circ} n, x = 31 5^{\circ} + 36 0^{\circ} n, x = 13 5^{\circ} + 36 0^{\circ} n, x = 22 5^{\circ} + 36 0^{\circ} n

Solution steps

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

Subtract

3 tan^{2} (x) - 1

from both sides

sec^{2} (x) - 3 tan^{2} (x) + 1 = 0

Rewrite using trig identities

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

Use the Pythagorean identity:

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

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

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

Simplify

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

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

Expand

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

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

Apply the distributive law:

a (b - c) = ab - a c

a = - 3, b = sec^{2} (x), c = 1

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

Apply minus-plus rules

- (- a) = a

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

Multiply the numbers:

3 \cdot 1 = 3

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

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

Simplify

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

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

Add similar elements:

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

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

Group like terms

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

Add the numbers:

1 + 3 = 4

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

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

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

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

Solve by substitution

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

Let:

sec (x) = u

4 - 2 u^{2} = 0

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

4 - 2 u^{2} = 0

Move

4

to the right side

4 - 2 u^{2} = 0

Subtract

4

from both sides

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

Simplify

- 2 u^{2} = - 4

- 2 u^{2} = - 4

Divide both sides by

- 2

- 2 u^{2} = - 4

Divide both sides by

- 2

\frac{- 2 u ^{2}}{- 2} = \frac{- 4}{- 2}

Simplify

\frac{- 2 u ^{2}}{- 2} = \frac{- 4}{- 2}

Simplify

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

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

Apply the fraction rule:

\frac{- a}{- b} = \frac{a}{b}

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

Divide the numbers:

\frac{2}{2} = 1

= u^{2}

Simplify

\frac{- 4}{- 2} : 2

\frac{- 4}{- 2}

Apply the fraction rule:

\frac{- a}{- b} = \frac{a}{b}

= \frac{4}{2}

Divide the numbers:

\frac{4}{2} = 2

= 2

u^{2} = 2

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 (x)

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

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

sec (x) = 2 : x = \frac{π}{4} + 2 πn, x = \frac{7 π}{4} + 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}{4} + 2 πn, x = \frac{7 π}{4} + 2 πn

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

sec (x) = - 2 : x = \frac{3 π}{4} + 2 πn, x = \frac{5 π}{4} + 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{3 π}{4} + 2 πn, x = \frac{5 π}{4} + 2 πn

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

Combine all the solutions

x = \frac{π}{4} + 2 πn, x = \frac{7 π}{4} + 2 πn, x = \frac{3 π}{4} + 2 πn, x = \frac{5 π}{4} + 2 πn

Popular Examples

csc(x)+cot(x)=(sqrt(3))/3

csc (x) + cot (x) = \frac{3}{3}

sin(θ)+1=cos(θ)

sin (θ) + 1 = cos (θ)

sqrt(2)sin^2(θ)-sin(θ)=0

2 sin^{2} (θ) - sin (θ) = 0

2sin^2(x)=2-sqrt(3)cos(x)

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

tan^2(x)= 3/2 sec(x)

tan^{2} (x) = \frac{3}{2} sec (x)