What is the general solution for 2sec^2(θ)+3=4+sec^2(θ) ?

The general solution for 2sec^2(θ)+3=4+sec^2(θ) is θ=2pin,θ=pi+2pin

English

Español

Português

Français

Deutsch

Italiano

Русский

中文(简体)

한국어

日本語

Tiếng Việt

עברית

العربية

Popular >

2sec^2(θ)+3=4+sec^2(θ)

Solution

2 sec^{2} (θ) + 3 = 4 + sec^{2} (θ)

Solution

θ = 2 πn, θ = π + 2 πn

Degrees

θ = 0^{\circ} + 36 0^{\circ} n, θ = 18 0^{\circ} + 36 0^{\circ} n

Solution steps

2 sec^{2} (θ) + 3 = 4 + sec^{2} (θ)

Solve by substitution

2 sec^{2} (θ) + 3 = 4 + sec^{2} (θ)

Let:

sec (θ) = u

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

2 u^{2} + 3 = 4 + u^{2} : u = 1, u = - 1

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

Move

3

to the right side

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

Subtract

3

from both sides

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

Simplify

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

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

Move

u^{2}

to the left side

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

Subtract

u^{2}

from both sides

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

Simplify

u^{2} = 1

u^{2} = 1

For

x^{2} = f (a)

the solutions are

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

u = 1, u = - 1

1 = 1

1

Apply rule

1 = 1

= 1

- 1 = - 1

- 1

Apply rule

1 = 1

= - 1

u = 1, u = - 1

Substitute back

u = sec (θ)

sec (θ) = 1, sec (θ) = - 1

sec (θ) = 1, sec (θ) = - 1

sec (θ) = 1 : θ = 2 πn

sec (θ) = 1

General solutions for

sec (θ) = 1

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}

θ = 0 + 2 πn

θ = 0 + 2 πn

Solve

θ = 0 + 2 πn : θ = 2 πn

θ = 0 + 2 πn

0 + 2 πn = 2 πn

θ = 2 πn

θ = 2 πn

sec (θ) = - 1 : θ = π + 2 πn

sec (θ) = - 1

General solutions for

sec (θ) = - 1

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}

θ = π + 2 πn

θ = π + 2 πn

Combine all the solutions

θ = 2 πn, θ = π + 2 πn

Popular Examples

3cos^2(x)-5cos(x)-2=0

3 cos^{2} (x) - 5 cos (x) - 2 = 0

cos(2x)+1=-2cos(x)

cos (2 x) + 1 = - 2 cos (x)

sin(θ)=(-1)/2 ,0<= θ<= 4pi

sin (θ) = \frac{- 1}{2}, 0 \leq θ \leq 4 π

4cos^2(x)+2cos(x)=2,0<= x<= 2pi

4 cos^{2} (x) + 2 cos (x) = 2, 0 \leq x \leq 2 π

arctan(x)+arctan(2x)= pi/4

arctan (x) + arctan (2 x) = \frac{π}{4}

Frequently Asked Questions (FAQ)

What is the general solution for 2sec^2(θ)+3=4+sec^2(θ) ?
The general solution for 2sec^2(θ)+3=4+sec^2(θ) is θ=2pin,θ=pi+2pin