en

English

Español

Português

Français

Deutsch

Italiano

Русский

中文(简体)

한국어

日本語

Tiếng Việt

עברית

العربية

Popular >

2sec^2(θ)-3=tan^2(θ)

Solution

2 sec^{2} (θ) - 3 = tan^{2} (θ)

Solution

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

+1

Degrees

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

Solution steps

2 sec^{2} (θ) - 3 = tan^{2} (θ)

Subtract

tan^{2} (θ)

from both sides

2 sec^{2} (θ) - 3 - tan^{2} (θ) = 0

Rewrite using trig identities

- 3 - tan^{2} (θ) + 2 sec^{2} (θ)

Use the Pythagorean identity:

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

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

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

Simplify

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

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

- (sec^{2} (θ) - 1) : - sec^{2} (θ) + 1

- (sec^{2} (θ) - 1)

Distribute parentheses

= - (sec^{2} (θ)) - (- 1)

Apply minus-plus rules

- (- a) = a, - (a) = - a

= - sec^{2} (θ) + 1

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

Simplify

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

- 3 - sec^{2} (θ) + 1 + 2 sec^{2} (θ)

Group like terms

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

Add similar elements:

- sec^{2} (θ) + 2 sec^{2} (θ) = sec^{2} (θ)

= sec^{2} (θ) - 3 + 1

Add/Subtract the numbers:

- 3 + 1 = - 2

= sec^{2} (θ) - 2

= sec^{2} (θ) - 2

= sec^{2} (θ) - 2

- 2 + sec^{2} (θ) = 0

Solve by substitution

- 2 + sec^{2} (θ) = 0

Let:

sec (θ) = u

- 2 + u^{2} = 0

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

- 2 + u^{2} = 0

Move

2

to the right side

- 2 + u^{2} = 0

Add

2

to both sides

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

Simplify

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 (θ)

sec (θ) = 2, sec (θ) = - 2

sec (θ) = 2, sec (θ) = - 2

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

sec (θ) = 2

General solutions for

sec (θ) = 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}

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

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

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

sec (θ) = - 2

General solutions for

sec (θ) = - 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}

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

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

Combine all the solutions

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

Popular Examples

sin(pi-x)+cos(x)=0

sin (π - x) + cos (x) = 0

sin(x)=-sin(2x)

sin (x) = - sin (2 x)

- 3 = 1 + 4 tan (θ)

2cos(2θ)-3cos(θ)-2=0

2 cos (2 θ) - 3 cos (θ) - 2 = 0

tan(2θ)+6tan(θ)+8=0

tan (2 θ) + 6 tan (θ) + 8 = 0