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Popular Trigonometry >

2tan^2(θ)=sec^2(θ)+2

Solution

2 tan^{2} (θ) = sec^{2} (θ) + 2

Solution

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

Degrees

θ = 6 0^{\circ} + 36 0^{\circ} n, θ = 30 0^{\circ} + 36 0^{\circ} n, θ = 12 0^{\circ} + 36 0^{\circ} n, θ = 24 0^{\circ} + 36 0^{\circ} n

Solution steps

2 tan^{2} (θ) = sec^{2} (θ) + 2

Subtract

sec^{2} (θ) + 2

from both sides

2 tan^{2} (θ) - sec^{2} (θ) - 2 = 0

Rewrite using trig identities

- 2 - sec^{2} (θ) + 2 tan^{2} (θ)

Use the Pythagorean identity:

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

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

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

Simplify

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

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

Expand

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

2 (sec^{2} (θ) - 1)

Apply the distributive law:

a (b - c) = ab - a c

a = 2, b = sec^{2} (θ), c = 1

= 2 sec^{2} (θ) - 2 \cdot 1

Multiply the numbers:

2 \cdot 1 = 2

= 2 sec^{2} (θ) - 2

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

Simplify

- 2 - sec^{2} (θ) + 2 sec^{2} (θ) - 2 : sec^{2} (θ) - 4

- 2 - sec^{2} (θ) + 2 sec^{2} (θ) - 2

Add similar elements:

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

= - 2 + sec^{2} (θ) - 2

Group like terms

= sec^{2} (θ) - 2 - 2

Subtract the numbers:

- 2 - 2 = - 4

= sec^{2} (θ) - 4

= sec^{2} (θ) - 4

= sec^{2} (θ) - 4

- 4 + sec^{2} (θ) = 0

Solve by substitution

- 4 + sec^{2} (θ) = 0

Let:

sec (θ) = u

- 4 + u^{2} = 0

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

- 4 + u^{2} = 0

Move

4

to the right side

- 4 + u^{2} = 0

Add

4

to both sides

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

Simplify

u^{2} = 4

u^{2} = 4

For

x^{2} = f (a)

the solutions are

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

u = 4, u = - 4

4 = 2

4

Factor the number:

4 = 2^{2}

= 2^{2}

Apply radical rule:

n a^{n} = a

2^{2} = 2

= 2

- 4 = - 2

- 4

4 = 2

4

Factor the number:

4 = 2^{2}

= 2^{2}

Apply radical rule:

n a^{n} = a

2^{2} = 2

= 2

= - 2

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

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

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

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

Combine all the solutions

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

Graph

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