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Popular Trigonometry >

sec^2(θ)tan(θ)=2tan(θ),0<= θ<= 2pi

Solution

sec^{2} (θ) tan (θ) = 2 tan (θ), 0 \leq θ \leq 2 π

Solution

θ = 0, θ = π, θ = 2 π, θ = \frac{3 π}{4}, θ = \frac{5 π}{4}, θ = \frac{π}{4}, θ = \frac{7 π}{4}

Degrees

θ = 0^{\circ}, θ = 18 0^{\circ}, θ = 36 0^{\circ}, θ = 13 5^{\circ}, θ = 22 5^{\circ}, θ = 4 5^{\circ}, θ = 31 5^{\circ}

Solution steps

sec^{2} (θ) tan (θ) = 2 tan (θ), 0 \leq θ \leq 2 π

Subtract

2 tan (θ)

from both sides

sec^{2} (θ) tan (θ) - 2 tan (θ) = 0

Factor

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

sec^{2} (θ) tan (θ) - 2 tan (θ)

Factor out common term

tan (θ)

= tan (θ) (sec^{2} (θ) - 2)

Factor

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

sec^{2} (θ) - 2

Apply radical rule:

a = (a)^{2}

2 = (2)^{2}

= sec^{2} (θ) - (2)^{2}

Apply Difference of Two Squares Formula:

x^{2} - y^{2} = (x + y) (x - y)

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

= (sec (θ) + 2) (sec (θ) - 2)

= tan (θ) (sec (θ) + 2) (sec (θ) - 2)

tan (θ) (sec (θ) + 2) (sec (θ) - 2) = 0

Solving each part separately

tan (θ) = 0 or sec (θ) + 2 = 0 or sec (θ) - 2 = 0

tan (θ) = 0, 0 \leq θ \leq 2 π : θ = 0, θ = π, θ = 2 π

tan (θ) = 0, 0 \leq θ \leq 2 π

General solutions for

tan (θ) = 0

tan (x)

periodicity table with

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} tan (x) 0 \frac{3}{3} 13 \pm \infty - 3 - 1 - \frac{3}{3}

θ = 0 + πn

θ = 0 + πn

Solve

θ = 0 + πn : θ = πn

θ = 0 + πn

0 + πn = πn

θ = πn

θ = πn

Solutions for the range

0 \leq θ \leq 2 π

θ = 0, θ = π, θ = 2 π

sec (θ) + 2 = 0, 0 \leq θ \leq 2 π : θ = \frac{3 π}{4}, θ = \frac{5 π}{4}

sec (θ) + 2 = 0, 0 \leq θ \leq 2 π

Move

2

to the right side

sec (θ) + 2 = 0

Subtract

2

from both sides

sec (θ) + 2 - 2 = 0 - 2

Simplify

sec (θ) = - 2

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

Solutions for the range

0 \leq θ \leq 2 π

θ = \frac{3 π}{4}, θ = \frac{5 π}{4}

sec (θ) - 2 = 0, 0 \leq θ \leq 2 π : θ = \frac{π}{4}, θ = \frac{7 π}{4}

sec (θ) - 2 = 0, 0 \leq θ \leq 2 π

Move

2

to the right side

sec (θ) - 2 = 0

Add

2

to both sides

sec (θ) - 2 + 2 = 0 + 2

Simplify

sec (θ) = 2

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

Solutions for the range

0 \leq θ \leq 2 π

θ = \frac{π}{4}, θ = \frac{7 π}{4}

Combine all the solutions

θ = 0, θ = π, θ = 2 π, θ = \frac{3 π}{4}, θ = \frac{5 π}{4}, θ = \frac{π}{4}, θ = \frac{7 π}{4}

Graph

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