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Popular Trigonometry >

2sec(u)sin(u)+2=4sin(u)+sec(u)

Solution

2 sec (u) sin (u) + 2 = 4 sin (u) + sec (u)

Solution

u = \frac{π}{6} + 2 πn, u = \frac{5 π}{6} + 2 πn, u = \frac{π}{3} + 2 πn, u = \frac{5 π}{3} + 2 πn

+1

Degrees

u = 3 0^{\circ} + 36 0^{\circ} n, u = 15 0^{\circ} + 36 0^{\circ} n, u = 6 0^{\circ} + 36 0^{\circ} n, u = 30 0^{\circ} + 36 0^{\circ} n

Solution steps

2 sec (u) sin (u) + 2 = 4 sin (u) + sec (u)

Subtract

4 sin (u) + sec (u)

from both sides

2 sec (u) sin (u) + 2 - 4 sin (u) - sec (u) = 0

Factor

2 sec (u) sin (u) + 2 - 4 sin (u) - sec (u) : (- 1 + 2 sin (u)) (sec (u) - 2)

2 sec (u) sin (u) + 2 - 4 sin (u) - sec (u)

Factor

2 sec (u) sin (u) - sec (u) : sec (u) (2 sin (u) - 1)

2 sec (u) sin (u) - sec (u)

Factor out common term

sec (u)

= sec (u) (2 sin (u) - 1)

Factor

2 - 4 sin (u) : 2 (1 - 2 sin (u))

2 - 4 sin (u)

Rewrite as

= 2 \cdot 1 - 2 \cdot 2 sin (u)

Factor out common term

2

= 2 (1 - 2 sin (u))

= sec (u) (2 sin (u) - 1) + 2 (1 - 2 sin (u))

Rewrite as

= (- 1 + 2 sin (u)) sec (u) - 2 (- 1 + 2 sin (u))

Factor out common term

(- 1 + 2 sin (u))

= (- 1 + 2 sin (u)) (sec (u) - 2)

(- 1 + 2 sin (u)) (sec (u) - 2) = 0

Solving each part separately

- 1 + 2 sin (u) = 0 or sec (u) - 2 = 0

- 1 + 2 sin (u) = 0 : u = \frac{π}{6} + 2 πn, u = \frac{5 π}{6} + 2 πn

- 1 + 2 sin (u) = 0

Move

1

to the right side

- 1 + 2 sin (u) = 0

Add

1

to both sides

- 1 + 2 sin (u) + 1 = 0 + 1

Simplify

2 sin (u) = 1

2 sin (u) = 1

Divide both sides by

2

2 sin (u) = 1

Divide both sides by

2

\frac{2 sin ( u )}{2} = \frac{1}{2}

Simplify

sin (u) = \frac{1}{2}

sin (u) = \frac{1}{2}

General solutions for

sin (u) = \frac{1}{2}

sin (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} sin (x) 0 \frac{1}{2} \frac{2}{2} \frac{3}{2} 1 \frac{3}{2} \frac{2}{2} \frac{1}{2} 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} sin (x) 0 - \frac{1}{2} - \frac{2}{2} - \frac{3}{2} - 1 - \frac{3}{2} - \frac{2}{2} - \frac{1}{2}

u = \frac{π}{6} + 2 πn, u = \frac{5 π}{6} + 2 πn

u = \frac{π}{6} + 2 πn, u = \frac{5 π}{6} + 2 πn

sec (u) - 2 = 0 : u = \frac{π}{3} + 2 πn, u = \frac{5 π}{3} + 2 πn

sec (u) - 2 = 0

Move

2

to the right side

sec (u) - 2 = 0

Add

2

to both sides

sec (u) - 2 + 2 = 0 + 2

Simplify

sec (u) = 2

sec (u) = 2

General solutions for

sec (u) = 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}

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

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

Combine all the solutions

u = \frac{π}{6} + 2 πn, u = \frac{5 π}{6} + 2 πn, u = \frac{π}{3} + 2 πn, u = \frac{5 π}{3} + 2 πn

Graph

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