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Popular >

2sin(x)sec(x)-2sqrt(3)sin(x)=0

Solution

2 sin (x) sec (x) - 23 sin (x) = 0

x = 2 πn, x = π + 2 πn, x = 0.95531 \dots + 2 πn, x = 2 π - 0.95531 \dots + 2 πn

Degrees

x = 0^{\circ} + 36 0^{\circ} n, x = 18 0^{\circ} + 36 0^{\circ} n, x = 54.73561 \dots^{\circ} + 36 0^{\circ} n, x = 305.26438 \dots^{\circ} + 36 0^{\circ} n

Solution steps

2 sin (x) sec (x) - 23 sin (x) = 0

Factor

2 sin (x) sec (x) - 23 sin (x) : 2 sin (x) (sec (x) - 3)

2 sin (x) sec (x) - 23 sin (x)

Factor out common term

2 sin (x)

= 2 sin (x) (sec (x) - 3)

2 sin (x) (sec (x) - 3) = 0

Solving each part separately

sin (x) = 0 or sec (x) - 3 = 0

sin (x) = 0 : x = 2 πn, x = π + 2 πn

sin (x) = 0

General solutions for

sin (x) = 0

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}

x = 0 + 2 πn, x = π + 2 πn

x = 0 + 2 πn, x = π + 2 πn

Solve

x = 0 + 2 πn : x = 2 πn

x = 0 + 2 πn

0 + 2 πn = 2 πn

x = 2 πn

x = 2 πn, x = π + 2 πn

sec (x) - 3 = 0 : x = arcsec (3) + 2 πn, x = 2 π - arcsec (3) + 2 πn

sec (x) - 3 = 0

Move

3

to the right side

sec (x) - 3 = 0

Add

3

to both sides

sec (x) - 3 + 3 = 0 + 3

Simplify

sec (x) = 3

sec (x) = 3

Apply trig inverse properties

sec (x) = 3

General solutions for

sec (x) = 3

sec (x) = a \Rightarrow x = arcsec (a) + 2 πn, x = 2 π - arcsec (a) + 2 πn

x = arcsec (3) + 2 πn, x = 2 π - arcsec (3) + 2 πn

x = arcsec (3) + 2 πn, x = 2 π - arcsec (3) + 2 πn

Combine all the solutions

x = 2 πn, x = π + 2 πn, x = arcsec (3) + 2 πn, x = 2 π - arcsec (3) + 2 πn

Show solutions in decimal form

x = 2 πn, x = π + 2 πn, x = 0.95531 \dots + 2 πn, x = 2 π - 0.95531 \dots + 2 πn

cot (a) sec (a) = cos (a)

cos (2 x) 2 cos (x) = 2 (2 cos^{2} (x) - 1) 2 cos (x)

4 sin^{2} (x) = 4 cos (x) + 1

2 tan^{2} (x) + 3 tan (x) - 2 = 0

sin^{2} (2 x) - \frac{1}{2 3} sin (4 x) = \frac{1}{2}