What is the general solution for sin(a)tan(a)=sin(a) ?

The general solution for sin(a)tan(a)=sin(a) is a=2pin,a=pi+2pin,a= pi/4+pin

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sin(a)tan(a)=sin(a)

sin (a) tan (a) = sin (a)

a = 2 πn, a = π + 2 πn, a = \frac{π}{4} + πn

Degrees

a = 0^{\circ} + 36 0^{\circ} n, a = 18 0^{\circ} + 36 0^{\circ} n, a = 4 5^{\circ} + 18 0^{\circ} n

Solution steps

sin (a) tan (a) = sin (a)

Subtract

sin (a)

from both sides

sin (a) tan (a) - sin (a) = 0

Factor

sin (a) tan (a) - sin (a) : sin (a) (tan (a) - 1)

sin (a) tan (a) - sin (a)

Factor out common term

sin (a)

= sin (a) (tan (a) - 1)

sin (a) (tan (a) - 1) = 0

Solving each part separately

sin (a) = 0 or tan (a) - 1 = 0

sin (a) = 0 : a = 2 πn, a = π + 2 πn

sin (a) = 0

General solutions for

sin (a) = 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}

a = 0 + 2 πn, a = π + 2 πn

a = 0 + 2 πn, a = π + 2 πn

Solve

a = 0 + 2 πn : a = 2 πn

a = 0 + 2 πn

0 + 2 πn = 2 πn

a = 2 πn

a = 2 πn, a = π + 2 πn

tan (a) - 1 = 0 : a = \frac{π}{4} + πn

tan (a) - 1 = 0

Move

1

to the right side

tan (a) - 1 = 0

Add

1

to both sides

tan (a) - 1 + 1 = 0 + 1

Simplify

tan (a) = 1

tan (a) = 1

General solutions for

tan (a) = 1

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}

a = \frac{π}{4} + πn

a = \frac{π}{4} + πn

Combine all the solutions

a = 2 πn, a = π + 2 πn, a = \frac{π}{4} + πn

2 tan (x) + sec^{2} (x) = 0

cos (x) = \frac{1}{9}

sin (θ) + 0.069 = 0

tan (θ) = 1.3

tan (x) = 0.25

What is the general solution for sin(a)tan(a)=sin(a) ?
The general solution for sin(a)tan(a)=sin(a) is a=2pin,a=pi+2pin,a= pi/4+pin