What is the general solution for 3sin^2(t)=2cos(t)sin(t) ?

The general solution for 3sin^2(t)=2cos(t)sin(t) is t=2pin,t=pi+2pin,t=0.58800…+pin

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Popular >

3sin^2(t)=2cos(t)sin(t)

Solution

3 sin^{2} (t) = 2 cos (t) sin (t)

Solution

t = 2 πn, t = π + 2 πn, t = 0.58800 \dots + πn

Degrees

t = 0^{\circ} + 36 0^{\circ} n, t = 18 0^{\circ} + 36 0^{\circ} n, t = 33.69006 \dots^{\circ} + 18 0^{\circ} n

Solution steps

3 sin^{2} (t) = 2 cos (t) sin (t)

Subtract

2 cos (t) sin (t)

from both sides

3 sin^{2} (t) - 2 cos (t) sin (t) = 0

Factor

3 sin^{2} (t) - 2 cos (t) sin (t) : sin (t) (3 sin (t) - 2 cos (t))

3 sin^{2} (t) - 2 cos (t) sin (t)

Apply exponent rule:

a^{b + c} = a^{b} a^{c}

sin^{2} (t) = sin (t) sin (t)

= 3 sin (t) sin (t) - 2 sin (t) cos (t)

Factor out common term

sin (t)

= sin (t) (3 sin (t) - 2 cos (t))

sin (t) (3 sin (t) - 2 cos (t)) = 0

Solving each part separately

sin (t) = 0 or 3 sin (t) - 2 cos (t) = 0

sin (t) = 0 : t = 2 πn, t = π + 2 πn

sin (t) = 0

General solutions for

sin (t) = 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}

t = 0 + 2 πn, t = π + 2 πn

t = 0 + 2 πn, t = π + 2 πn

Solve

t = 0 + 2 πn : t = 2 πn

t = 0 + 2 πn

0 + 2 πn = 2 πn

t = 2 πn

t = 2 πn, t = π + 2 πn

3 sin (t) - 2 cos (t) = 0 : t = arctan (\frac{2}{3}) + πn

3 sin (t) - 2 cos (t) = 0

Rewrite using trig identities

3 sin (t) - 2 cos (t) = 0

Divide both sides by

cos (t), cos (t) \neq = 0

\frac{3 sin ( t ) - 2 cos ( t )}{cos ( t )} = \frac{0}{cos ( t )}

Simplify

\frac{3 sin ( t )}{cos ( t )} - 2 = 0

Use the basic trigonometric identity:

\frac{sin ( x )}{cos ( x )} = tan (x)

3 tan (t) - 2 = 0

3 tan (t) - 2 = 0

Move

2

to the right side

3 tan (t) - 2 = 0

Add

2

to both sides

3 tan (t) - 2 + 2 = 0 + 2

Simplify

3 tan (t) = 2

3 tan (t) = 2

Divide both sides by

3

3 tan (t) = 2

Divide both sides by

3

\frac{3 tan ( t )}{3} = \frac{2}{3}

Simplify

tan (t) = \frac{2}{3}

tan (t) = \frac{2}{3}

Apply trig inverse properties

tan (t) = \frac{2}{3}

General solutions for

tan (t) = \frac{2}{3}

tan (x) = a \Rightarrow x = arctan (a) + πn

t = arctan (\frac{2}{3}) + πn

t = arctan (\frac{2}{3}) + πn

Combine all the solutions

t = 2 πn, t = π + 2 πn, t = arctan (\frac{2}{3}) + πn

Show solutions in decimal form

t = 2 πn, t = π + 2 πn, t = 0.58800 \dots + πn

Popular Examples

sin(x+pi/4)+sin(x-pi/4)=0

sin (x + \frac{π}{4}) + sin (x - \frac{π}{4}) = 0

cos(x)(tan(x)+(cos(x))/(sin(x)))=sqrt(2)

cos (x) (tan (x) + \frac{cos ( x )}{sin ( x )}) = 2

2cos(2x)+sin(x)= 1/2

2 cos (2 x) + sin (x) = \frac{1}{2}

2cos^2(x)+cos(x)+1=0

2 cos^{2} (x) + cos (x) + 1 = 0

cos^2(x/2)-sin^2(x/2)=1-sin(x)

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

Frequently Asked Questions (FAQ)

What is the general solution for 3sin^2(t)=2cos(t)sin(t) ?
The general solution for 3sin^2(t)=2cos(t)sin(t) is t=2pin,t=pi+2pin,t=0.58800…+pin