What is the general solution for 5sin(x)=3sin(x)+cos(x) ?

The general solution for 5sin(x)=3sin(x)+cos(x) is x=0.46364…+pin

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5sin(x)=3sin(x)+cos(x)

5 sin (x) = 3 sin (x) + cos (x)

x = 0.46364 \dots + πn

Degrees

x = 26.56505 \dots^{\circ} + 18 0^{\circ} n

Solution steps

5 sin (x) = 3 sin (x) + cos (x)

Subtract

3 sin (x) + cos (x)

from both sides

2 sin (x) - cos (x) = 0

Rewrite using trig identities

2 sin (x) - cos (x) = 0

Divide both sides by

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

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

Simplify

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

Use the basic trigonometric identity:

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

2 tan (x) - 1 = 0

2 tan (x) - 1 = 0

Move

1

to the right side

2 tan (x) - 1 = 0

Add

1

to both sides

2 tan (x) - 1 + 1 = 0 + 1

Simplify

2 tan (x) = 1

2 tan (x) = 1

Divide both sides by

2

2 tan (x) = 1

Divide both sides by

2

\frac{2 tan ( x )}{2} = \frac{1}{2}

Simplify

tan (x) = \frac{1}{2}

tan (x) = \frac{1}{2}

Apply trig inverse properties

tan (x) = \frac{1}{2}

General solutions for

tan (x) = \frac{1}{2}

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

x = arctan (\frac{1}{2}) + πn

x = arctan (\frac{1}{2}) + πn

Show solutions in decimal form

x = 0.46364 \dots + πn

3 - 3 cos (5 x) = 3 cos (5 x)

6 tan^{2} (θ) + 11 tan (θ) = 4 tan (θ) - 2

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

2 cos^{2} (θ) - sin (θ) = 2

θ, 2 sin^{2} (θ) = 1 - sin (θ)

What is the general solution for 5sin(x)=3sin(x)+cos(x) ?
The general solution for 5sin(x)=3sin(x)+cos(x) is x=0.46364…+pin