What is the general solution for cos(φ)=sqrt(3)sin(φ) ?

The general solution for cos(φ)=sqrt(3)sin(φ) is φ= pi/6+pin

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cos(φ)=sqrt(3)sin(φ)

cos (φ) = 3 sin (φ)

φ = \frac{π}{6} + πn

Degrees

φ = 3 0^{\circ} + 18 0^{\circ} n

Solution steps

cos (φ) = 3 sin (φ)

Rewrite using trig identities

cos (φ) = 3 sin (φ)

Divide both sides by

cos (φ), cos (φ) \neq = 0

\frac{cos ( φ )}{cos ( φ )} = \frac{3 sin ( φ )}{cos ( φ )}

Simplify

1 = \frac{3 sin ( φ )}{cos ( φ )}

Use the basic trigonometric identity:

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

1 = 3 tan (φ)

Switch sides

3 tan (φ) = 1

3 tan (φ) = 1

Divide both sides by

3

3 tan (φ) = 1

Divide both sides by

3

\frac{3 tan ( φ )}{3} = \frac{1}{3}

Simplify

\frac{3 tan ( φ )}{3} = \frac{1}{3}

Simplify

\frac{3 tan ( φ )}{3} : tan (φ)

\frac{3 tan ( φ )}{3}

Cancel the common factor:

3

= tan (φ)

Simplify

\frac{1}{3} : \frac{3}{3}

\frac{1}{3}

Multiply by the conjugate

\frac{3}{3}

= \frac{1 \cdot 3}{3 3}

1 \cdot 3 = 3

33 = 3

33

Apply radical rule:

a a = a

33 = 3

= 3

= \frac{3}{3}

tan (φ) = \frac{3}{3}

tan (φ) = \frac{3}{3}

tan (φ) = \frac{3}{3}

General solutions for

tan (φ) = \frac{3}{3}

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}

φ = \frac{π}{6} + πn

φ = \frac{π}{6} + πn

5 sin (x) - 3 cos (2 x) + 3 = 0

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

2 cos^{2} (x) + cos (x) = 1, 0 \leq x < 2 π

sin (β) = - 0, 8 (θ \in β v c) s = sec (β) - tan (β)

cos (\frac{2 x - π}{17}) = 0

What is the general solution for cos(φ)=sqrt(3)sin(φ) ?
The general solution for cos(φ)=sqrt(3)sin(φ) is φ= pi/6+pin