What is the general solution for 4sin(θ)=sqrt(3)sec(θ) ?

The general solution for 4sin(θ)=sqrt(3)sec(θ) is θ= pi/6+pin,θ= pi/3+pin

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4sin(θ)=sqrt(3)sec(θ)

Solution

4 sin (θ) = 3 sec (θ)

Solution

θ = \frac{π}{6} + πn, θ = \frac{π}{3} + πn

Degrees

θ = 3 0^{\circ} + 18 0^{\circ} n, θ = 6 0^{\circ} + 18 0^{\circ} n

Solution steps

4 sin (θ) = 3 sec (θ)

Subtract

3 sec (θ)

from both sides

4 sin (θ) - 3 sec (θ) = 0

Express with sin, cos

4 sin (θ) - sec (θ) 3

Use the basic trigonometric identity:

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

= 4 sin (θ) - \frac{1}{cos ( θ )} 3

Simplify

4 sin (θ) - \frac{1}{cos ( θ )} 3 : \frac{4 sin ( θ ) cos ( θ ) - 3}{cos ( θ )}

4 sin (θ) - \frac{1}{cos ( θ )} 3

\frac{1}{cos ( θ )} 3 = \frac{3}{cos ( θ )}

\frac{1}{cos ( θ )} 3

Multiply fractions:

a \cdot \frac{b}{c} = \frac{a \cdot b}{c}

= \frac{1 \cdot 3}{cos ( θ )}

Multiply:

1 \cdot 3 = 3

= \frac{3}{cos ( θ )}

= 4 sin (θ) - \frac{3}{cos ( θ )}

Convert element to fraction:

4 sin (θ) = \frac{4 sin ( θ ) cos ( θ )}{cos ( θ )}

= \frac{4 sin ( θ ) cos ( θ )}{cos ( θ )} - \frac{3}{cos ( θ )}

Since the denominators are equal, combine the fractions:

\frac{a}{c} \pm \frac{b}{c} = \frac{a \pm b}{c}

= \frac{4 sin ( θ ) cos ( θ ) - 3}{cos ( θ )}

= \frac{4 sin ( θ ) cos ( θ ) - 3}{cos ( θ )}

\frac{- 3 + 4 cos ( θ ) sin ( θ )}{cos ( θ )} = 0

\frac{f ( x )}{g ( x )} = 0 \Rightarrow f (x) = 0

- 3 + 4 cos (θ) sin (θ) = 0

Rewrite using trig identities

- 3 + 4 cos (θ) sin (θ)

Use the Double Angle identity:

2 sin (x) cos (x) = sin (2 x)

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

= - 3 + 4 \cdot \frac{sin ( 2 θ )}{2}

- 3 + 4 \cdot \frac{sin ( 2 θ )}{2} = 0

4 \cdot \frac{sin ( 2 θ )}{2} = 2 sin (2 θ)

4 \cdot \frac{sin ( 2 θ )}{2}

Multiply fractions:

a \cdot \frac{b}{c} = \frac{a \cdot b}{c}

= \frac{sin ( 2 θ ) \cdot 4}{2}

Divide the numbers:

\frac{4}{2} = 2

= 2 sin (2 θ)

- 3 + 2 sin (2 θ) = 0

Move

3

to the right side

- 3 + 2 sin (2 θ) = 0

Add

3

to both sides

- 3 + 2 sin (2 θ) + 3 = 0 + 3

Simplify

2 sin (2 θ) = 3

2 sin (2 θ) = 3

Divide both sides by

2

2 sin (2 θ) = 3

Divide both sides by

2

\frac{2 sin ( 2 θ )}{2} = \frac{3}{2}

Simplify

sin (2 θ) = \frac{3}{2}

sin (2 θ) = \frac{3}{2}

General solutions for

sin (2 θ) = \frac{3}{2}

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}

2 θ = \frac{π}{3} + 2 πn, 2 θ = \frac{2 π}{3} + 2 πn

2 θ = \frac{π}{3} + 2 πn, 2 θ = \frac{2 π}{3} + 2 πn

Solve

2 θ = \frac{π}{3} + 2 πn : θ = \frac{π}{6} + πn

2 θ = \frac{π}{3} + 2 πn

Divide both sides by

2

2 θ = \frac{π}{3} + 2 πn

Divide both sides by

2

\frac{2 θ}{2} = \frac{\frac{π}{3}}{2} + \frac{2 πn}{2}

Simplify

\frac{2 θ}{2} = \frac{\frac{π}{3}}{2} + \frac{2 πn}{2}

Simplify

\frac{2 θ}{2} : θ

\frac{2 θ}{2}

Divide the numbers:

\frac{2}{2} = 1

= θ

Simplify

\frac{\frac{π}{3}}{2} + \frac{2 πn}{2} : \frac{π}{6} + πn

\frac{\frac{π}{3}}{2} + \frac{2 πn}{2}

\frac{\frac{π}{3}}{2} = \frac{π}{6}

\frac{\frac{π}{3}}{2}

Apply the fraction rule:

\frac{\frac{b}{c}}{a} = \frac{b}{c \cdot a}

= \frac{π}{3 \cdot 2}

Multiply the numbers:

3 \cdot 2 = 6

= \frac{π}{6}

\frac{2 πn}{2} = πn

\frac{2 πn}{2}

Divide the numbers:

\frac{2}{2} = 1

= πn

= \frac{π}{6} + πn

θ = \frac{π}{6} + πn

θ = \frac{π}{6} + πn

θ = \frac{π}{6} + πn

Solve

2 θ = \frac{2 π}{3} + 2 πn : θ = \frac{π}{3} + πn

2 θ = \frac{2 π}{3} + 2 πn

Divide both sides by

2

2 θ = \frac{2 π}{3} + 2 πn

Divide both sides by

2

\frac{2 θ}{2} = \frac{\frac{2 π}{3}}{2} + \frac{2 πn}{2}

Simplify

\frac{2 θ}{2} = \frac{\frac{2 π}{3}}{2} + \frac{2 πn}{2}

Simplify

\frac{2 θ}{2} : θ

\frac{2 θ}{2}

Divide the numbers:

\frac{2}{2} = 1

= θ

Simplify

\frac{\frac{2 π}{3}}{2} + \frac{2 πn}{2} : \frac{π}{3} + πn

\frac{\frac{2 π}{3}}{2} + \frac{2 πn}{2}

\frac{\frac{2 π}{3}}{2} = \frac{π}{3}

\frac{\frac{2 π}{3}}{2}

Apply the fraction rule:

\frac{\frac{b}{c}}{a} = \frac{b}{c \cdot a}

= \frac{2 π}{3 \cdot 2}

Multiply the numbers:

3 \cdot 2 = 6

= \frac{2 π}{6}

Cancel the common factor:

2

= \frac{π}{3}

\frac{2 πn}{2} = πn

\frac{2 πn}{2}

Divide the numbers:

\frac{2}{2} = 1

= πn

= \frac{π}{3} + πn

θ = \frac{π}{3} + πn

θ = \frac{π}{3} + πn

θ = \frac{π}{3} + πn

θ = \frac{π}{6} + πn, θ = \frac{π}{3} + πn

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Frequently Asked Questions (FAQ)

What is the general solution for 4sin(θ)=sqrt(3)sec(θ) ?
The general solution for 4sin(θ)=sqrt(3)sec(θ) is θ= pi/6+pin,θ= pi/3+pin