What is the general solution for 4sin(y)= 1/(cos(y)) ?

The general solution for 4sin(y)= 1/(cos(y)) is y= pi/(12)+pin,y=(5pi)/(12)+pin

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4sin(y)= 1/(cos(y))

Solution

4 sin (y) = \frac{1}{cos ( y )}

Solution

y = \frac{π}{12} + πn, y = \frac{5 π}{12} + πn

Degrees

y = 1 5^{\circ} + 18 0^{\circ} n, y = 7 5^{\circ} + 18 0^{\circ} n

Solution steps

4 sin (y) = \frac{1}{cos ( y )}

Subtract

\frac{1}{cos ( y )}

from both sides

4 sin (y) - \frac{1}{cos ( y )} = 0

Simplify

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

4 sin (y) - \frac{1}{cos ( y )}

Convert element to fraction:

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

= \frac{4 sin ( y ) cos ( y )}{cos ( y )} - \frac{1}{cos ( y )}

Since the denominators are equal, combine the fractions:

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

= \frac{4 sin ( y ) cos ( y ) - 1}{cos ( y )}

\frac{4 sin ( y ) cos ( y ) - 1}{cos ( y )} = 0

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

4 sin (y) cos (y) - 1 = 0

Rewrite using trig identities

4 sin (y) cos (y) - 1

Use the Double Angle identity:

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

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

= - 1 + 4 \cdot \frac{sin ( 2 y )}{2}

- 1 + 4 \cdot \frac{sin ( 2 y )}{2} = 0

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

4 \cdot \frac{sin ( 2 y )}{2}

Multiply fractions:

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

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

Divide the numbers:

\frac{4}{2} = 2

= 2 sin (2 y)

- 1 + 2 sin (2 y) = 0

Move

1

to the right side

- 1 + 2 sin (2 y) = 0

Add

1

to both sides

- 1 + 2 sin (2 y) + 1 = 0 + 1

Simplify

2 sin (2 y) = 1

2 sin (2 y) = 1

Divide both sides by

2

2 sin (2 y) = 1

Divide both sides by

2

\frac{2 sin ( 2 y )}{2} = \frac{1}{2}

Simplify

sin (2 y) = \frac{1}{2}

sin (2 y) = \frac{1}{2}

General solutions for

sin (2 y) = \frac{1}{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 y = \frac{π}{6} + 2 πn, 2 y = \frac{5 π}{6} + 2 πn

2 y = \frac{π}{6} + 2 πn, 2 y = \frac{5 π}{6} + 2 πn

Solve

2 y = \frac{π}{6} + 2 πn : y = \frac{π}{12} + πn

2 y = \frac{π}{6} + 2 πn

Divide both sides by

2

2 y = \frac{π}{6} + 2 πn

Divide both sides by

2

\frac{2 y}{2} = \frac{\frac{π}{6}}{2} + \frac{2 πn}{2}

Simplify

\frac{2 y}{2} = \frac{\frac{π}{6}}{2} + \frac{2 πn}{2}

Simplify

\frac{2 y}{2} : y

\frac{2 y}{2}

Divide the numbers:

\frac{2}{2} = 1

= y

Simplify

\frac{\frac{π}{6}}{2} + \frac{2 πn}{2} : \frac{π}{12} + πn

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

\frac{\frac{π}{6}}{2} = \frac{π}{12}

\frac{\frac{π}{6}}{2}

Apply the fraction rule:

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

= \frac{π}{6 \cdot 2}

Multiply the numbers:

6 \cdot 2 = 12

= \frac{π}{12}

\frac{2 πn}{2} = πn

\frac{2 πn}{2}

Divide the numbers:

\frac{2}{2} = 1

= πn

= \frac{π}{12} + πn

y = \frac{π}{12} + πn

y = \frac{π}{12} + πn

y = \frac{π}{12} + πn

Solve

2 y = \frac{5 π}{6} + 2 πn : y = \frac{5 π}{12} + πn

2 y = \frac{5 π}{6} + 2 πn

Divide both sides by

2

2 y = \frac{5 π}{6} + 2 πn

Divide both sides by

2

\frac{2 y}{2} = \frac{\frac{5 π}{6}}{2} + \frac{2 πn}{2}

Simplify

\frac{2 y}{2} = \frac{\frac{5 π}{6}}{2} + \frac{2 πn}{2}

Simplify

\frac{2 y}{2} : y

\frac{2 y}{2}

Divide the numbers:

\frac{2}{2} = 1

= y

Simplify

\frac{\frac{5 π}{6}}{2} + \frac{2 πn}{2} : \frac{5 π}{12} + πn

\frac{\frac{5 π}{6}}{2} + \frac{2 πn}{2}

\frac{\frac{5 π}{6}}{2} = \frac{5 π}{12}

\frac{\frac{5 π}{6}}{2}

Apply the fraction rule:

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

= \frac{5 π}{6 \cdot 2}

Multiply the numbers:

6 \cdot 2 = 12

= \frac{5 π}{12}

\frac{2 πn}{2} = πn

\frac{2 πn}{2}

Divide the numbers:

\frac{2}{2} = 1

= πn

= \frac{5 π}{12} + πn

y = \frac{5 π}{12} + πn

y = \frac{5 π}{12} + πn

y = \frac{5 π}{12} + πn

y = \frac{π}{12} + πn, y = \frac{5 π}{12} + πn

Popular Examples

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

2cos(θ)=2sin(θ)

2 cos (θ) = 2 sin (θ)

arcsin(x)= pi/8

arcsin (x) = \frac{π}{8}

5tan(x)=8

5 tan (x) = 8

3sin(b)+3=5sin(b)+2

3 sin (b) + 3 = 5 sin (b) + 2

Frequently Asked Questions (FAQ)

What is the general solution for 4sin(y)= 1/(cos(y)) ?
The general solution for 4sin(y)= 1/(cos(y)) is y= pi/(12)+pin,y=(5pi)/(12)+pin