What is the general solution for 2cos(x)=csc(x) ?

The general solution for 2cos(x)=csc(x) is x= pi/4+pin

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2cos(x)=csc(x)

Solution

2 cos (x) = csc (x)

Solution

x = \frac{π}{4} + πn

Degrees

x = 4 5^{\circ} + 18 0^{\circ} n

Solution steps

2 cos (x) = csc (x)

Subtract

csc (x)

from both sides

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

Express with sin, cos

- csc (x) + 2 cos (x)

Use the basic trigonometric identity:

csc (x) = \frac{1}{sin ( x )}

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

Simplify

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

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

Convert element to fraction:

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

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

Since the denominators are equal, combine the fractions:

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

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

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

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

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

- 1 + 2 cos (x) sin (x) = 0

Rewrite using trig identities

- 1 + 2 cos (x) sin (x)

Use the Double Angle identity:

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

= - 1 + sin (2 x)

- 1 + sin (2 x) = 0

Move

1

to the right side

- 1 + sin (2 x) = 0

Add

1

to both sides

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

Simplify

sin (2 x) = 1

sin (2 x) = 1

General solutions for

sin (2 x) = 1

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 x = \frac{π}{2} + 2 πn

2 x = \frac{π}{2} + 2 πn

Solve

2 x = \frac{π}{2} + 2 πn : x = \frac{π}{4} + πn

2 x = \frac{π}{2} + 2 πn

Divide both sides by

2

2 x = \frac{π}{2} + 2 πn

Divide both sides by

2

\frac{2 x}{2} = \frac{\frac{π}{2}}{2} + \frac{2 πn}{2}

Simplify

\frac{2 x}{2} = \frac{\frac{π}{2}}{2} + \frac{2 πn}{2}

Simplify

\frac{2 x}{2} : x

\frac{2 x}{2}

Divide the numbers:

\frac{2}{2} = 1

= x

Simplify

\frac{\frac{π}{2}}{2} + \frac{2 πn}{2} : \frac{π}{4} + πn

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

\frac{\frac{π}{2}}{2} = \frac{π}{4}

\frac{\frac{π}{2}}{2}

Apply the fraction rule:

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

= \frac{π}{2 \cdot 2}

Multiply the numbers:

2 \cdot 2 = 4

= \frac{π}{4}

\frac{2 πn}{2} = πn

\frac{2 πn}{2}

Divide the numbers:

\frac{2}{2} = 1

= πn

= \frac{π}{4} + πn

x = \frac{π}{4} + πn

x = \frac{π}{4} + πn

x = \frac{π}{4} + πn

x = \frac{π}{4} + πn

Popular Examples

cos(θ)=0.3

cos (θ) = 0.3

sin^5(x)=4sin(x)

sin^{5} (x) = 4 sin (x)

tan(x)=2+sqrt(3)

tan (x) = 2 + 3

sin(a)= 11/61

sin (a) = \frac{11}{61}

4sin(2θ)=2

4 sin (2 θ) = 2

Frequently Asked Questions (FAQ)

What is the general solution for 2cos(x)=csc(x) ?
The general solution for 2cos(x)=csc(x) is x= pi/4+pin