What is the general solution for 2csc(x)+7/(cos(x))=0 ?

The general solution for 2csc(x)+7/(cos(x))=0 is x=2.86329…+pin

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Popular Trigonometry >

2csc(x)+7/(cos(x))=0

Solution

2 csc (x) + \frac{7}{cos ( x )} = 0

Solution

x = 2.86329 \dots + πn

Degrees

x = 164.05460 \dots^{\circ} + 18 0^{\circ} n

Solution steps

2 csc (x) + \frac{7}{cos ( x )} = 0

Simplify

2 csc (x) + \frac{7}{cos ( x )} : \frac{2 csc ( x ) cos ( x ) + 7}{cos ( x )}

2 csc (x) + \frac{7}{cos ( x )}

Convert element to fraction:

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

= \frac{2 csc ( x ) cos ( x )}{cos ( x )} + \frac{7}{cos ( x )}

Since the denominators are equal, combine the fractions:

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

= \frac{2 csc ( x ) cos ( x ) + 7}{cos ( x )}

\frac{2 csc ( x ) cos ( x ) + 7}{cos ( x )} = 0

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

2 csc (x) cos (x) + 7 = 0

Rewrite using trig identities

2 csc (x) cos (x) + 7

csc (x) cos (x) = cot (x)

csc (x) cos (x)

Express with sin, cos

csc (x) cos (x)

Use the basic trigonometric identity:

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

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

Simplify

\frac{1}{sin ( x )} cos (x) : \frac{cos ( x )}{sin ( x )}

\frac{1}{sin ( x )} cos (x)

Multiply fractions:

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

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

Multiply:

1 \cdot cos (x) = cos (x)

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

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

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

Use the basic trigonometric identity:

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

= cot (x)

= 7 + 2 cot (x)

7 + 2 cot (x) = 0

Move

7

to the right side

7 + 2 cot (x) = 0

Subtract

7

from both sides

7 + 2 cot (x) - 7 = 0 - 7

Simplify

2 cot (x) = - 7

2 cot (x) = - 7

Divide both sides by

2

2 cot (x) = - 7

Divide both sides by

2

\frac{2 cot ( x )}{2} = \frac{- 7}{2}

Simplify

cot (x) = - \frac{7}{2}

cot (x) = - \frac{7}{2}

Apply trig inverse properties

cot (x) = - \frac{7}{2}

General solutions for

cot (x) = - \frac{7}{2}

cot (x) = - a \Rightarrow x = arccot (- a) + πn

x = arccot (- \frac{7}{2}) + πn

x = arccot (- \frac{7}{2}) + πn

Show solutions in decimal form

x = 2.86329 \dots + πn

Graph

Popular Examples

csc(θ)= 13/6

(tan(θ)cot(θ))/(sec^2(θ))=cot(θ)

sin(y)=(sqrt(3))/2

9sin^2(x)+3cos(x)-7=0

3sec(2x+3)=4

Frequently Asked Questions (FAQ)

What is the general solution for 2csc(x)+7/(cos(x))=0 ?
The general solution for 2csc(x)+7/(cos(x))=0 is x=2.86329…+pin