What is the general solution for 2csc(θ)+3sec(θ)=0 ?

The general solution for 2csc(θ)+3sec(θ)=0 is θ=-0.58800…+pin

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Popular Trigonometry >

2csc(θ)+3sec(θ)=0

Solution

2 csc (θ) + 3 sec (θ) = 0

Solution

θ = - 0.58800 \dots + πn

Degrees

θ = - 33.69006 \dots^{\circ} + 18 0^{\circ} n

Solution steps

2 csc (θ) + 3 sec (θ) = 0

Express with sin, cos

2 csc (θ) + 3 sec (θ)

Use the basic trigonometric identity:

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

= 2 \cdot \frac{1}{sin ( θ )} + 3 sec (θ)

Use the basic trigonometric identity:

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

= 2 \cdot \frac{1}{sin ( θ )} + 3 \cdot \frac{1}{cos ( θ )}

Simplify

2 \cdot \frac{1}{sin ( θ )} + 3 \cdot \frac{1}{cos ( θ )} : \frac{2 cos ( θ ) + 3 sin ( θ )}{sin ( θ ) cos ( θ )}

2 \cdot \frac{1}{sin ( θ )} + 3 \cdot \frac{1}{cos ( θ )}

2 \cdot \frac{1}{sin ( θ )} = \frac{2}{sin ( θ )}

2 \cdot \frac{1}{sin ( θ )}

Multiply fractions:

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

= \frac{1 \cdot 2}{sin ( θ )}

Multiply the numbers:

1 \cdot 2 = 2

= \frac{2}{sin ( θ )}

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

3 \cdot \frac{1}{cos ( θ )}

Multiply fractions:

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

= \frac{1 \cdot 3}{cos ( θ )}

Multiply the numbers:

1 \cdot 3 = 3

= \frac{3}{cos ( θ )}

= \frac{2}{sin ( θ )} + \frac{3}{cos ( θ )}

Least Common Multiplier of

sin (θ), cos (θ) : sin (θ) cos (θ)

sin (θ), cos (θ)

Lowest Common Multiplier (LCM)

Compute an expression comprised of factors that appear either in

sin (θ)

cos (θ)

= sin (θ) cos (θ)

Adjust Fractions based on the LCM

Multiply each numerator by the same amount needed to multiply its

corresponding denominator to turn it into the LCM

sin (θ) cos (θ)

For

\frac{2}{sin ( θ )} :

multiply the denominator and numerator by

cos (θ)

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

For

\frac{3}{cos ( θ )} :

multiply the denominator and numerator by

sin (θ)

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

= \frac{2 cos ( θ )}{sin ( θ ) cos ( θ )} + \frac{3 sin ( θ )}{cos ( θ ) sin ( θ )}

Since the denominators are equal, combine the fractions:

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

= \frac{2 cos ( θ ) + 3 sin ( θ )}{sin ( θ ) cos ( θ )}

= \frac{2 cos ( θ ) + 3 sin ( θ )}{sin ( θ ) cos ( θ )}

\frac{2 cos ( θ ) + 3 sin ( θ )}{cos ( θ ) sin ( θ )} = 0

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

2 cos (θ) + 3 sin (θ) = 0

Rewrite using trig identities

2 cos (θ) + 3 sin (θ) = 0

Divide both sides by

cos (θ), cos (θ) \neq = 0

\frac{2 cos ( θ ) + 3 sin ( θ )}{cos ( θ )} = \frac{0}{cos ( θ )}

Simplify

2 + \frac{3 sin ( θ )}{cos ( θ )} = 0

Use the basic trigonometric identity:

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

2 + 3 tan (θ) = 0

2 + 3 tan (θ) = 0

Move

2

to the right side

2 + 3 tan (θ) = 0

Subtract

2

from both sides

2 + 3 tan (θ) - 2 = 0 - 2

Simplify

3 tan (θ) = - 2

3 tan (θ) = - 2

Divide both sides by

3

3 tan (θ) = - 2

Divide both sides by

3

\frac{3 tan ( θ )}{3} = \frac{- 2}{3}

Simplify

tan (θ) = - \frac{2}{3}

tan (θ) = - \frac{2}{3}

Apply trig inverse properties

tan (θ) = - \frac{2}{3}

General solutions for

tan (θ) = - \frac{2}{3}

tan (x) = - a \Rightarrow x = arctan (- a) + πn

θ = arctan (- \frac{2}{3}) + πn

θ = arctan (- \frac{2}{3}) + πn

Show solutions in decimal form

θ = - 0.58800 \dots + πn

Graph

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Frequently Asked Questions (FAQ)

What is the general solution for 2csc(θ)+3sec(θ)=0 ?
The general solution for 2csc(θ)+3sec(θ)=0 is θ=-0.58800…+pin