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

The general solution for sec(x)=2csc(x) is x=1.10714…+pin

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Popular >

sec(x)=2csc(x)

Solution

sec (x) = 2 csc (x)

Solution

x = 1.10714 \dots + πn

Degrees

x = 63.43494 \dots^{\circ} + 18 0^{\circ} n

Solution steps

sec (x) = 2 csc (x)

Subtract

2 csc (x)

from both sides

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

Express with sin, cos

sec (x) - 2 csc (x)

Use the basic trigonometric identity:

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

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

Use the basic trigonometric identity:

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

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

Simplify

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

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

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

2 \cdot \frac{1}{sin ( x )}

Multiply fractions:

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

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

Multiply the numbers:

1 \cdot 2 = 2

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

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

Least Common Multiplier of

cos (x), sin (x) : cos (x) sin (x)

cos (x), sin (x)

Lowest Common Multiplier (LCM)

Compute an expression comprised of factors that appear either in

cos (x)

sin (x)

= cos (x) sin (x)

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

cos (x) sin (x)

For

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

multiply the denominator and numerator by

sin (x)

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

For

\frac{2}{sin ( x )} :

multiply the denominator and numerator by

cos (x)

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

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

Since the denominators are equal, combine the fractions:

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

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

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

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

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

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

Rewrite using trig identities

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

Divide both sides by

cos (x), cos (x) \neq = 0

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

Simplify

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

Use the basic trigonometric identity:

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

tan (x) - 2 = 0

tan (x) - 2 = 0

Move

2

to the right side

tan (x) - 2 = 0

Add

2

to both sides

tan (x) - 2 + 2 = 0 + 2

Simplify

tan (x) = 2

tan (x) = 2

Apply trig inverse properties

tan (x) = 2

General solutions for

tan (x) = 2

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

x = arctan (2) + πn

x = arctan (2) + πn

Show solutions in decimal form

x = 1.10714 \dots + πn

Popular Examples

cos^2(x)=0.75

cos^{2} (x) = 0.75

tan(x)= pi/2

tan (x) = \frac{π}{2}

tan(x)= pi/4

tan (x) = \frac{π}{4}

13cos(x)+7=-cos(x)

13 cos (x) + 7 = - cos (x)

cos^2(x)=0.25

cos^{2} (x) = 0.25

Frequently Asked Questions (FAQ)

What is the general solution for sec(x)=2csc(x) ?
The general solution for sec(x)=2csc(x) is x=1.10714…+pin