What is the general solution for sec(x)=(sin(x))/(cos(x)) ?

The general solution for sec(x)=(sin(x))/(cos(x)) is No Solution for x\in\mathbb{R}

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Popular >

sec(x)=(sin(x))/(cos(x))

Solution

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

Solution

No Solution for x \in R

Solution steps

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

Subtract

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

from both sides

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

Simplify

sec (x) - \frac{sin ( x )}{cos ( x )} : \frac{sec ( x ) cos ( x ) - sin ( x )}{cos ( x )}

sec (x) - \frac{sin ( x )}{cos ( x )}

Convert element to fraction:

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

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

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

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

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

Rewrite using trig identities

sec (x) cos (x) - sin (x)

sec (x) cos (x) = 1

sec (x) cos (x)

Express with sin, cos

sec (x) cos (x)

Use the basic trigonometric identity:

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

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

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

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

Multiply fractions:

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

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

Cancel the common factor:

cos (x)

= 1

= 1

= - sin (x) + 1

- sin (x) + 1 = 0

Move

1

to the right side

- sin (x) + 1 = 0

Subtract

1

from both sides

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

Simplify

- sin (x) = - 1

- sin (x) = - 1

Divide both sides by

- 1

- sin (x) = - 1

Divide both sides by

- 1

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

Simplify

sin (x) = 1

sin (x) = 1

General solutions for

sin (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}

x = \frac{π}{2} + 2 πn

x = \frac{π}{2} + 2 πn

Since the equation is undefined for:

\frac{π}{2} + 2 πn

No Solution for x \in R

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(cos(x))/2 =(-sqrt(3))/2

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solvefor b,a=arccos((a^2-b^2-c^2)/(-2bc))solve for

b, a = arccos (\frac{a ^{2} - b ^{2} - c ^{2}}{- 2 b c})

Frequently Asked Questions (FAQ)

What is the general solution for sec(x)=(sin(x))/(cos(x)) ?
The general solution for sec(x)=(sin(x))/(cos(x)) is No Solution for x\in\mathbb{R}