What is the general solution for 3tan(x)= 1/(cos(x)) ?

The general solution for 3tan(x)= 1/(cos(x)) is x=0.33983…+2pin,x=pi-0.33983…+2pin

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Popular Trigonometry >

3tan(x)= 1/(cos(x))

Solution

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

Solution

x = 0.33983 \dots + 2 πn, x = π - 0.33983 \dots + 2 πn

Degrees

x = 19.47122 \dots^{\circ} + 36 0^{\circ} n, x = 160.52877 \dots^{\circ} + 36 0^{\circ} n

Solution steps

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

Subtract

\frac{1}{cos ( x )}

from both sides

3 tan (x) - \frac{1}{cos ( x )} = 0

Simplify

3 tan (x) - \frac{1}{cos ( x )} : \frac{3 tan ( x ) cos ( x ) - 1}{cos ( x )}

3 tan (x) - \frac{1}{cos ( x )}

Convert element to fraction:

3 tan (x) = \frac{3 tan ( x ) cos ( x )}{cos ( x )}

= \frac{3 tan ( x ) cos ( x )}{cos ( x )} - \frac{1}{cos ( x )}

Since the denominators are equal, combine the fractions:

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

= \frac{3 tan ( x ) cos ( x ) - 1}{cos ( x )}

\frac{3 tan ( x ) cos ( x ) - 1}{cos ( x )} = 0

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

3 tan (x) cos (x) - 1 = 0

Rewrite using trig identities

3 tan (x) cos (x) - 1

cos (x) tan (x) = sin (x)

cos (x) tan (x)

Use the basic trigonometric identity:

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

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

Simplify

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

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

Multiply fractions:

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

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

Cancel the common factor:

cos (x)

= sin (x)

= sin (x)

= - 1 + 3 sin (x)

- 1 + 3 sin (x) = 0

Move

1

to the right side

- 1 + 3 sin (x) = 0

Add

1

to both sides

- 1 + 3 sin (x) + 1 = 0 + 1

Simplify

3 sin (x) = 1

3 sin (x) = 1

Divide both sides by

3

3 sin (x) = 1

Divide both sides by

3

\frac{3 sin ( x )}{3} = \frac{1}{3}

Simplify

sin (x) = \frac{1}{3}

sin (x) = \frac{1}{3}

Apply trig inverse properties

sin (x) = \frac{1}{3}

General solutions for

sin (x) = \frac{1}{3}

sin (x) = a \Rightarrow x = arcsin (a) + 2 πn, x = π - arcsin (a) + 2 πn

x = arcsin (\frac{1}{3}) + 2 πn, x = π - arcsin (\frac{1}{3}) + 2 πn

x = arcsin (\frac{1}{3}) + 2 πn, x = π - arcsin (\frac{1}{3}) + 2 πn

Show solutions in decimal form

x = 0.33983 \dots + 2 πn, x = π - 0.33983 \dots + 2 πn

Graph

Popular Examples

cos(x)-sin(x)=0,-pi<= x<= pi

Frequently Asked Questions (FAQ)

What is the general solution for 3tan(x)= 1/(cos(x)) ?
The general solution for 3tan(x)= 1/(cos(x)) is x=0.33983…+2pin,x=pi-0.33983…+2pin