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Popular >

y= 1/(3cos^3(x))-1/(cos(x))

Solution

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

Solution

Period : 2 π

Domain : 2 πn \leq x < \frac{π}{2} + 2 πn or \frac{π}{2} + 2 πn < x < \frac{3 π}{2} + 2 πn or \frac{3 π}{2} + 2 πn < x < 2 π + 2 πn

Range : - \infty < f (x) < \infty

X Intercepts : (arccos (\frac{1}{3}) + 2 πn, 0), (2 π - arccos (\frac{1}{3}) + 2 πn, 0), (arccos (- \frac{1}{3}) + 2 πn, 0), (- arccos (- \frac{1}{3}) + 2 π + 2 πn, 0), Y Intercepts : (0, - \frac{2}{3})

Asymptotes : Vertical x = \frac{3 π}{2} + 2 πn, x = \frac{π}{2} + 2 πn

Extreme Points : Minimum (2 πn, - \frac{2}{3}), Maximum (π + 2 πn, \frac{2}{3})

+1

Interval Notation

Domain : [2 πn, \frac{π}{2} + 2 πn) \cup (\frac{π}{2} + 2 πn, \frac{3 π}{2} + 2 πn) \cup (\frac{3 π}{2} + 2 πn, 2 π + 2 πn)

Range : (- \infty, \infty)

Solution steps

Periodicity of

\frac{1}{3 cos ^{3} ( x )} - \frac{1}{cos ( x )} : 2 π

Domain of

\frac{1}{3 cos ^{3} ( x )} - \frac{1}{cos ( x )} : 2 πn \leq x < \frac{π}{2} + 2 πn or \frac{π}{2} + 2 πn < x < \frac{3 π}{2} + 2 πn or \frac{3 π}{2} + 2 πn < x < 2 π + 2 πn

Range of

\frac{1}{3 cos ^{3} ( x )} - \frac{1}{cos ( x )} : - \infty < f (x) < \infty

Axis interception points of

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

X Intercepts

: (arccos (\frac{1}{3}) + 2 πn, 0), (2 π - arccos (\frac{1}{3}) + 2 πn, 0), (arccos (- \frac{1}{3}) + 2 πn, 0), (- arccos (- \frac{1}{3}) + 2 π + 2 πn, 0),

Y Intercepts

: (0, - \frac{2}{3})

Asymptotes of

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

Vertical

: x = \frac{3 π}{2} + 2 πn, x = \frac{π}{2} + 2 πn

Extreme Points of

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

Minimum

(2 πn, - \frac{2}{3}),

Maximum

(π + 2 πn, \frac{2}{3})

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Frequently Asked Questions (FAQ)

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