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Popular Trigonometry >

5sec^3(x)-20sec(x)=0

Solution

5 sec^{3} (x) - 20 sec (x) = 0

Solution

x = \frac{2 π}{3} + 2 πn, x = \frac{4 π}{3} + 2 πn, x = \frac{π}{3} + 2 πn, x = \frac{5 π}{3} + 2 πn

+1

Degrees

x = 12 0^{\circ} + 36 0^{\circ} n, x = 24 0^{\circ} + 36 0^{\circ} n, x = 6 0^{\circ} + 36 0^{\circ} n, x = 30 0^{\circ} + 36 0^{\circ} n

Solution steps

5 sec^{3} (x) - 20 sec (x) = 0

Solve by substitution

5 sec^{3} (x) - 20 sec (x) = 0

Let:

sec (x) = u

5 u^{3} - 20 u = 0

5 u^{3} - 20 u = 0 : u = 0, u = - 2, u = 2

5 u^{3} - 20 u = 0

Factor

5 u^{3} - 20 u : 5 u (u + 2) (u - 2)

5 u^{3} - 20 u

Factor out common term

5 u : 5 u (u^{2} - 4)

5 u^{3} - 20 u

Apply exponent rule:

a^{b + c} = a^{b} a^{c}

u^{3} = u^{2} u

= 5 u^{2} u - 20 u

Rewrite

20

as

5 \cdot 4

= 5 u^{2} u - 5 \cdot 4 u

Factor out common term

5 u

= 5 u (u^{2} - 4)

= 5 u (u^{2} - 4)

Factor

u^{2} - 4 : (u + 2) (u - 2)

u^{2} - 4

Rewrite

4

as

2^{2}

= u^{2} - 2^{2}

Apply Difference of Two Squares Formula:

x^{2} - y^{2} = (x + y) (x - y)

u^{2} - 2^{2} = (u + 2) (u - 2)

= (u + 2) (u - 2)

= 5 u (u + 2) (u - 2)

5 u (u + 2) (u - 2) = 0

Using the Zero Factor Principle:

If

ab = 0

then

a = 0

or

b = 0

u = 0 or u + 2 = 0 or u - 2 = 0

Solve

u + 2 = 0 : u = - 2

u + 2 = 0

Move

2

to the right side

u + 2 = 0

Subtract

2

from both sides

u + 2 - 2 = 0 - 2

Simplify

u = - 2

u = - 2

Solve

u - 2 = 0 : u = 2

u - 2 = 0

Move

2

to the right side

u - 2 = 0

Add

2

to both sides

u - 2 + 2 = 0 + 2

Simplify

u = 2

u = 2

The solutions are

u = 0, u = - 2, u = 2

Substitute back

u = sec (x)

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

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

sec (x) = 0 :

No Solution

sec (x) = 0

sec (x) \leq - 1 or sec (x) \geq 1

No Solution

sec (x) = - 2 : x = \frac{2 π}{3} + 2 πn, x = \frac{4 π}{3} + 2 πn

sec (x) = - 2

General solutions for

sec (x) = - 2

sec (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} sec (x) 1 \frac{2 3}{3} 22 Undefined - 2 - 2 - \frac{2 3}{3} 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} sec (x) - 1 - \frac{2 3}{3} - 2 - 2 Undefined 22 \frac{2 3}{3}

x = \frac{2 π}{3} + 2 πn, x = \frac{4 π}{3} + 2 πn

x = \frac{2 π}{3} + 2 πn, x = \frac{4 π}{3} + 2 πn

sec (x) = 2 : x = \frac{π}{3} + 2 πn, x = \frac{5 π}{3} + 2 πn

sec (x) = 2

General solutions for

sec (x) = 2

sec (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} sec (x) 1 \frac{2 3}{3} 22 Undefined - 2 - 2 - \frac{2 3}{3} 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} sec (x) - 1 - \frac{2 3}{3} - 2 - 2 Undefined 22 \frac{2 3}{3}

x = \frac{π}{3} + 2 πn, x = \frac{5 π}{3} + 2 πn

x = \frac{π}{3} + 2 πn, x = \frac{5 π}{3} + 2 πn

Combine all the solutions

x = \frac{2 π}{3} + 2 πn, x = \frac{4 π}{3} + 2 πn, x = \frac{π}{3} + 2 πn, x = \frac{5 π}{3} + 2 πn

Graph

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