What is the general solution for cos^3(x)= 5/8 ?

The general solution for cos^3(x)= 5/8 is x=0.54526…+2pin,x=2pi-0.54526…+2pin

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cos^3(x)= 5/8

Solution

cos^{3} (x) = \frac{5}{8}

Solution

x = 0.54526 \dots + 2 πn, x = 2 π - 0.54526 \dots + 2 πn

Degrees

x = 31.24159 \dots^{\circ} + 36 0^{\circ} n, x = 328.75840 \dots^{\circ} + 36 0^{\circ} n

Solution steps

cos^{3} (x) = \frac{5}{8}

Solve by substitution

cos^{3} (x) = \frac{5}{8}

Let:

cos (x) = u

u^{3} = \frac{5}{8}

u^{3} = \frac{5}{8}

For

x^{3} = f (a)

the solutions are

Apply radical rule:

assuming

a \geq 0, b \geq 0

Factor the number:

8 = 2^{3}

Apply radical rule:

= 2

Simplify

Apply radical rule:

assuming

a \geq 0, b \geq 0

Factor the number:

8 = 2^{3}

Apply radical rule:

= 2

Multiply fractions:

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

Multiply the numbers:

2 \cdot 2 = 4

Rewrite

in standard complex form:

Expand

Apply the distributive law:

a (b + c) = ab + a c

Apply minus-plus rules

+ (- a) = - a

Multiply:

Apply the fraction rule:

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

Simplify

Apply radical rule:

assuming

a \geq 0, b \geq 0

Factor the number:

8 = 2^{3}

Apply radical rule:

= 2

Multiply fractions:

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

Multiply the numbers:

2 \cdot 2 = 4

Rewrite

in standard complex form:

Expand

Apply the distributive law:

a (b - c) = ab - a c

Apply minus-plus rules

+ (- a) = - a

Multiply:

Apply the fraction rule:

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

Substitute back

u = cos (x)

Apply trig inverse properties

General solutions for

cos (x) = a \Rightarrow x = arccos (a) + 2 πn, x = 2 π - arccos (a) + 2 πn

No Solution

No Solution

No Solution

No Solution

Combine all the solutions

Show solutions in decimal form

x = 0.54526 \dots + 2 πn, x = 2 π - 0.54526 \dots + 2 πn

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Frequently Asked Questions (FAQ)

What is the general solution for cos^3(x)= 5/8 ?
The general solution for cos^3(x)= 5/8 is x=0.54526…+2pin,x=2pi-0.54526…+2pin