What is the general solution for cos(x)=(sqrt(125))/(sqrt(174)) ?

The general solution for cos(x)=(sqrt(125))/(sqrt(174)) is No Solution for x\in\mathbb{R}

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cos(x)=(sqrt(125))/(sqrt(174))

cos (x^{\circ}) = \frac{125}{174}

No Solution for x \in R

Solution steps

cos (x^{\circ}) = \frac{125}{174}

Simplify

\frac{125}{174} : \frac{5 870}{174}

\frac{125}{174}

125 = 55

125

Prime factorization of

125 : 5^{3}

125

125

divides by

5125 = 25 \cdot 5

= 5 \cdot 25

25

divides by

525 = 5 \cdot 5

= 5 \cdot 5 \cdot 5

5

is a prime number, therefore no further factorization is possible

= 5 \cdot 5 \cdot 5

= 5^{3}

= 5^{3}

Apply exponent rule:

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

= 5^{2} \cdot 5

Apply radical rule:

= 5 5^{2}

Apply radical rule:

5^{2} = 5

= 55

= \frac{5 5}{174}

Rationalize

\frac{5 5}{174} : \frac{5 870}{174}

\frac{5 5}{174}

Multiply by the conjugate

\frac{174}{174}

= \frac{5 5 174}{174 174}

55174 = 5870

55174

Apply radical rule:

a b = a \cdot b

5174 = 5 \cdot 174

= 5 5 \cdot 174

Multiply the numbers:

5 \cdot 174 = 870

= 5870

174174 = 174

174174

Apply radical rule:

a a = a

174174 = 174

= 174

= \frac{5 870}{174}

= \frac{5 870}{174}

cos (x^{\circ}) = \frac{5 870}{174}

Apply trig inverse properties

cos (x^{\circ}) = \frac{5 870}{174}

General solutions for

cos (x^{\circ}) = \frac{5 870}{174}

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

x^{\circ} = arccos (\frac{5 870}{174}) + 2 πn, x^{\circ} = 2 π - arccos (\frac{5 870}{174}) + 2 πn

x^{\circ} = arccos (\frac{5 870}{174}) + 2 πn, x^{\circ} = 2 π - arccos (\frac{5 870}{174}) + 2 πn

No Solution for x \in R

csc (θ) = 23

2 csc^{2} (x) + 5 cot (x) = 5

60 sin (4 5^{\circ}) = 400 sin (θ)

(1 + tan (x)) (5 sin (x) - 2) = 0

arccos (e - x) = π^{3}

What is the general solution for cos(x)=(sqrt(125))/(sqrt(174)) ?
The general solution for cos(x)=(sqrt(125))/(sqrt(174)) is No Solution for x\in\mathbb{R}