What is the general solution for 5/(sqrt(21)*\sqrt{30)}=cos(x) ?

The general solution for 5/(sqrt(21)*\sqrt{30)}=cos(x) is x=1.37024…+2pin,x=2pi-1.37024…+2pin

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5/(sqrt(21)*\sqrt{30)}=cos(x)

Solution

\frac{5}{21 \cdot 30} = cos (x)

Solution

x = 1.37024 \dots + 2 πn, x = 2 π - 1.37024 \dots + 2 πn

Degrees

x = 78.50954 \dots^{\circ} + 36 0^{\circ} n, x = 281.49045 \dots^{\circ} + 36 0^{\circ} n

Solution steps

\frac{5}{21 30} = cos (x)

Switch sides

cos (x) = \frac{5}{21 30}

Simplify

\frac{5}{21 30} : \frac{70}{42}

\frac{5}{21 30}

Factor

30 : 235

Factor

30 = 2 \cdot 3 \cdot 5

= 2 \cdot 3 \cdot 5

Apply radical rule:

= 235

= \frac{5}{21 2 3 5}

Cancel

\frac{5}{21 2 3 5} : \frac{5}{21 2 3}

\frac{5}{21 2 3 5}

Apply radical rule:

5 = 5^{\frac{1}{2}}

= \frac{5}{21 2 3 \cdot 5 ^{\frac{1}{2}}}

Apply exponent rule:

\frac{x ^{a}}{x ^{b}} = x^{a - b}

\frac{5 ^{1}}{5 ^{\frac{1}{2}}} = 5^{1 - \frac{1}{2}}

= \frac{5 ^{1 - \frac{1}{2}}}{21 2 3}

Subtract the numbers:

1 - \frac{1}{2} = \frac{1}{2}

= \frac{5 ^{\frac{1}{2}}}{21 2 3}

Apply radical rule:

5^{\frac{1}{2}} = 5

= \frac{5}{21 2 3}

= \frac{5}{21 2 3}

Simplify

2123 : 372

2123

Factor integer

21 = 3 \cdot 7

= 3 \cdot 7 23

Apply radical rule:

3 \cdot 7 = 37

= 3723

Apply radical rule:

a a = a

33 = 3

= 372

= \frac{5}{3 7 2}

Simplify

72 \cdot 3 : 314

72 \cdot 3

Apply radical rule:

a b = a \cdot b

72 = 7 \cdot 2

= 3 7 \cdot 2

Multiply the numbers:

7 \cdot 2 = 14

= 314

= \frac{5}{3 14}

Rationalize

\frac{5}{3 14} : \frac{70}{42}

\frac{5}{3 14}

Multiply by the conjugate

\frac{14}{14}

= \frac{5 14}{14 \cdot 3 14}

514 = 70

514

Apply radical rule:

a b = a \cdot b

514 = 5 \cdot 14

= 5 \cdot 14

Multiply the numbers:

5 \cdot 14 = 70

= 70

14 \cdot 314 = 42

14 \cdot 314

Apply radical rule:

a a = a

1414 = 14

= 3 \cdot 14

Multiply the numbers:

3 \cdot 14 = 42

= 42

= \frac{70}{42}

= \frac{70}{42}

cos (x) = \frac{70}{42}

Apply trig inverse properties

cos (x) = \frac{70}{42}

General solutions for

cos (x) = \frac{70}{42}

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

x = arccos (\frac{70}{42}) + 2 πn, x = 2 π - arccos (\frac{70}{42}) + 2 πn

x = arccos (\frac{70}{42}) + 2 πn, x = 2 π - arccos (\frac{70}{42}) + 2 πn

Show solutions in decimal form

x = 1.37024 \dots + 2 πn, x = 2 π - 1.37024 \dots + 2 πn

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Frequently Asked Questions (FAQ)

What is the general solution for 5/(sqrt(21)*\sqrt{30)}=cos(x) ?
The general solution for 5/(sqrt(21)*\sqrt{30)}=cos(x) is x=1.37024…+2pin,x=2pi-1.37024…+2pin