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Popular >

cos^2(x)= 3/(4*5cos^2(x))

Solution

cos^{2} (x) = \frac{3}{4 \cdot 5 cos ^{2} ( x )}

Solution

x = 0.89907 \dots + 2 πn, x = 2 π - 0.89907 \dots + 2 πn, x = 2.24251 \dots + 2 πn, x = - 2.24251 \dots + 2 πn

Degrees

x = 51.51329 \dots^{\circ} + 36 0^{\circ} n, x = 308.48670 \dots^{\circ} + 36 0^{\circ} n, x = 128.48670 \dots^{\circ} + 36 0^{\circ} n, x = - 128.48670 \dots^{\circ} + 36 0^{\circ} n

Solution steps

cos^{2} (x) = \frac{3}{4 \cdot 5 cos ^{2} ( x )}

Solve by substitution

cos^{2} (x) = \frac{3}{4 \cdot 5 cos ^{2} ( x )}

Let:

cos (x) = u

u^{2} = \frac{3}{4 \cdot 5 u ^{2}}

u^{2} = \frac{3}{4 \cdot 5 u ^{2}} : u = \frac{15}{10}, u = - \frac{15}{10}, u = i \frac{15}{10}, u = - i \frac{15}{10}

u^{2} = \frac{3}{4 \cdot 5 u ^{2}}

Simplify

\frac{3}{4 \cdot 5 u ^{2}} : \frac{3}{20 u ^{2}}

\frac{3}{4 \cdot 5 u ^{2}}

Multiply the numbers:

4 \cdot 5 = 20

= \frac{3}{20 u ^{2}}

u^{2} = \frac{3}{20 u ^{2}}

Multiply both sides by

u^{2}

u^{2} = \frac{3}{20 u ^{2}}

Multiply both sides by

u^{2}

u^{2} u^{2} = \frac{3}{20 u ^{2}} u^{2}

Simplify

u^{2} u^{2} : u^{4}

u^{2} u^{2} = \frac{3}{20 u ^{2}} u^{2}

Apply exponent rule:

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

u^{2} u^{2} = u^{2 + 2}

= u^{2 + 2}

Add the numbers:

2 + 2 = 4

= u^{4}

u^{4} = \frac{3}{20}

u^{4} = \frac{3}{20}

Solve

u^{4} = \frac{3}{20} : u = \frac{15}{10}, u = - \frac{15}{10}, u = i \frac{15}{10}, u = - i \frac{15}{10}

u^{4} = \frac{3}{20}

Rewrite the equation with

v = u^{2}

and

v^{2} = u^{4}

v^{2} = \frac{3}{20}

Solve

v^{2} = \frac{3}{20} : v = \frac{3}{20}, v = - \frac{3}{20}

v^{2} = \frac{3}{20}

For

(g (x))^{2} = f (a)

the solutions are

g (x) = f (a), - f (a)

v = \frac{3}{20}, v = - \frac{3}{20}

v = \frac{3}{20}, v = - \frac{3}{20}

Substitute back

v = u^{2},

solve for

u

Solve

u^{2} = \frac{3}{20} : u = \frac{15}{10}, u = - \frac{15}{10}

u^{2} = \frac{3}{20}

Simplify

\frac{3}{20} : \frac{15}{10}

\frac{3}{20}

Apply radical rule:

assuming

a \geq 0, b \geq 0

= \frac{3}{20}

20 = 25

20

Prime factorization of

20 : 2^{2} \cdot 5

20

20

divides by

220 = 10 \cdot 2

= 2 \cdot 10

10

divides by

210 = 5 \cdot 2

= 2 \cdot 2 \cdot 5

2, 5

are all prime numbers, therefore no further factorization is possible

= 2 \cdot 2 \cdot 5

= 2^{2} \cdot 5

= 2^{2} \cdot 5

Apply radical rule:

= 5 2^{2}

Apply radical rule:

2^{2} = 2

= 25

= \frac{3}{2 5}

Rationalize

\frac{3}{2 5} : \frac{15}{10}

\frac{3}{2 5}

Multiply by the conjugate

\frac{5}{5}

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

35 = 15

35

Apply radical rule:

a b = a \cdot b

35 = 3 \cdot 5

= 3 \cdot 5

Multiply the numbers:

3 \cdot 5 = 15

= 15

255 = 10

255

Apply radical rule:

a a = a

55 = 5

= 2 \cdot 5

Multiply the numbers:

2 \cdot 5 = 10

= 10

= \frac{15}{10}

= \frac{15}{10}

u^{2} = \frac{15}{10}

For

x^{2} = f (a)

the solutions are

x = f (a), - f (a)

u = \frac{15}{10}, u = - \frac{15}{10}

Solve

u^{2} = - \frac{3}{20} : u = i \frac{15}{10}, u = - i \frac{15}{10}

u^{2} = - \frac{3}{20}

Simplify

- \frac{3}{20} : - \frac{15}{10}

- \frac{3}{20}

Simplify

\frac{3}{20} : \frac{3}{2 5}

\frac{3}{20}

Apply radical rule:

assuming

a \geq 0, b \geq 0

= \frac{3}{20}

20 = 25

20

Prime factorization of

20 : 2^{2} \cdot 5

20

20

divides by

220 = 10 \cdot 2

= 2 \cdot 10

10

divides by

210 = 5 \cdot 2

= 2 \cdot 2 \cdot 5

2, 5

are all prime numbers, therefore no further factorization is possible

= 2 \cdot 2 \cdot 5

= 2^{2} \cdot 5

= 2^{2} \cdot 5

Apply radical rule:

= 5 2^{2}

Apply radical rule:

2^{2} = 2

= 25

= \frac{3}{2 5}

= - \frac{3}{2 5}

Rationalize

- \frac{3}{2 5} : - \frac{15}{10}

- \frac{3}{2 5}

Multiply by the conjugate

\frac{5}{5}

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

35 = 15

35

Apply radical rule:

a b = a \cdot b

35 = 3 \cdot 5

= 3 \cdot 5

Multiply the numbers:

3 \cdot 5 = 15

= 15

255 = 10

255

Apply radical rule:

a a = a

55 = 5

= 2 \cdot 5

Multiply the numbers:

2 \cdot 5 = 10

= 10

= - \frac{15}{10}

= - \frac{15}{10}

u^{2} = - \frac{15}{10}

For

x^{2} = f (a)

the solutions are

x = f (a), - f (a)

u = - \frac{15}{10}, u = - - \frac{15}{10}

Simplify

- \frac{15}{10} : i \frac{15}{10}

- \frac{15}{10}

Apply radical rule:

- a = - 1 a

- \frac{15}{10} = - 1 \frac{15}{10}

= - 1 \frac{15}{10}

Apply imaginary number rule:

- 1 = i

= i \frac{15}{10}

Rewrite

i \frac{15}{10}

in standard complex form:

\frac{15}{10} i

i \frac{15}{10}

\frac{15}{10} = \frac{3}{2 5}

\frac{15}{10}

\frac{15}{10} = \frac{3}{2 5}

\frac{15}{10}

Factor

15 : 35

Factor

15 = 3 \cdot 5

= 3 \cdot 5

Apply radical rule:

= 35

Factor

10 : 2 \cdot 5

Factor

10 = 2 \cdot 5

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

Cancel

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

\frac{3 5}{2 \cdot 5}

Apply radical rule:

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

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

Apply exponent rule:

\frac{x ^{a}}{x ^{b}} = \frac{1}{x ^{b - a}}

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

= \frac{3}{2 \cdot 5 ^{- \frac{1}{2} + 1}}

Subtract the numbers:

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

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

Apply radical rule:

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

= \frac{3}{2 5}

= \frac{3}{2 5}

= \frac{3}{2 5}

= i \frac{3}{2 5}

\frac{3}{2 5} = \frac{15}{10}

\frac{3}{2 5}

\frac{3}{2 5} = \frac{15}{10}

\frac{3}{2 5}

Multiply by the conjugate

\frac{5}{5}

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

35 = 15

35

Apply radical rule:

a b = a \cdot b

35 = 3 \cdot 5

= 3 \cdot 5

Multiply the numbers:

3 \cdot 5 = 15

= 15

255 = 10

255

Apply radical rule:

a a = a

55 = 5

= 2 \cdot 5

Multiply the numbers:

2 \cdot 5 = 10

= 10

= \frac{15}{10}

= \frac{15}{10}

= \frac{15}{10} i

= \frac{15}{10} i

Simplify

- - \frac{15}{10} : - i \frac{15}{10}

- - \frac{15}{10}

Simplify

- \frac{15}{10} : i \frac{15}{10}

- \frac{15}{10}

Apply radical rule:

- a = - 1 a

- \frac{15}{10} = - 1 \frac{15}{10}

= - 1 \frac{15}{10}

Apply imaginary number rule:

- 1 = i

= i \frac{15}{10}

= - i \frac{15}{10}

Rewrite

- i \frac{15}{10}

in standard complex form:

- \frac{15}{10} i

- i \frac{15}{10}

\frac{15}{10} = \frac{3}{2 5}

\frac{15}{10}

\frac{15}{10} = \frac{3}{2 5}

\frac{15}{10}

Factor

15 : 35

Factor

15 = 3 \cdot 5

= 3 \cdot 5

Apply radical rule:

= 35

Factor

10 : 2 \cdot 5

Factor

10 = 2 \cdot 5

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

Cancel

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

\frac{3 5}{2 \cdot 5}

Apply radical rule:

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

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

Apply exponent rule:

\frac{x ^{a}}{x ^{b}} = \frac{1}{x ^{b - a}}

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

= \frac{3}{2 \cdot 5 ^{- \frac{1}{2} + 1}}

Subtract the numbers:

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

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

Apply radical rule:

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

= \frac{3}{2 5}

= \frac{3}{2 5}

= \frac{3}{2 5}

= - i \frac{3}{2 5}

- \frac{3}{2 5} = - \frac{15}{10}

- \frac{3}{2 5}

\frac{3}{2 5} = \frac{15}{10}

\frac{3}{2 5}

Multiply by the conjugate

\frac{5}{5}

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

35 = 15

35

Apply radical rule:

a b = a \cdot b

35 = 3 \cdot 5

= 3 \cdot 5

Multiply the numbers:

3 \cdot 5 = 15

= 15

255 = 10

255

Apply radical rule:

a a = a

55 = 5

= 2 \cdot 5

Multiply the numbers:

2 \cdot 5 = 10

= 10

= \frac{15}{10}

= - \frac{15}{10}

= - \frac{15}{10} i

= - \frac{15}{10} i

u = i \frac{15}{10}, u = - i \frac{15}{10}

The solutions are

u = \frac{15}{10}, u = - \frac{15}{10}, u = i \frac{15}{10}, u = - i \frac{15}{10}

u = \frac{15}{10}, u = - \frac{15}{10}, u = i \frac{15}{10}, u = - i \frac{15}{10}

Verify Solutions

Find undefined (singularity) points:

u = 0

Take the denominator(s) of

\frac{3}{45 u ^{2}}

and compare to zero

Solve

45 u^{2} = 0 : u = 0

4 \cdot 5 u^{2} = 0

Divide both sides by

20

4 \cdot 5 u^{2} = 0

Divide both sides by

20

4 \cdot 5 u^{2} = 0

Divide both sides by

20

\frac{4 \cdot 5 u ^{2}}{20} = \frac{0}{20}

Simplify

u^{2} = 0

u^{2} = 0

Apply rule

x^{n} = 0 \Rightarrow x = 0

u = 0

The following points are undefined

u = 0

Combine undefined points with solutions:

u = \frac{15}{10}, u = - \frac{15}{10}, u = i \frac{15}{10}, u = - i \frac{15}{10}

Substitute back

u = cos (x)

cos (x) = \frac{15}{10}, cos (x) = - \frac{15}{10}, cos (x) = i \frac{15}{10}, cos (x) = - i \frac{15}{10}

cos (x) = \frac{15}{10}, cos (x) = - \frac{15}{10}, cos (x) = i \frac{15}{10}, cos (x) = - i \frac{15}{10}

cos (x) = \frac{15}{10} : x = arccos \frac{15}{10} + 2 πn, x = 2 π - arccos \frac{15}{10} + 2 πn

cos (x) = \frac{15}{10}

Apply trig inverse properties

cos (x) = \frac{15}{10}

General solutions for

cos (x) = \frac{15}{10}

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

x = arccos \frac{15}{10} + 2 πn, x = 2 π - arccos \frac{15}{10} + 2 πn

x = arccos \frac{15}{10} + 2 πn, x = 2 π - arccos \frac{15}{10} + 2 πn

cos (x) = - \frac{15}{10} : x = arccos - \frac{15}{10} + 2 πn, x = - arccos - \frac{15}{10} + 2 πn

cos (x) = - \frac{15}{10}

Apply trig inverse properties

cos (x) = - \frac{15}{10}

General solutions for

cos (x) = - \frac{15}{10}

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

x = arccos - \frac{15}{10} + 2 πn, x = - arccos - \frac{15}{10} + 2 πn

x = arccos - \frac{15}{10} + 2 πn, x = - arccos - \frac{15}{10} + 2 πn

cos (x) = i \frac{15}{10} :

No Solution

cos (x) = i \frac{15}{10}

No Solution

cos (x) = - i \frac{15}{10} :

No Solution

cos (x) = - i \frac{15}{10}

No Solution

Combine all the solutions

x = arccos \frac{15}{10} + 2 πn, x = 2 π - arccos \frac{15}{10} + 2 πn, x = arccos - \frac{15}{10} + 2 πn, x = - arccos - \frac{15}{10} + 2 πn

Show solutions in decimal form

x = 0.89907 \dots + 2 πn, x = 2 π - 0.89907 \dots + 2 πn, x = 2.24251 \dots + 2 πn, x = - 2.24251 \dots + 2 πn

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