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Popular >

5cos^2(x)=4

Solution

5 cos^{2} (x) = 4

Solution

x = 0.46364 \dots + 2 πn, x = 2 π - 0.46364 \dots + 2 πn, x = 2.67794 \dots + 2 πn, x = - 2.67794 \dots + 2 πn

+1

Degrees

x = 26.56505 \dots^{\circ} + 36 0^{\circ} n, x = 333.43494 \dots^{\circ} + 36 0^{\circ} n, x = 153.43494 \dots^{\circ} + 36 0^{\circ} n, x = - 153.43494 \dots^{\circ} + 36 0^{\circ} n

Solution steps

5 cos^{2} (x) = 4

Solve by substitution

5 cos^{2} (x) = 4

Let:

cos (x) = u

5 u^{2} = 4

5 u^{2} = 4 : u = \frac{2 5}{5}, u = - \frac{2 5}{5}

5 u^{2} = 4

Divide both sides by

5

5 u^{2} = 4

Divide both sides by

5

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

Simplify

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

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

For

x^{2} = f (a)

the solutions are

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

u = \frac{4}{5}, u = - \frac{4}{5}

\frac{4}{5} = \frac{2 5}{5}

\frac{4}{5}

Apply radical rule:

assuming

a \geq 0, b \geq 0

= \frac{4}{5}

4 = 2

4

Factor the number:

4 = 2^{2}

= 2^{2}

Apply radical rule:

2^{2} = 2

= 2

= \frac{2}{5}

Rationalize

\frac{2}{5} : \frac{2 5}{5}

\frac{2}{5}

Multiply by the conjugate

\frac{5}{5}

= \frac{2 5}{5 5}

55 = 5

55

Apply radical rule:

a a = a

55 = 5

= 5

= \frac{2 5}{5}

= \frac{2 5}{5}

- \frac{4}{5} = - \frac{2 5}{5}

- \frac{4}{5}

Simplify

\frac{4}{5} : \frac{2}{5}

\frac{4}{5}

Apply radical rule:

assuming

a \geq 0, b \geq 0

= \frac{4}{5}

4 = 2

4

Factor the number:

4 = 2^{2}

= 2^{2}

Apply radical rule:

2^{2} = 2

= 2

= \frac{2}{5}

= - \frac{2}{5}

Rationalize

- \frac{2}{5} : - \frac{2 5}{5}

- \frac{2}{5}

Multiply by the conjugate

\frac{5}{5}

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

55 = 5

55

Apply radical rule:

a a = a

55 = 5

= 5

= - \frac{2 5}{5}

= - \frac{2 5}{5}

u = \frac{2 5}{5}, u = - \frac{2 5}{5}

Substitute back

u = cos (x)

cos (x) = \frac{2 5}{5}, cos (x) = - \frac{2 5}{5}

cos (x) = \frac{2 5}{5}, cos (x) = - \frac{2 5}{5}

cos (x) = \frac{2 5}{5} : x = arccos (\frac{2 5}{5}) + 2 πn, x = 2 π - arccos (\frac{2 5}{5}) + 2 πn

cos (x) = \frac{2 5}{5}

Apply trig inverse properties

cos (x) = \frac{2 5}{5}

General solutions for

cos (x) = \frac{2 5}{5}

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

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

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

cos (x) = - \frac{2 5}{5} : x = arccos (- \frac{2 5}{5}) + 2 πn, x = - arccos (- \frac{2 5}{5}) + 2 πn

cos (x) = - \frac{2 5}{5}

Apply trig inverse properties

cos (x) = - \frac{2 5}{5}

General solutions for

cos (x) = - \frac{2 5}{5}

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

x = arccos (- \frac{2 5}{5}) + 2 πn, x = - arccos (- \frac{2 5}{5}) + 2 πn

x = arccos (- \frac{2 5}{5}) + 2 πn, x = - arccos (- \frac{2 5}{5}) + 2 πn

Combine all the solutions

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

Show solutions in decimal form

x = 0.46364 \dots + 2 πn, x = 2 π - 0.46364 \dots + 2 πn, x = 2.67794 \dots + 2 πn, x = - 2.67794 \dots + 2 πn

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