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Popular >

9cos^2(x)=2

Solution

9 cos^{2} (x) = 2

Solution

x = 1.07991 \dots + 2 πn, x = 2 π - 1.07991 \dots + 2 πn, x = 2.06167 \dots + 2 πn, x = - 2.06167 \dots + 2 πn

+1

Degrees

x = 61.87449 \dots^{\circ} + 36 0^{\circ} n, x = 298.12550 \dots^{\circ} + 36 0^{\circ} n, x = 118.12550 \dots^{\circ} + 36 0^{\circ} n, x = - 118.12550 \dots^{\circ} + 36 0^{\circ} n

Solution steps

9 cos^{2} (x) = 2

Solve by substitution

9 cos^{2} (x) = 2

Let:

cos (x) = u

9 u^{2} = 2

9 u^{2} = 2 : u = \frac{2}{3}, u = - \frac{2}{3}

9 u^{2} = 2

Divide both sides by

9

9 u^{2} = 2

Divide both sides by

9

\frac{9 u ^{2}}{9} = \frac{2}{9}

Simplify

u^{2} = \frac{2}{9}

u^{2} = \frac{2}{9}

For

x^{2} = f (a)

the solutions are

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

u = \frac{2}{9}, u = - \frac{2}{9}

\frac{2}{9} = \frac{2}{3}

\frac{2}{9}

Apply radical rule:

assuming

a \geq 0, b \geq 0

= \frac{2}{9}

9 = 3

9

Factor the number:

9 = 3^{2}

= 3^{2}

Apply radical rule:

3^{2} = 3

= 3

= \frac{2}{3}

- \frac{2}{9} = - \frac{2}{3}

- \frac{2}{9}

Simplify

\frac{2}{9} : \frac{2}{3}

\frac{2}{9}

Apply radical rule:

assuming

a \geq 0, b \geq 0

= \frac{2}{9}

9 = 3

9

Factor the number:

9 = 3^{2}

= 3^{2}

Apply radical rule:

3^{2} = 3

= 3

= \frac{2}{3}

= - \frac{2}{3}

u = \frac{2}{3}, u = - \frac{2}{3}

Substitute back

u = cos (x)

cos (x) = \frac{2}{3}, cos (x) = - \frac{2}{3}

cos (x) = \frac{2}{3}, cos (x) = - \frac{2}{3}

cos (x) = \frac{2}{3} : x = arccos (\frac{2}{3}) + 2 πn, x = 2 π - arccos (\frac{2}{3}) + 2 πn

cos (x) = \frac{2}{3}

Apply trig inverse properties

cos (x) = \frac{2}{3}

General solutions for

cos (x) = \frac{2}{3}

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

x = arccos (\frac{2}{3}) + 2 πn, x = 2 π - arccos (\frac{2}{3}) + 2 πn

x = arccos (\frac{2}{3}) + 2 πn, x = 2 π - arccos (\frac{2}{3}) + 2 πn

cos (x) = - \frac{2}{3} : x = arccos (- \frac{2}{3}) + 2 πn, x = - arccos (- \frac{2}{3}) + 2 πn

cos (x) = - \frac{2}{3}

Apply trig inverse properties

cos (x) = - \frac{2}{3}

General solutions for

cos (x) = - \frac{2}{3}

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

x = arccos (- \frac{2}{3}) + 2 πn, x = - arccos (- \frac{2}{3}) + 2 πn

x = arccos (- \frac{2}{3}) + 2 πn, x = - arccos (- \frac{2}{3}) + 2 πn

Combine all the solutions

x = arccos (\frac{2}{3}) + 2 πn, x = 2 π - arccos (\frac{2}{3}) + 2 πn, x = arccos (- \frac{2}{3}) + 2 πn, x = - arccos (- \frac{2}{3}) + 2 πn

Show solutions in decimal form

x = 1.07991 \dots + 2 πn, x = 2 π - 1.07991 \dots + 2 πn, x = 2.06167 \dots + 2 πn, x = - 2.06167 \dots + 2 πn

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