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Popular >

9/2 cos^2(x)-2=0

Solution

\frac{9}{2} cos^{2} (x) - 2 = 0

Solution

x = 0.84106 \dots + 2 πn, x = 2 π - 0.84106 \dots + 2 πn, x = 2.30052 \dots + 2 πn, x = - 2.30052 \dots + 2 πn

+1

Degrees

x = 48.18968 \dots^{\circ} + 36 0^{\circ} n, x = 311.81031 \dots^{\circ} + 36 0^{\circ} n, x = 131.81031 \dots^{\circ} + 36 0^{\circ} n, x = - 131.81031 \dots^{\circ} + 36 0^{\circ} n

Solution steps

\frac{9}{2} cos^{2} (x) - 2 = 0

Solve by substitution

\frac{9}{2} cos^{2} (x) - 2 = 0

Let:

cos (x) = u

\frac{9}{2} u^{2} - 2 = 0

\frac{9}{2} u^{2} - 2 = 0 : u = \frac{2}{3}, u = - \frac{2}{3}

\frac{9}{2} u^{2} - 2 = 0

Move

2

to the right side

\frac{9}{2} u^{2} - 2 = 0

Add

2

to both sides

\frac{9}{2} u^{2} - 2 + 2 = 0 + 2

Simplify

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

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

Multiply both sides by

2

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

Multiply both sides by

2

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

Simplify

9 u^{2} = 4

9 u^{2} = 4

Divide both sides by

9

9 u^{2} = 4

Divide both sides by

9

\frac{9 u ^{2}}{9} = \frac{4}{9}

Simplify

u^{2} = \frac{4}{9}

u^{2} = \frac{4}{9}

For

x^{2} = f (a)

the solutions are

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

u = \frac{4}{9}, u = - \frac{4}{9}

\frac{4}{9} = \frac{2}{3}

\frac{4}{9}

Apply radical rule:

assuming

a \geq 0, b \geq 0

= \frac{4}{9}

9 = 3

9

Factor the number:

9 = 3^{2}

= 3^{2}

Apply radical rule:

3^{2} = 3

= 3

= \frac{4}{3}

4 = 2

4

Factor the number:

4 = 2^{2}

= 2^{2}

Apply radical rule:

2^{2} = 2

= 2

= \frac{2}{3}

- \frac{4}{9} = - \frac{2}{3}

- \frac{4}{9}

Simplify

\frac{4}{9} : \frac{2}{3}

\frac{4}{9}

Apply radical rule:

assuming

a \geq 0, b \geq 0

= \frac{4}{9}

9 = 3

9

Factor the number:

9 = 3^{2}

= 3^{2}

Apply radical rule:

3^{2} = 3

= 3

= \frac{4}{3}

4 = 2

4

Factor the number:

4 = 2^{2}

= 2^{2}

Apply radical rule:

2^{2} = 2

= 2

= \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 = 0.84106 \dots + 2 πn, x = 2 π - 0.84106 \dots + 2 πn, x = 2.30052 \dots + 2 πn, x = - 2.30052 \dots + 2 πn

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