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Popular >

16cos^2(x)-9=0

Solution

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

Solution

x = 0.72273 \dots + 2 πn, x = 2 π - 0.72273 \dots + 2 πn, x = 2.41885 \dots + 2 πn, x = - 2.41885 \dots + 2 πn

+1

Degrees

x = 41.40962 \dots^{\circ} + 36 0^{\circ} n, x = 318.59037 \dots^{\circ} + 36 0^{\circ} n, x = 138.59037 \dots^{\circ} + 36 0^{\circ} n, x = - 138.59037 \dots^{\circ} + 36 0^{\circ} n

Solution steps

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

Solve by substitution

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

Let:

cos (x) = u

16 u^{2} - 9 = 0

16 u^{2} - 9 = 0 : u = \frac{3}{4}, u = - \frac{3}{4}

16 u^{2} - 9 = 0

Move

9

to the right side

16 u^{2} - 9 = 0

Add

9

to both sides

16 u^{2} - 9 + 9 = 0 + 9

Simplify

16 u^{2} = 9

16 u^{2} = 9

Divide both sides by

16

16 u^{2} = 9

Divide both sides by

16

\frac{16 u ^{2}}{16} = \frac{9}{16}

Simplify

u^{2} = \frac{9}{16}

u^{2} = \frac{9}{16}

For

x^{2} = f (a)

the solutions are

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

u = \frac{9}{16}, u = - \frac{9}{16}

\frac{9}{16} = \frac{3}{4}

\frac{9}{16}

Apply radical rule:

assuming

a \geq 0, b \geq 0

= \frac{9}{16}

16 = 4

16

Factor the number:

16 = 4^{2}

= 4^{2}

Apply radical rule:

4^{2} = 4

= 4

= \frac{9}{4}

9 = 3

9

Factor the number:

9 = 3^{2}

= 3^{2}

Apply radical rule:

3^{2} = 3

= 3

= \frac{3}{4}

- \frac{9}{16} = - \frac{3}{4}

- \frac{9}{16}

Simplify

\frac{9}{16} : \frac{3}{4}

\frac{9}{16}

Apply radical rule:

assuming

a \geq 0, b \geq 0

= \frac{9}{16}

16 = 4

16

Factor the number:

16 = 4^{2}

= 4^{2}

Apply radical rule:

4^{2} = 4

= 4

= \frac{9}{4}

9 = 3

9

Factor the number:

9 = 3^{2}

= 3^{2}

Apply radical rule:

3^{2} = 3

= 3

= \frac{3}{4}

= - \frac{3}{4}

u = \frac{3}{4}, u = - \frac{3}{4}

Substitute back

u = cos (x)

cos (x) = \frac{3}{4}, cos (x) = - \frac{3}{4}

cos (x) = \frac{3}{4}, cos (x) = - \frac{3}{4}

cos (x) = \frac{3}{4} : x = arccos (\frac{3}{4}) + 2 πn, x = 2 π - arccos (\frac{3}{4}) + 2 πn

cos (x) = \frac{3}{4}

Apply trig inverse properties

cos (x) = \frac{3}{4}

General solutions for

cos (x) = \frac{3}{4}

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

x = arccos (\frac{3}{4}) + 2 πn, x = 2 π - arccos (\frac{3}{4}) + 2 πn

x = arccos (\frac{3}{4}) + 2 πn, x = 2 π - arccos (\frac{3}{4}) + 2 πn

cos (x) = - \frac{3}{4} : x = arccos (- \frac{3}{4}) + 2 πn, x = - arccos (- \frac{3}{4}) + 2 πn

cos (x) = - \frac{3}{4}

Apply trig inverse properties

cos (x) = - \frac{3}{4}

General solutions for

cos (x) = - \frac{3}{4}

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

x = arccos (- \frac{3}{4}) + 2 πn, x = - arccos (- \frac{3}{4}) + 2 πn

x = arccos (- \frac{3}{4}) + 2 πn, x = - arccos (- \frac{3}{4}) + 2 πn

Combine all the solutions

x = arccos (\frac{3}{4}) + 2 πn, x = 2 π - arccos (\frac{3}{4}) + 2 πn, x = arccos (- \frac{3}{4}) + 2 πn, x = - arccos (- \frac{3}{4}) + 2 πn

Show solutions in decimal form

x = 0.72273 \dots + 2 πn, x = 2 π - 0.72273 \dots + 2 πn, x = 2.41885 \dots + 2 πn, x = - 2.41885 \dots + 2 πn

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