What is the general solution for 0.08=0.12cos^2(3.922t) ?

The general solution for 0.08=0.12cos^2(3.922t) is

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0.08=0.12cos^2(3.922t)

Solution

0.08 = 0.12 cos^{2} (3.922 t)

Solution

t = \frac{0.61547 \dots}{3.922} + \frac{2 πn}{3.922}, t = \frac{2 π}{3.922} - \frac{0.61547 \dots}{3.922} + \frac{2 πn}{3.922}, t = \frac{2.52611 \dots}{3.922} + \frac{2 πn}{3.922}, t = - \frac{2.52611 \dots}{3.922} + \frac{2 πn}{3.922}

Degrees

t = 8.99143 \dots^{\circ} + 91.78990 \dots^{\circ} n, t = 82.79847 \dots^{\circ} + 91.78990 \dots^{\circ} n, t = 36.90352 \dots^{\circ} + 91.78990 \dots^{\circ} n, t = - 36.90352 \dots^{\circ} + 91.78990 \dots^{\circ} n

Solution steps

0.08 = 0.12 cos^{2} (3.922 t)

Switch sides

0.12 cos^{2} (3.922 t) = 0.08

Solve by substitution

0.12 cos^{2} (3.922 t) = 0.08

Let:

cos (3.922 t) = u

0.12 u^{2} = 0.08

0.12 u^{2} = 0.08 : u = \frac{2}{3}, u = - \frac{2}{3}

0.12 u^{2} = 0.08

Multiply both sides by

100

0.12 u^{2} = 0.08

To eliminate decimal points, multiply by 10 for every digit after the decimal pointThere are

2

digits to the right of the decimal point, therefore multiply by

100

0.12 u^{2} \cdot 100 = 0.08 \cdot 100

Refine

12 u^{2} = 8

12 u^{2} = 8

Divide both sides by

12

12 u^{2} = 8

Divide both sides by

12

\frac{12 u ^{2}}{12} = \frac{8}{12}

Simplify

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

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

For

x^{2} = f (a)

the solutions are

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

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

Substitute back

u = cos (3.922 t)

cos (3.922 t) = \frac{2}{3}, cos (3.922 t) = - \frac{2}{3}

cos (3.922 t) = \frac{2}{3}, cos (3.922 t) = - \frac{2}{3}

cos (3.922 t) = \frac{2}{3} : t = \frac{arccos ( \frac{2}{3} )}{3.922} + \frac{2 πn}{3.922}, t = \frac{2 π}{3.922} - \frac{arccos ( \frac{2}{3} )}{3.922} + \frac{2 πn}{3.922}

cos (3.922 t) = \frac{2}{3}

Apply trig inverse properties

cos (3.922 t) = \frac{2}{3}

General solutions for

cos (3.922 t) = \frac{2}{3}

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

3.922 t = arccos (\frac{2}{3}) + 2 πn, 3.922 t = 2 π - arccos (\frac{2}{3}) + 2 πn

3.922 t = arccos (\frac{2}{3}) + 2 πn, 3.922 t = 2 π - arccos (\frac{2}{3}) + 2 πn

Solve

3.922 t = arccos (\frac{2}{3}) + 2 πn : t = \frac{arccos ( \frac{2}{3} )}{3.922} + \frac{2 πn}{3.922}

3.922 t = arccos (\frac{2}{3}) + 2 πn

Divide both sides by

3.922

3.922 t = arccos (\frac{2}{3}) + 2 πn

Divide both sides by

3.922

\frac{3.922 t}{3.922} = \frac{arccos ( \frac{2}{3} )}{3.922} + \frac{2 πn}{3.922}

Simplify

t = \frac{arccos ( \frac{2}{3} )}{3.922} + \frac{2 πn}{3.922}

t = \frac{arccos ( \frac{2}{3} )}{3.922} + \frac{2 πn}{3.922}

Solve

3.922 t = 2 π - arccos (\frac{2}{3}) + 2 πn : t = \frac{2 π}{3.922} - \frac{arccos ( \frac{2}{3} )}{3.922} + \frac{2 πn}{3.922}

3.922 t = 2 π - arccos (\frac{2}{3}) + 2 πn

Divide both sides by

3.922

3.922 t = 2 π - arccos (\frac{2}{3}) + 2 πn

Divide both sides by

3.922

\frac{3.922 t}{3.922} = \frac{2 π}{3.922} - \frac{arccos ( \frac{2}{3} )}{3.922} + \frac{2 πn}{3.922}

Simplify

t = \frac{2 π}{3.922} - \frac{arccos ( \frac{2}{3} )}{3.922} + \frac{2 πn}{3.922}

t = \frac{2 π}{3.922} - \frac{arccos ( \frac{2}{3} )}{3.922} + \frac{2 πn}{3.922}

t = \frac{arccos ( \frac{2}{3} )}{3.922} + \frac{2 πn}{3.922}, t = \frac{2 π}{3.922} - \frac{arccos ( \frac{2}{3} )}{3.922} + \frac{2 πn}{3.922}

cos (3.922 t) = - \frac{2}{3} : t = \frac{arccos ( - \frac{2}{3} )}{3.922} + \frac{2 πn}{3.922}, t = - \frac{arccos ( - \frac{2}{3} )}{3.922} + \frac{2 πn}{3.922}

cos (3.922 t) = - \frac{2}{3}

Apply trig inverse properties

cos (3.922 t) = - \frac{2}{3}

General solutions for

cos (3.922 t) = - \frac{2}{3}

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

3.922 t = arccos (- \frac{2}{3}) + 2 πn, 3.922 t = - arccos (- \frac{2}{3}) + 2 πn

3.922 t = arccos (- \frac{2}{3}) + 2 πn, 3.922 t = - arccos (- \frac{2}{3}) + 2 πn

Solve

3.922 t = arccos (- \frac{2}{3}) + 2 πn : t = \frac{arccos ( - \frac{2}{3} )}{3.922} + \frac{2 πn}{3.922}

3.922 t = arccos (- \frac{2}{3}) + 2 πn

Divide both sides by

3.922

3.922 t = arccos (- \frac{2}{3}) + 2 πn

Divide both sides by

3.922

\frac{3.922 t}{3.922} = \frac{arccos ( - \frac{2}{3} )}{3.922} + \frac{2 πn}{3.922}

Simplify

t = \frac{arccos ( - \frac{2}{3} )}{3.922} + \frac{2 πn}{3.922}

t = \frac{arccos ( - \frac{2}{3} )}{3.922} + \frac{2 πn}{3.922}

Solve

3.922 t = - arccos (- \frac{2}{3}) + 2 πn : t = - \frac{arccos ( - \frac{2}{3} )}{3.922} + \frac{2 πn}{3.922}

3.922 t = - arccos (- \frac{2}{3}) + 2 πn

Divide both sides by

3.922

3.922 t = - arccos (- \frac{2}{3}) + 2 πn

Divide both sides by

3.922

\frac{3.922 t}{3.922} = - \frac{arccos ( - \frac{2}{3} )}{3.922} + \frac{2 πn}{3.922}

Simplify

t = - \frac{arccos ( - \frac{2}{3} )}{3.922} + \frac{2 πn}{3.922}

t = - \frac{arccos ( - \frac{2}{3} )}{3.922} + \frac{2 πn}{3.922}

t = \frac{arccos ( - \frac{2}{3} )}{3.922} + \frac{2 πn}{3.922}, t = - \frac{arccos ( - \frac{2}{3} )}{3.922} + \frac{2 πn}{3.922}

Combine all the solutions

t = \frac{arccos ( \frac{2}{3} )}{3.922} + \frac{2 πn}{3.922}, t = \frac{2 π}{3.922} - \frac{arccos ( \frac{2}{3} )}{3.922} + \frac{2 πn}{3.922}, t = \frac{arccos ( - \frac{2}{3} )}{3.922} + \frac{2 πn}{3.922}, t = - \frac{arccos ( - \frac{2}{3} )}{3.922} + \frac{2 πn}{3.922}

Show solutions in decimal form

t = \frac{0.61547 \dots}{3.922} + \frac{2 πn}{3.922}, t = \frac{2 π}{3.922} - \frac{0.61547 \dots}{3.922} + \frac{2 πn}{3.922}, t = \frac{2.52611 \dots}{3.922} + \frac{2 πn}{3.922}, t = - \frac{2.52611 \dots}{3.922} + \frac{2 πn}{3.922}

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Frequently Asked Questions (FAQ)

What is the general solution for 0.08=0.12cos^2(3.922t) ?
The general solution for 0.08=0.12cos^2(3.922t) is