What is the general solution for cos^2(s)= 3/4 ?

The general solution for cos^2(s)= 3/4 is s= pi/6+2pin,s=(11pi)/6+2pin,s=(5pi)/6+2pin,s=(7pi)/6+2pin

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cos^2(s)= 3/4

Solution

cos^{2} (s) = \frac{3}{4}

Solution

s = \frac{π}{6} + 2 πn, s = \frac{11 π}{6} + 2 πn, s = \frac{5 π}{6} + 2 πn, s = \frac{7 π}{6} + 2 πn

Degrees

s = 3 0^{\circ} + 36 0^{\circ} n, s = 33 0^{\circ} + 36 0^{\circ} n, s = 15 0^{\circ} + 36 0^{\circ} n, s = 21 0^{\circ} + 36 0^{\circ} n

Solution steps

cos^{2} (s) = \frac{3}{4}

Solve by substitution

cos^{2} (s) = \frac{3}{4}

Let:

cos (s) = u

u^{2} = \frac{3}{4}

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

u^{2} = \frac{3}{4}

For

x^{2} = f (a)

the solutions are

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

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

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

\frac{3}{4}

Apply radical rule:

n \frac{a}{b} = \frac{n a}{n b},

assuming

a \geq 0, b \geq 0

= \frac{3}{4}

4 = 2

4

Factor the number:

4 = 2^{2}

= 2^{2}

Apply radical rule:

n a^{n} = a

2^{2} = 2

= 2

= \frac{3}{2}

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

- \frac{3}{4}

Simplify

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

\frac{3}{4}

Apply radical rule:

n \frac{a}{b} = \frac{n a}{n b},

assuming

a \geq 0, b \geq 0

= \frac{3}{4}

4 = 2

4

Factor the number:

4 = 2^{2}

= 2^{2}

Apply radical rule:

n a^{n} = a

2^{2} = 2

= 2

= \frac{3}{2}

= - \frac{3}{2}

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

Substitute back

u = cos (s)

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

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

cos (s) = \frac{3}{2} : s = \frac{π}{6} + 2 πn, s = \frac{11 π}{6} + 2 πn

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

General solutions for

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

cos (x)

periodicity table with

2 πn

cycle:

x 0 \frac{π}{6} \frac{π}{4} \frac{π}{3} \frac{π}{2} \frac{2 π}{3} \frac{3 π}{4} \frac{5 π}{6} cos (x) 1 \frac{3}{2} \frac{2}{2} \frac{1}{2} 0 - \frac{1}{2} - \frac{2}{2} - \frac{3}{2} x π \frac{7 π}{6} \frac{5 π}{4} \frac{4 π}{3} \frac{3 π}{2} \frac{5 π}{3} \frac{7 π}{4} \frac{11 π}{6} cos (x) - 1 - \frac{3}{2} - \frac{2}{2} - \frac{1}{2} 0 \frac{1}{2} \frac{2}{2} \frac{3}{2}

s = \frac{π}{6} + 2 πn, s = \frac{11 π}{6} + 2 πn

s = \frac{π}{6} + 2 πn, s = \frac{11 π}{6} + 2 πn

cos (s) = - \frac{3}{2} : s = \frac{5 π}{6} + 2 πn, s = \frac{7 π}{6} + 2 πn

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

General solutions for

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

cos (x)

periodicity table with

2 πn

cycle:

x 0 \frac{π}{6} \frac{π}{4} \frac{π}{3} \frac{π}{2} \frac{2 π}{3} \frac{3 π}{4} \frac{5 π}{6} cos (x) 1 \frac{3}{2} \frac{2}{2} \frac{1}{2} 0 - \frac{1}{2} - \frac{2}{2} - \frac{3}{2} x π \frac{7 π}{6} \frac{5 π}{4} \frac{4 π}{3} \frac{3 π}{2} \frac{5 π}{3} \frac{7 π}{4} \frac{11 π}{6} cos (x) - 1 - \frac{3}{2} - \frac{2}{2} - \frac{1}{2} 0 \frac{1}{2} \frac{2}{2} \frac{3}{2}

s = \frac{5 π}{6} + 2 πn, s = \frac{7 π}{6} + 2 πn

s = \frac{5 π}{6} + 2 πn, s = \frac{7 π}{6} + 2 πn

Combine all the solutions

s = \frac{π}{6} + 2 πn, s = \frac{11 π}{6} + 2 πn, s = \frac{5 π}{6} + 2 πn, s = \frac{7 π}{6} + 2 πn

Graph

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Frequently Asked Questions (FAQ)

What is the general solution for cos^2(s)= 3/4 ?
The general solution for cos^2(s)= 3/4 is s= pi/6+2pin,s=(11pi)/6+2pin,s=(5pi)/6+2pin,s=(7pi)/6+2pin