What is the general solution for sec^2(a)= 4/3 ?

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

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sec^2(a)= 4/3

Solution

sec^{2} (a) = \frac{4}{3}

Solution

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

Degrees

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

Solution steps

sec^{2} (a) = \frac{4}{3}

Solve by substitution

sec^{2} (a) = \frac{4}{3}

Let:

sec (a) = u

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

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

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

For

x^{2} = f (a)

the solutions are

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

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

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

\frac{4}{3}

Apply radical rule:

assuming

a \geq 0, b \geq 0

= \frac{4}{3}

4 = 2

4

Factor the number:

4 = 2^{2}

= 2^{2}

Apply radical rule:

2^{2} = 2

= 2

= \frac{2}{3}

Rationalize

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

\frac{2}{3}

Multiply by the conjugate

\frac{3}{3}

= \frac{2 3}{3 3}

33 = 3

33

Apply radical rule:

a a = a

33 = 3

= 3

= \frac{2 3}{3}

= \frac{2 3}{3}

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

- \frac{4}{3}

Simplify

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

\frac{4}{3}

Apply radical rule:

assuming

a \geq 0, b \geq 0

= \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}

Rationalize

- \frac{2}{3} : - \frac{2 3}{3}

- \frac{2}{3}

Multiply by the conjugate

\frac{3}{3}

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

33 = 3

33

Apply radical rule:

a a = a

33 = 3

= 3

= - \frac{2 3}{3}

= - \frac{2 3}{3}

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

Substitute back

u = sec (a)

sec (a) = \frac{2 3}{3}, sec (a) = - \frac{2 3}{3}

sec (a) = \frac{2 3}{3}, sec (a) = - \frac{2 3}{3}

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

sec (a) = \frac{2 3}{3}

General solutions for

sec (a) = \frac{2 3}{3}

sec (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} sec (x) 1 \frac{2 3}{3} 22 Undefined - 2 - 2 - \frac{2 3}{3} 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} sec (x) - 1 - \frac{2 3}{3} - 2 - 2 Undefined 22 \frac{2 3}{3}

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

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

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

sec (a) = - \frac{2 3}{3}

General solutions for

sec (a) = - \frac{2 3}{3}

sec (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} sec (x) 1 \frac{2 3}{3} 22 Undefined - 2 - 2 - \frac{2 3}{3} 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} sec (x) - 1 - \frac{2 3}{3} - 2 - 2 Undefined 22 \frac{2 3}{3}

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

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

Combine all the solutions

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

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Frequently Asked Questions (FAQ)

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