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Popular Trigonometry >

sec^2(x)=(3sqrt(3))/pi

Solution

sec^{2} (x) = \frac{3 3}{π}

Solution

x = 0.68001 \dots + 2 πn, x = 2 π - 0.68001 \dots + 2 πn, x = 2.46157 \dots + 2 πn, x = - 2.46157 \dots + 2 πn

+1

Degrees

x = 38.96227 \dots^{\circ} + 36 0^{\circ} n, x = 321.03772 \dots^{\circ} + 36 0^{\circ} n, x = 141.03772 \dots^{\circ} + 36 0^{\circ} n, x = - 141.03772 \dots^{\circ} + 36 0^{\circ} n

Solution steps

sec^{2} (x) = \frac{3 3}{π}

Solve by substitution

sec^{2} (x) = \frac{3 3}{π}

Let:

sec (x) = u

u^{2} = \frac{3 3}{π}

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

u^{2} = \frac{3 3}{π}

For

x^{2} = f (a)

the solutions are

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

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

Substitute back

u = sec (x)

sec (x) = \frac{3 3}{π}, sec (x) = - \frac{3 3}{π}

sec (x) = \frac{3 3}{π}, sec (x) = - \frac{3 3}{π}

sec (x) = \frac{3 3}{π} : x = arcsec \frac{3 3}{π} + 2 πn, x = 2 π - arcsec \frac{3 3}{π} + 2 πn

sec (x) = \frac{3 3}{π}

Apply trig inverse properties

sec (x) = \frac{3 3}{π}

General solutions for

sec (x) = \frac{3 3}{π}

sec (x) = a \Rightarrow x = arcsec (a) + 2 πn, x = 2 π - arcsec (a) + 2 πn

x = arcsec \frac{3 3}{π} + 2 πn, x = 2 π - arcsec \frac{3 3}{π} + 2 πn

x = arcsec \frac{3 3}{π} + 2 πn, x = 2 π - arcsec \frac{3 3}{π} + 2 πn

sec (x) = - \frac{3 3}{π} : x = arcsec - \frac{3 3}{π} + 2 πn, x = - arcsec - \frac{3 3}{π} + 2 πn

sec (x) = - \frac{3 3}{π}

Apply trig inverse properties

sec (x) = - \frac{3 3}{π}

General solutions for

sec (x) = - \frac{3 3}{π}

sec (x) = - a \Rightarrow x = arcsec (- a) + 2 πn, x = - arcsec (- a) + 2 πn

x = arcsec - \frac{3 3}{π} + 2 πn, x = - arcsec - \frac{3 3}{π} + 2 πn

x = arcsec - \frac{3 3}{π} + 2 πn, x = - arcsec - \frac{3 3}{π} + 2 πn

Combine all the solutions

x = arcsec \frac{3 3}{π} + 2 πn, x = 2 π - arcsec \frac{3 3}{π} + 2 πn, x = arcsec - \frac{3 3}{π} + 2 πn, x = - arcsec - \frac{3 3}{π} + 2 πn

Show solutions in decimal form

x = 0.68001 \dots + 2 πn, x = 2 π - 0.68001 \dots + 2 πn, x = 2.46157 \dots + 2 πn, x = - 2.46157 \dots + 2 πn

Graph

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