What is the value of arcsec(csc((5pi)/3)) ?

The value of arcsec(csc((5pi)/3)) is (5pi)/6

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arcsec(csc((5pi)/3))

Solution

arcsec (csc (\frac{5 π}{3}))

Solution

\frac{5 π}{6}

Decimal Notation

150

Solution steps

arcsec (csc (\frac{5 π}{3}))

Rewrite using trig identities:

csc (\frac{5 π}{3}) = - \frac{2 3}{3}

csc (\frac{5 π}{3})

Rewrite using trig identities:

\frac{1}{sin ( \frac{5 π}{3} )}

csc (\frac{5 π}{3})

Use the basic trigonometric identity:

csc (x) = \frac{1}{sin ( x )}

= \frac{1}{sin ( \frac{5 π}{3} )}

= \frac{1}{sin ( \frac{5 π}{3} )}

Rewrite using trig identities:

sin (\frac{5 π}{3}) = - \frac{3}{2}

sin (\frac{5 π}{3})

Rewrite using trig identities:

sin (π) cos (\frac{2 π}{3}) + cos (π) sin (\frac{2 π}{3})

sin (\frac{5 π}{3})

Write

sin (\frac{5 π}{3})

sin (π + \frac{2 π}{3})

= sin (π + \frac{2 π}{3})

Use the Angle Sum identity:

sin (s + t) = sin (s) cos (t) + cos (s) sin (t)

= sin (π) cos (\frac{2 π}{3}) + cos (π) sin (\frac{2 π}{3})

= sin (π) cos (\frac{2 π}{3}) + cos (π) sin (\frac{2 π}{3})

Use the following trivial identity:

sin (π) = 0

sin (π)

sin (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} sin (x) 0 \frac{1}{2} \frac{2}{2} \frac{3}{2} 1 \frac{3}{2} \frac{2}{2} \frac{1}{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} sin (x) 0 - \frac{1}{2} - \frac{2}{2} - \frac{3}{2} - 1 - \frac{3}{2} - \frac{2}{2} - \frac{1}{2}

= 0

Use the following trivial identity:

cos (\frac{2 π}{3}) = - \frac{1}{2}

cos (\frac{2 π}{3})

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}

= - \frac{1}{2}

Use the following trivial identity:

cos (π) = (- 1)

cos (π)

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}

= (- 1)

Use the following trivial identity:

sin (\frac{2 π}{3}) = \frac{3}{2}

sin (\frac{2 π}{3})

sin (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} sin (x) 0 \frac{1}{2} \frac{2}{2} \frac{3}{2} 1 \frac{3}{2} \frac{2}{2} \frac{1}{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} sin (x) 0 - \frac{1}{2} - \frac{2}{2} - \frac{3}{2} - 1 - \frac{3}{2} - \frac{2}{2} - \frac{1}{2}

= \frac{3}{2}

= 0 \cdot (- \frac{1}{2}) + (- 1) \frac{3}{2}

Simplify

= - \frac{3}{2}

= \frac{1}{- \frac{3}{2}}

Simplify

\frac{1}{- \frac{3}{2}} : - \frac{2 3}{3}

\frac{1}{- \frac{3}{2}}

Apply the fraction rule:

\frac{a}{- b} = - \frac{a}{b}

= - \frac{1}{\frac{3}{2}}

Apply the fraction rule:

\frac{1}{\frac{b}{c}} = \frac{c}{b}

\frac{1}{\frac{3}{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}

= - \frac{2 3}{3}

= arcsec (- \frac{2 3}{3})

= arcsec (- \frac{2 3}{3})

Use the following trivial identity:

arcsec (- \frac{2 3}{3}) = \frac{5 π}{6}

arcsec (- \frac{2 3}{3})

x 1 \frac{2 3}{3} 22 - 2 - 2 - \frac{2 3}{3} - 1 arcsec (x) 0 \frac{π}{6} \frac{π}{4} \frac{π}{3} \frac{2 π}{3} \frac{3 π}{4} \frac{5 π}{6} π arcsec (x) 0^{\circ} 3 0^{\circ} 4 5^{\circ} 6 0^{\circ} 12 0^{\circ} 13 5^{\circ} 15 0^{\circ} 18 0^{\circ}

= \frac{5 π}{6}

= \frac{5 π}{6}

Popular Examples

sin^2(pi)+cos(pi)

sin^{2} (π) + cos (π)

cot(arcsec(5/3)+(arcsin(5/13)))

cot (arcsec (\frac{5}{3}) + (arcsin (\frac{5}{13})))

e^{2sin(pi/6)}

e^{2 s i n (\frac{π}{6})}

cot((13pi)/(12))

cot (\frac{13 π}{12})

cot(arcsec(41/9))

cot (arcsec (\frac{41}{9}))

Frequently Asked Questions (FAQ)

What is the value of arcsec(csc((5pi)/3)) ?
The value of arcsec(csc((5pi)/3)) is (5pi)/6