What is the value of -csc^2((5pi)/3) ?

The value of -csc^2((5pi)/3) is -4/3

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Popular >

-csc^2((5pi)/3)

Solution

- csc^{2} (\frac{5 π}{3})

Solution

- \frac{4}{3}

Decimal Notation

- 1.33333 \dots

Solution steps

- csc^{2} (\frac{5 π}{3})

Rewrite using trig identities:

csc^{2} (\frac{5 π}{3}) = 1 + cot^{2} (\frac{2 π}{3})

csc^{2} (\frac{5 π}{3})

Use the Pythagorean identity:

csc^{2} (x) = 1 + cot^{2} (x)

= 1 + cot^{2} (\frac{5 π}{3})

cot (\frac{5 π}{3}) = cot (\frac{2 π}{3})

cot (\frac{5 π}{3})

Rewrite

\frac{5 π}{3}

π + \frac{2 π}{3}

= cot (π + \frac{2 π}{3})

Apply the periodicity of

cot

cot (x + π) = cot (x)

cot (π + \frac{2 π}{3}) = cot (\frac{2 π}{3})

= cot (\frac{2 π}{3})

= 1 + cot^{2} (\frac{2 π}{3})

= - (1 + cot^{2} (\frac{2 π}{3}))

Simplify

= - 1 - cot^{2} (\frac{2 π}{3})

Use the following trivial identity:

cot (\frac{2 π}{3}) = - \frac{3}{3}

cot (\frac{2 π}{3})

cot (x)

periodicity table with

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} cot (x) \mp \infty 31 \frac{3}{3} 0 - \frac{3}{3} - 1 - 3

= - \frac{3}{3}

= - 1 - (- \frac{3}{3})^{2}

Simplify

- 1 - (- \frac{3}{3})^{2} : - \frac{4}{3}

- 1 - (- \frac{3}{3})^{2}

(- \frac{3}{3})^{2} = \frac{1}{3}

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

Apply exponent rule:

(- a)^{n} = a^{n},

n

is even

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

= (\frac{3}{3})^{2}

Apply exponent rule:

(\frac{a}{b})^{c} = \frac{a ^{c}}{b ^{c}}

= \frac{( 3 ) ^{2}}{3 ^{2}}

(3)^{2} : 3

Apply radical rule:

a = a^{\frac{1}{2}}

= (3^{\frac{1}{2}})^{2}

Apply exponent rule:

(a^{b})^{c} = a^{b c}

= 3^{\frac{1}{2} \cdot 2}

\frac{1}{2} \cdot 2 = 1

\frac{1}{2} \cdot 2

Multiply fractions:

a \cdot \frac{b}{c} = \frac{a \cdot b}{c}

= \frac{1 \cdot 2}{2}

Cancel the common factor:

2

= 1

= 3

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

Cancel the common factor:

3

= \frac{1}{3}

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

Convert element to fraction:

1 = \frac{1 \cdot 3}{3}

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

Since the denominators are equal, combine the fractions:

\frac{a}{c} \pm \frac{b}{c} = \frac{a \pm b}{c}

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

- 1 \cdot 3 - 1 = - 4

- 1 \cdot 3 - 1

Multiply the numbers:

1 \cdot 3 = 3

= - 3 - 1

Subtract the numbers:

- 3 - 1 = - 4

= - 4

= \frac{- 4}{3}

Apply the fraction rule:

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

= - \frac{4}{3}

= - \frac{4}{3}

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Frequently Asked Questions (FAQ)

What is the value of -csc^2((5pi)/3) ?
The value of -csc^2((5pi)/3) is -4/3