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

The value of 5csc^2(-pi/4) is 10

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5csc^2(-pi/4)

5 csc^{2} (- \frac{π}{4})

10

Solution steps

5 csc^{2} (- \frac{π}{4})

Rewrite using trig identities:

csc (- \frac{π}{4}) = - 2

csc (- \frac{π}{4})

Use the following property:

csc (- x) = - csc (x)

csc (- \frac{π}{4}) = - csc (\frac{π}{4})

= - csc (\frac{π}{4})

Use the following trivial identity:

csc (\frac{π}{4}) = 2

csc (\frac{π}{4})

csc (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} csc (x) Undefiend 22 \frac{2 3}{3} 1 \frac{2 3}{3} 22 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} csc (x) Undefiend - 2 - 2 - \frac{2 3}{3} - 1 - \frac{2 3}{3} - 2 - 2

= 2

= - 2

= 5 (- 2)^{2}

Simplify

5 (- 2)^{2} : 10

5 (- 2)^{2}

(- 2)^{2} = 2

(- 2)^{2}

Apply exponent rule:

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

n

is even

(- 2)^{2} = (2)^{2}

= (2)^{2}

Apply radical rule:

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

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

Apply exponent rule:

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

= 2^{\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

= 2

= 5 \cdot 2

Multiply the numbers:

5 \cdot 2 = 10

= 10

= 10

3 cos^{2} (\frac{π}{2})

1 - sin (2)

cos (10 2^{\circ})

cos^{2} (3 4^{\circ})

cot (arccos (\frac{4}{5}))