What is the value of 3sin(-(10pi)/7)csc(-(24pi)/7) ?

The value of 3sin(-(10pi)/7)csc(-(24pi)/7) is 3

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3sin(-(10pi)/7)csc(-(24pi)/7)

3 sin (- \frac{10 π}{7}) csc (- \frac{24 π}{7})

3

Solution steps

3 sin (- \frac{10 π}{7}) csc (- \frac{24 π}{7})

Use the following property:

csc (- x) = - csc (x)

csc (- \frac{24 π}{7}) = - csc (\frac{24 π}{7})

= 3 sin (- \frac{10 π}{7}) (- csc (\frac{24 π}{7}))

Use the following property:

sin (- x) = - sin (x)

sin (- \frac{10 π}{7}) = - sin (\frac{10 π}{7})

= 3 (- sin (\frac{10 π}{7})) (- csc (\frac{24 π}{7}))

csc (\frac{24 π}{7}) = csc (\frac{10 π}{7})

csc (\frac{24 π}{7})

Rewrite

\frac{24 π}{7}

2 π + \frac{10 π}{7}

= csc (2 π + \frac{10 π}{7})

Apply the periodicity of

csc

csc (x + 2 π) = csc (x)

csc (2 π + \frac{10 π}{7}) = csc (\frac{10 π}{7})

= csc (\frac{10 π}{7})

= 3 (- sin (\frac{10 π}{7})) (- csc (\frac{10 π}{7}))

Simplify

= 3 sin (\frac{10 π}{7}) csc (\frac{10 π}{7})

Rewrite using trig identities:

sin (\frac{10 π}{7}) = \frac{1}{csc ( \frac{10 π}{7} )}

sin (\frac{10 π}{7})

Use the basic trigonometric identity:

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

= \frac{1}{csc ( \frac{10 π}{7} )}

= 3 \cdot \frac{1}{csc ( \frac{10 π}{7} )} csc (\frac{10 π}{7})

3 \cdot \frac{1}{csc ( \frac{10 π}{7} )} csc (\frac{10 π}{7}) = 3

3 \cdot \frac{1}{csc ( \frac{10 π}{7} )} csc (\frac{10 π}{7})

Multiply fractions:

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

= \frac{1 \cdot 3 csc ( \frac{10 π}{7} )}{csc ( \frac{10 π}{7} )}

Cancel the common factor:

csc (\frac{10 π}{7})

= 1 \cdot 3

Multiply the numbers:

1 \cdot 3 = 3

= 3

= 3

sin (9 9^{\circ})

arctan (1000)

50 cos (4 5^{\circ})

sin (- \frac{7 π}{3})

tan (20 0^{\circ})

What is the value of 3sin(-(10pi)/7)csc(-(24pi)/7) ?
The value of 3sin(-(10pi)/7)csc(-(24pi)/7) is 3