What is the value of csc(-510) ?

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csc(-510)

csc (- 51 0^{\circ})

- 2

Solution steps

csc (- 51 0^{\circ})

Use the following property:

csc (- x) = - csc (x)

csc (- 51 0^{\circ}) = - csc (51 0^{\circ})

= - csc (51 0^{\circ})

csc (51 0^{\circ}) = csc (15 0^{\circ})

csc (51 0^{\circ})

Rewrite

51 0^{\circ}

36 0^{\circ} + 15 0^{\circ}

= csc (36 0^{\circ} + 15 0^{\circ})

Apply the periodicity of

csc

csc (x + 36 0^{\circ}) = csc (x)

csc (36 0^{\circ} + 15 0^{\circ}) = csc (15 0^{\circ})

= csc (15 0^{\circ})

= - csc (15 0^{\circ})

Rewrite using trig identities:

csc (15 0^{\circ}) = 2

csc (15 0^{\circ})

Rewrite using trig identities:

\frac{1}{sin ( 15 0 ^{\circ} )}

csc (15 0^{\circ})

Use the basic trigonometric identity:

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

= \frac{1}{sin ( 15 0 ^{\circ} )}

= \frac{1}{sin ( 15 0 ^{\circ} )}

Use the following trivial identity:

sin (15 0^{\circ}) = \frac{1}{2}

sin (15 0^{\circ})

sin (x)

periodicity table with

36 0^{\circ} n

cycle:

x 0 3 0^{\circ} 4 5^{\circ} 6 0^{\circ} 9 0^{\circ} 12 0^{\circ} 13 5^{\circ} 15 0^{\circ} sin (x) 0 \frac{1}{2} \frac{2}{2} \frac{3}{2} 1 \frac{3}{2} \frac{2}{2} \frac{1}{2} x 18 0^{\circ} 21 0^{\circ} 22 5^{\circ} 24 0^{\circ} 27 0^{\circ} 30 0^{\circ} 31 5^{\circ} 33 0^{\circ} sin (x) 0 - \frac{1}{2} - \frac{2}{2} - \frac{3}{2} - 1 - \frac{3}{2} - \frac{2}{2} - \frac{1}{2}

= \frac{1}{2}

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

Simplify

= 2

= - 2

sin (arcsin (1) + arccos (1))

40 \cdot \frac{sin ( 13 5 ^{\circ} )}{sin ( 10 5 ^{\circ} )}

8 sin (29 0^{\circ}) + 6 sin (6 0^{\circ})

arccos (cos (\frac{π}{12}))

\frac{10}{sin ( 3 6 ^{\circ} )}