What is the value of csc(285) ?

The value of csc(285) is -sqrt(2)(sqrt(3)-1)

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Popular >

csc(285)

Solution

csc (28 5^{\circ})

Solution

- 2 (3 - 1)

Decimal Notation

- 1.03527 \dots

Solution steps

csc (28 5^{\circ})

Rewrite using trig identities:

\frac{1}{sin ( 28 5 ^{\circ} )}

csc (28 5^{\circ})

Use the basic trigonometric identity:

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

= \frac{1}{sin ( 28 5 ^{\circ} )}

= \frac{1}{sin ( 28 5 ^{\circ} )}

Rewrite using trig identities:

sin (28 5^{\circ}) = \frac{2 ( - 1 - 3 )}{4}

sin (28 5^{\circ})

Rewrite using trig identities:

sin (15 0^{\circ}) cos (13 5^{\circ}) + cos (15 0^{\circ}) sin (13 5^{\circ})

sin (28 5^{\circ})

Write

sin (28 5^{\circ})

sin (15 0^{\circ} + 13 5^{\circ})

= sin (15 0^{\circ} + 13 5^{\circ})

Use the Angle Sum identity:

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

= sin (15 0^{\circ}) cos (13 5^{\circ}) + cos (15 0^{\circ}) sin (13 5^{\circ})

= sin (15 0^{\circ}) cos (13 5^{\circ}) + cos (15 0^{\circ}) sin (13 5^{\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}

Use the following trivial identity:

cos (13 5^{\circ}) = - \frac{2}{2}

cos (13 5^{\circ})

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

= - \frac{2}{2}

Use the following trivial identity:

cos (15 0^{\circ}) = - \frac{3}{2}

cos (15 0^{\circ})

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

= - \frac{3}{2}

Use the following trivial identity:

sin (13 5^{\circ}) = \frac{2}{2}

sin (13 5^{\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{2}{2}

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

Simplify

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

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

Remove parentheses:

(- a) = - a

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

Factor out common term

\frac{2}{2}

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

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

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

Apply rule

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

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

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

Multiply fractions:

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

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

Multiply the numbers:

2 \cdot 2 = 4

= \frac{2 ( - 1 - 3 )}{4}

= \frac{2 ( - 1 - 3 )}{4}

= \frac{1}{\frac{2 ( - 1 - 3 )}{4}}

Simplify

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

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

Apply the fraction rule:

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

= \frac{4}{2 ( - 1 - 3 )}

Factor

4 : 2^{2}

Factor

4 = 2^{2}

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

Cancel

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

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

Apply radical rule:

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

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

Apply exponent rule:

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

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

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

Subtract the numbers:

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

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

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

2^{\frac{3}{2}} = 22

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

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

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

Apply exponent rule:

x^{a + b} = x^{a} x^{b}

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

Refine

= 22

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

Rationalize

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

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

Multiply by the conjugate

\frac{- 1 + 3}{- 1 + 3}

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

(- 1 - 3) (- 1 + 3) = - 2

(- 1 - 3) (- 1 + 3)

Apply FOIL method:

(a + b) (c + d) = a c + a d + b c + b d

a = - 1, b = - 3, c = - 1, d = 3

= (- 1) (- 1) + (- 1) 3 + (- 3) (- 1) + (- 3) 3

Apply minus-plus rules

(- a) (- b) = ab, + (- a) = - a

= 1 \cdot 1 - 1 \cdot 3 + 1 \cdot 3 - 33

Simplify

1 \cdot 1 - 1 \cdot 3 + 1 \cdot 3 - 33 : - 2

1 \cdot 1 - 1 \cdot 3 + 1 \cdot 3 - 33

Add similar elements:

- 1 \cdot 3 + 1 \cdot 3 = 0

= 1 \cdot 1 - 33

Multiply the numbers:

1 \cdot 1 = 1

= 1 - 33

Apply radical rule:

a a = a

33 = 3

= 1 - 3

Subtract the numbers:

1 - 3 = - 2

= - 2

= - 2

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

Apply the fraction rule:

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

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

Divide the numbers:

\frac{2}{2} = 1

= - 2 (3 - 1)

= - 2 (3 - 1)

= - 2 (3 - 1)

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Frequently Asked Questions (FAQ)

What is the value of csc(285) ?
The value of csc(285) is -sqrt(2)(sqrt(3)-1)