What is the value of sin(60)(csc(60)-sin(60)) ?

The value of sin(60)(csc(60)-sin(60)) is 1/4

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sin(60)(csc(60)-sin(60))

Solution

sin (6 0^{\circ}) (csc (6 0^{\circ}) - sin (6 0^{\circ}))

Solution

\frac{1}{4}

Decimal Notation

0.25

Solution steps

sin (6 0^{\circ}) (csc (6 0^{\circ}) - sin (6 0^{\circ}))

Use the following trivial identity:

sin (6 0^{\circ}) = \frac{3}{2}

sin (6 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{3}{2}

Use the following trivial identity:

csc (6 0^{\circ}) = \frac{2 3}{3}

csc (6 0^{\circ})

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

= \frac{2 3}{3}

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

Simplify

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

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

Join

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

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

Least Common Multiplier of

3, 2 : 6

3, 2

Least Common Multiplier (LCM)

Prime factorization of

3 : 3

3

3

is a prime number, therefore no factorization is possible

= 3

Prime factorization of

2 : 2

2

2

is a prime number, therefore no factorization is possible

= 2

Multiply each factor the greatest number of times it occurs in either

3

2

= 3 \cdot 2

Multiply the numbers:

3 \cdot 2 = 6

= 6

Adjust Fractions based on the LCM

Multiply each numerator by the same amount needed to multiply its

corresponding denominator to turn it into the LCM

6

For

\frac{2 3}{3} :

multiply the denominator and numerator by

2

\frac{2 3}{3} = \frac{2 3 \cdot 2}{3 \cdot 2} = \frac{4 3}{6}

For

\frac{3}{2} :

multiply the denominator and numerator by

3

\frac{3}{2} = \frac{3 \cdot 3}{2 \cdot 3} = \frac{3 \cdot 3}{6}

= \frac{4 3}{6} - \frac{3 \cdot 3}{6}

Since the denominators are equal, combine the fractions:

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

= \frac{4 3 - 3 \cdot 3}{6}

Add similar elements:

43 - 33 = 3

= \frac{3}{6}

Factor

6 : 2 \cdot 3

Factor

6 = 2 \cdot 3

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

Cancel

\frac{3}{2 \cdot 3} : \frac{1}{2 3}

\frac{3}{2 \cdot 3}

Apply radical rule:

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

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

Apply exponent rule:

\frac{x ^{a}}{x ^{b}} = \frac{1}{x ^{b - a}}

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

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

Subtract the numbers:

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

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

Apply radical rule:

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

= \frac{1}{2 3}

= \frac{1}{2 3}

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

Multiply fractions:

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

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

Cancel the common factor:

3

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

Multiply the numbers:

2 \cdot 2 = 4

= \frac{1}{4}

= \frac{1}{4}

Popular Examples

cos(1050)

cos (105 0^{\circ})

cos((27pi)/2)

cos (\frac{27 π}{2})

arccos(4/(sqrt(5)\sqrt{5)})

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

cos(45)-csc(30)

cos (4 5^{\circ}) - csc (3 0^{\circ})

cos((3pi)/4)+isin((3pi)/4)

cos (\frac{3 π}{4}) + i sin (\frac{3 π}{4})

Frequently Asked Questions (FAQ)

What is the value of sin(60)(csc(60)-sin(60)) ?
The value of sin(60)(csc(60)-sin(60)) is 1/4