What is the value of sin(45)+csc(60) ?

The value of sin(45)+csc(60) is (3sqrt(2)+4sqrt(3))/6

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sin(45)+csc(60)

Solution

sin (4 5^{\circ}) + csc (6 0^{\circ})

Solution

\frac{3 2 + 4 3}{6}

Decimal Notation

1.86180 \dots

Solution steps

sin (4 5^{\circ}) + csc (6 0^{\circ})

Use the following trivial identity:

sin (4 5^{\circ}) = \frac{2}{2}

sin (4 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}

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{2}{2} + \frac{2 3}{3}

Simplify

\frac{2}{2} + \frac{2 3}{3} : \frac{3 2 + 4 3}{6}

\frac{2}{2} + \frac{2 3}{3}

Least Common Multiplier of

2, 3 : 6

2, 3

Least Common Multiplier (LCM)

Prime factorization of

2 : 2

2

2

is a prime number, therefore no factorization is possible

= 2

Prime factorization of

3 : 3

3

3

is a prime number, therefore no factorization is possible

= 3

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

2

3

= 2 \cdot 3

Multiply the numbers:

2 \cdot 3 = 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}{2} :

multiply the denominator and numerator by

3

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

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

Since the denominators are equal, combine the fractions:

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

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

Factor

23 + 43 : 3 (6 + 4)

2 \cdot 3 + 43

3 = 33

= 233 + 43

Factor out common term

3

= 3 (23 + 4)

Refine

= 3 (4 + 6)

= \frac{3 ( 6 + 4 )}{6}

Factor

6 : 2 \cdot 3

Factor

6 = 2 \cdot 3

= \frac{3 ( 4 + 6 )}{2 \cdot 3}

Cancel

\frac{3 ( 6 + 4 )}{2 \cdot 3} : \frac{6 + 4}{2 3}

\frac{3 ( 6 + 4 )}{2 \cdot 3}

Apply radical rule:

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

= \frac{3 ^{\frac{1}{2}} ( 4 + 6 )}{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{6 + 4}{2 \cdot 3 ^{- \frac{1}{2} + 1}}

Subtract the numbers:

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

= \frac{6 + 4}{2 \cdot 3 ^{\frac{1}{2}}}

Apply radical rule:

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

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

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

Rationalize

\frac{6 + 4}{2 3} : \frac{3 2 + 4 3}{6}

\frac{6 + 4}{2 3}

Multiply by the conjugate

\frac{3}{3}

= \frac{( 6 + 4 ) 3}{2 3 3}

(6 + 4) 3 = 32 + 43

(6 + 4) 3

= 3 (6 + 4)

Apply the distributive law:

a (b + c) = ab + a c

a = 3, b = 6, c = 4

= 36 + 3 \cdot 4

= 36 + 43

36 = 32

36

Factor integer

6 = 3 \cdot 2

= 3 3 \cdot 2

Apply radical rule:

3 \cdot 2 = 32

= 332

Apply radical rule:

a a = a

33 = 3

= 32

= 32 + 43

233 = 6

233

Apply radical rule:

a a = a

33 = 3

= 2 \cdot 3

Multiply the numbers:

2 \cdot 3 = 6

= 6

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

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

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

Popular Examples

tan(45)-cos(30)

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

arctan((6.25)/(9.8)+0.2)

arctan (\frac{6.25}{9.8} + 0.2)

cos(pi/2*3)

cos (\frac{π}{2} \cdot 3)

tan(30)*cos(30)

tan (3 0^{\circ}) \cdot cos (3 0^{\circ})

arctan(-90)

arctan (- 90)

Frequently Asked Questions (FAQ)

What is the value of sin(45)+csc(60) ?
The value of sin(45)+csc(60) is (3sqrt(2)+4sqrt(3))/6