What is the value of (2sin(60))/(cot(30)-4sin(45)) ?

The value of (2sin(60))/(cot(30)-4sin(45)) is -(3+2sqrt(6))/5

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Popular >

(2sin(60))/(cot(30)-4sin(45))

Solution

\frac{2 sin ( 6 0 ^{\circ} )}{cot ( 3 0 ^{\circ} ) - 4 sin ( 4 5 ^{\circ} )}

Solution

- \frac{3 + 2 6}{5}

Decimal Notation

- 1.57979 \dots

Solution steps

\frac{2 sin ( 6 0 ^{\circ} )}{cot ( 3 0 ^{\circ} ) - 4 sin ( 4 5 ^{\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:

cot (3 0^{\circ}) = 3

cot (3 0^{\circ})

cot (x)

periodicity table with

18 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} cot (x) \mp \infty 31 \frac{3}{3} 0 - \frac{3}{3} - 1 - 3

= 3

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}

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

Simplify

\frac{2 \cdot \frac{3}{2}}{3 - 4 \cdot \frac{2}{2}} : - \frac{3 + 2 6}{5}

\frac{2 \cdot \frac{3}{2}}{3 - 4 \cdot \frac{2}{2}}

4 \cdot \frac{2}{2} = 22

4 \cdot \frac{2}{2}

Multiply fractions:

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

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

Divide the numbers:

\frac{4}{2} = 2

= 22

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

Multiply

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

2 \cdot \frac{3}{2}

Multiply fractions:

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

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

Cancel the common factor:

2

= 3

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

Rationalize

\frac{3}{3 - 2 2} : - \frac{3 + 2 6}{5}

\frac{3}{3 - 2 2}

Multiply by the conjugate

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

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

3 (3 + 22) = 3 + 26

3 (3 + 22)

Apply the distributive law:

a (b + c) = ab + a c

a = 3, b = 3, c = 22

= 33 + 3 \cdot 22

= 33 + 232

Simplify

33 + 232 : 3 + 26

33 + 232

33 = 3

33

Apply radical rule:

a a = a

33 = 3

= 3

232 = 26

232

Apply radical rule:

a b = a \cdot b

32 = 3 \cdot 2

= 2 3 \cdot 2

Multiply the numbers:

3 \cdot 2 = 6

= 26

= 3 + 26

= 3 + 26

(3 - 22) (3 + 22) = - 5

(3 - 22) (3 + 22)

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

22

Apply exponent rule:

a^{b} \cdot a^{c} = a^{b + c}

22 = 2 \cdot 2^{\frac{1}{2}} = 2^{1 + \frac{1}{2}}

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

Join

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

1 + \frac{1}{2}

Convert element to fraction:

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

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

Since the denominators are equal, combine the fractions:

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

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

1 \cdot 2 + 1 = 3

1 \cdot 2 + 1

Multiply the numbers:

1 \cdot 2 = 2

= 2 + 1

Add the numbers:

2 + 1 = 3

= 3

= \frac{3}{2}

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

= (3 - 2^{\frac{3}{2}}) (22 + 3)

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

22

Apply exponent rule:

a^{b} \cdot a^{c} = a^{b + c}

22 = 2 \cdot 2^{\frac{1}{2}} = 2^{1 + \frac{1}{2}}

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

Join

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

1 + \frac{1}{2}

Convert element to fraction:

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

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

Since the denominators are equal, combine the fractions:

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

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

1 \cdot 2 + 1 = 3

1 \cdot 2 + 1

Multiply the numbers:

1 \cdot 2 = 2

= 2 + 1

Add the numbers:

2 + 1 = 3

= 3

= \frac{3}{2}

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

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

Apply Difference of Two Squares Formula:

(a - b) (a + b) = a^{2} - b^{2}

a = 3, b = 2^{\frac{3}{2}}

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

Simplify

(3)^{2} - (2^{\frac{3}{2}})^{2} : - 5

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

(3)^{2} = 3

(3)^{2}

Apply radical rule:

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

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

Apply exponent rule:

(a^{b})^{c} = a^{b c}

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

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

\frac{1}{2} \cdot 2

Multiply fractions:

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

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

Cancel the common factor:

2

= 1

= 3

(2^{\frac{3}{2}})^{2} = 8

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

Apply exponent rule:

(a^{b})^{c} = a^{b c}

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

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

\frac{3}{2} \cdot 2

Multiply fractions:

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

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

Cancel the common factor:

2

= 3

= 2^{3}

2^{3} = 8

= 8

= 3 - 8

Subtract the numbers:

3 - 8 = - 5

= - 5

= - 5

= \frac{3 + 2 6}{- 5}

Apply the fraction rule:

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

= - \frac{3 + 2 6}{5}

= - \frac{3 + 2 6}{5}

= - \frac{3 + 2 6}{5}

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Frequently Asked Questions (FAQ)

What is the value of (2sin(60))/(cot(30)-4sin(45)) ?
The value of (2sin(60))/(cot(30)-4sin(45)) is -(3+2sqrt(6))/5