What is the value of (sin((4pi)/3))/(tan(pi/4)+cos((5pi)/6)) ?

The value of (sin((4pi)/3))/(tan(pi/4)+cos((5pi)/6)) is -2sqrt(3)-3

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(sin((4pi)/3))/(tan(pi/4)+cos((5pi)/6))

Solution

\frac{sin ( \frac{4 π}{3} )}{tan ( \frac{π}{4} ) + cos ( \frac{5 π}{6} )}

Solution

- 23 - 3

Decimal Notation

- 6.46410 \dots

Solution steps

\frac{sin ( \frac{4 π}{3} )}{tan ( \frac{π}{4} ) + cos ( \frac{5 π}{6} )}

Rewrite using trig identities:

sin (\frac{4 π}{3}) = 2 sin (\frac{2 π}{3}) cos (\frac{2 π}{3})

sin (\frac{4 π}{3})

Write

sin (\frac{4 π}{3})

sin (2 \cdot \frac{2 π}{3})

= sin (2 \cdot \frac{2 π}{3})

Use the Double Angle identity:

sin (2 x) = 2 sin (x) cos (x)

= 2 sin (\frac{2 π}{3}) cos (\frac{2 π}{3})

= \frac{2 sin ( \frac{2 π}{3} ) cos ( \frac{2 π}{3} )}{tan ( \frac{π}{4} ) + cos ( \frac{5 π}{6} )}

Use the following trivial identity:

sin (\frac{2 π}{3}) = \frac{3}{2}

sin (\frac{2 π}{3})

sin (x)

periodicity table with

2 πn

cycle:

x 0 \frac{π}{6} \frac{π}{4} \frac{π}{3} \frac{π}{2} \frac{2 π}{3} \frac{3 π}{4} \frac{5 π}{6} sin (x) 0 \frac{1}{2} \frac{2}{2} \frac{3}{2} 1 \frac{3}{2} \frac{2}{2} \frac{1}{2} x π \frac{7 π}{6} \frac{5 π}{4} \frac{4 π}{3} \frac{3 π}{2} \frac{5 π}{3} \frac{7 π}{4} \frac{11 π}{6} 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:

cos (\frac{2 π}{3}) = - \frac{1}{2}

cos (\frac{2 π}{3})

cos (x)

periodicity table with

2 πn

cycle:

x 0 \frac{π}{6} \frac{π}{4} \frac{π}{3} \frac{π}{2} \frac{2 π}{3} \frac{3 π}{4} \frac{5 π}{6} cos (x) 1 \frac{3}{2} \frac{2}{2} \frac{1}{2} 0 - \frac{1}{2} - \frac{2}{2} - \frac{3}{2} x π \frac{7 π}{6} \frac{5 π}{4} \frac{4 π}{3} \frac{3 π}{2} \frac{5 π}{3} \frac{7 π}{4} \frac{11 π}{6} cos (x) - 1 - \frac{3}{2} - \frac{2}{2} - \frac{1}{2} 0 \frac{1}{2} \frac{2}{2} \frac{3}{2}

= - \frac{1}{2}

Use the following trivial identity:

tan (\frac{π}{4}) = 1

tan (\frac{π}{4})

tan (x)

periodicity table with

πn

cycle:

x 0 \frac{π}{6} \frac{π}{4} \frac{π}{3} \frac{π}{2} \frac{2 π}{3} \frac{3 π}{4} \frac{5 π}{6} tan (x) 0 \frac{3}{3} 13 \pm \infty - 3 - 1 - \frac{3}{3}

= 1

Use the following trivial identity:

cos (\frac{5 π}{6}) = - \frac{3}{2}

cos (\frac{5 π}{6})

cos (x)

periodicity table with

2 πn

cycle:

x 0 \frac{π}{6} \frac{π}{4} \frac{π}{3} \frac{π}{2} \frac{2 π}{3} \frac{3 π}{4} \frac{5 π}{6} cos (x) 1 \frac{3}{2} \frac{2}{2} \frac{1}{2} 0 - \frac{1}{2} - \frac{2}{2} - \frac{3}{2} x π \frac{7 π}{6} \frac{5 π}{4} \frac{4 π}{3} \frac{3 π}{2} \frac{5 π}{3} \frac{7 π}{4} \frac{11 π}{6} 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}

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

Simplify

\frac{2 \cdot \frac{3}{2} ( - \frac{1}{2} )}{1 - \frac{3}{2}} : - 23 - 3

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

Remove parentheses:

(- a) = - a

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

Apply the fraction rule:

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

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

Join

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

1 - \frac{3}{2}

Convert element to fraction:

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

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

Multiply the numbers:

1 \cdot 2 = 2

= \frac{2 - 3}{2}

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

Simplify

\frac{2 \cdot \frac{3}{2} \cdot \frac{1}{2}}{\frac{2 - 3}{2}} : \frac{4 \cdot \frac{1}{2} \cdot \frac{3}{2}}{2 - 3}

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

Apply the fraction rule:

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

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

Multiply the numbers:

2 \cdot 2 = 4

= \frac{4 \cdot \frac{1}{2} \cdot \frac{3}{2}}{2 - 3}

= - \frac{4 \cdot \frac{1}{2} \cdot \frac{3}{2}}{2 - 3}

Multiply

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

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

Multiply fractions:

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

= \frac{3 \cdot 1 \cdot 4}{2 \cdot 2}

Multiply the numbers:

1 \cdot 4 = 4

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

Multiply the numbers:

2 \cdot 2 = 4

= \frac{4 3}{4}

Divide the numbers:

\frac{4}{4} = 1

= 3

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

Rationalize

- \frac{3}{2 - 3} : - 23 - 3

- \frac{3}{2 - 3}

Multiply by the conjugate

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

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

3 (2 + 3) = 23 + 3

3 (2 + 3)

Apply the distributive law:

a (b + c) = ab + a c

a = 3, b = 2, c = 3

= 3 \cdot 2 + 33

= 23 + 33

Apply radical rule:

a a = a

33 = 3

= 23 + 3

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

(2 - 3) (2 + 3)

Apply Difference of Two Squares Formula:

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

a = 2, b = 3

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

Simplify

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

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

2^{2} = 4

2^{2}

2^{2} = 4

= 4

(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

= 4 - 3

Subtract the numbers:

4 - 3 = 1

= 1

= 1

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

Apply rule

\frac{a}{1} = a

= - (23 + 3)

Distribute parentheses

= - (23) - (3)

Apply minus-plus rules

+ (- a) = - a

= - 23 - 3

= - 23 - 3

= - 23 - 3

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Frequently Asked Questions (FAQ)

What is the value of (sin((4pi)/3))/(tan(pi/4)+cos((5pi)/6)) ?
The value of (sin((4pi)/3))/(tan(pi/4)+cos((5pi)/6)) is -2sqrt(3)-3