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

The value of (cos(pi/2)-cos((4pi)/3))/(2sin((5pi)/6)) is 1/2

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(cos(pi/2)-cos((4pi)/3))/(2sin((5pi)/6))

Solution

\frac{cos ( \frac{π}{2} ) - cos ( \frac{4 π}{3} )}{2 sin ( \frac{5 π}{6} )}

Solution

\frac{1}{2}

Decimal Notation

0.5

Solution steps

\frac{cos ( \frac{π}{2} ) - cos ( \frac{4 π}{3} )}{2 sin ( \frac{5 π}{6} )}

Rewrite using trig identities:

cos (\frac{4 π}{3}) = 1 - 2 sin^{2} (\frac{2 π}{3})

cos (\frac{4 π}{3})

Write

cos (\frac{4 π}{3})

cos (2 \cdot \frac{2 π}{3})

= cos (2 \cdot \frac{2 π}{3})

Use the Double Angle identity:

cos (2 x) = 1 - 2 sin^{2} (x)

= 1 - 2 sin^{2} (\frac{2 π}{3})

= \frac{cos ( \frac{π}{2} ) - ( 1 - 2 sin ^{2} ( \frac{2 π}{3} ) )}{2 sin ( \frac{5 π}{6} )}

Simplify

= \frac{cos ( \frac{π}{2} ) - 1 + 2 sin ^{2} ( \frac{2 π}{3} )}{2 sin ( \frac{5 π}{6} )}

Use the following trivial identity:

cos (\frac{π}{2}) = 0

cos (\frac{π}{2})

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}

= 0

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:

sin (\frac{5 π}{6}) = \frac{1}{2}

sin (\frac{5 π}{6})

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{1}{2}

= \frac{0 - 1 + 2 ( \frac{3}{2} ) ^{2}}{2 \cdot \frac{1}{2}}

Simplify

\frac{0 - 1 + 2 ( \frac{3}{2} ) ^{2}}{2 \cdot \frac{1}{2}} : \frac{1}{2}

\frac{0 - 1 + 2 ( \frac{3}{2} ) ^{2}}{2 \cdot \frac{1}{2}}

Subtract the numbers:

0 - 1 = - 1

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

Multiply

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

2 \cdot \frac{1}{2}

Multiply fractions:

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

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

Cancel the common factor:

2

= 1

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

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

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

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

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

Apply exponent rule:

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

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

(3)^{2} : 3

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

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

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

Multiply fractions:

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

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

Multiply the numbers:

3 \cdot 2 = 6

= \frac{6}{2 ^{2}}

Factor

6 : 2 \cdot 3

Factor

6 = 2 \cdot 3

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

Cancel the common factor:

2

= \frac{3}{2}

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

Join

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

\frac{3}{2} - 1

Convert element to fraction:

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

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

Since the denominators are equal, combine the fractions:

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

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

3 - 1 \cdot 2 = 1

3 - 1 \cdot 2

Multiply the numbers:

1 \cdot 2 = 2

= 3 - 2

Subtract the numbers:

3 - 2 = 1

= 1

= \frac{1}{2}

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

Apply the fraction rule:

\frac{a}{1} = a

= \frac{1}{2}

= \frac{1}{2}

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Frequently Asked Questions (FAQ)

What is the value of (cos(pi/2)-cos((4pi)/3))/(2sin((5pi)/6)) ?
The value of (cos(pi/2)-cos((4pi)/3))/(2sin((5pi)/6)) is 1/2