What is the value of -sin((5pi)/6)-cos((5pi)/6+(5pi)/6) ?

The value of -sin((5pi)/6)-cos((5pi)/6+(5pi)/6) is -1

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-sin((5pi)/6)-cos((5pi)/6+(5pi)/6)

Solution

- sin (\frac{5 π}{6}) - cos (\frac{5 π}{6} + \frac{5 π}{6})

Solution

- 1

Solution steps

- sin (\frac{5 π}{6}) - cos (\frac{5 π}{6} + \frac{5 π}{6})

Rewrite using trig identities:

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

cos (\frac{5 π}{6} + \frac{5 π}{6})

Use the Angle Sum identity:

cos (s + s) = cos (s) cos (s) - sin (s) sin (s)

= cos (\frac{5 π}{6}) cos (\frac{5 π}{6}) - sin (\frac{5 π}{6}) sin (\frac{5 π}{6})

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

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

cos (\frac{5 π}{6}) cos (\frac{5 π}{6}) = cos^{2} (\frac{5 π}{6})

cos (\frac{5 π}{6}) cos (\frac{5 π}{6})

Apply exponent rule:

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

cos (\frac{5 π}{6}) cos (\frac{5 π}{6}) = cos^{1 + 1} (\frac{5 π}{6})

= cos^{1 + 1} (\frac{5 π}{6})

Add the numbers:

1 + 1 = 2

= cos^{2} (\frac{5 π}{6})

sin (\frac{5 π}{6}) sin (\frac{5 π}{6}) = sin^{2} (\frac{5 π}{6})

sin (\frac{5 π}{6}) sin (\frac{5 π}{6})

Apply exponent rule:

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

sin (\frac{5 π}{6}) sin (\frac{5 π}{6}) = sin^{1 + 1} (\frac{5 π}{6})

= sin^{1 + 1} (\frac{5 π}{6})

Add the numbers:

1 + 1 = 2

= sin^{2} (\frac{5 π}{6})

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

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

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

Simplify

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

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}

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

Simplify

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

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

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

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

Apply exponent rule:

(- a)^{n} = a^{n},

n

is even

(- \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}}

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

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

Apply exponent rule:

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

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

Apply rule

1^{a} = 1

1^{2} = 1

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

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

Combine the fractions

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

Apply rule

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

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

Add/Subtract the numbers:

- 3 + 1 = - 2

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

Apply the fraction rule:

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

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

Cancel the common factor:

2

= - \frac{1}{2}

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

Add similar elements:

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

= - 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

= - 1

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Frequently Asked Questions (FAQ)

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