What is the value of sin((7pi)/8)-sin((7pi)/6)-cot(pi/3) ?

The value of sin((7pi)/8)-sin((7pi)/6)-cot(pi/3) is 1/2+(3sqrt(-\sqrt{2)+2}-2sqrt(3))/6

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sin((7pi)/8)-sin((7pi)/6)-cot(pi/3)

Solution

sin (\frac{7 π}{8}) - sin (\frac{7 π}{6}) - cot (\frac{π}{3})

Solution

\frac{1}{2} + \frac{3 - 2 + 2 - 2 3}{6}

Decimal

0.30533 \dots

Solution steps

sin (\frac{7 π}{8}) - sin (\frac{7 π}{6}) - cot (\frac{π}{3})

Rewrite using trig identities:

sin (\frac{7 π}{8}) = \frac{2 - 2}{2}

sin (\frac{7 π}{8})

Rewrite using trig identities:

sin (\frac{π}{8})

sin (\frac{7 π}{8})

Use the basic trigonometric identity:

sin (x) = sin (π - x)

= sin (π - \frac{7 π}{8})

Simplify:

π - \frac{7 π}{8} = \frac{π}{8}

π - \frac{7 π}{8}

Convert element to fraction:

π = \frac{π 8}{8}

= \frac{π 8}{8} - \frac{7 π}{8}

Since the denominators are equal, combine the fractions:

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

= \frac{π 8 - 7 π}{8}

Add similar elements:

8 π - 7 π = π

= \frac{π}{8}

= sin (\frac{π}{8})

= sin (\frac{π}{8})

Rewrite using trig identities:

\frac{1 - cos ( \frac{π}{4} )}{2}

sin (\frac{π}{8})

Write

sin (\frac{π}{8})

sin (\frac{\frac{π}{4}}{2})

= sin (\frac{\frac{π}{4}}{2})

Use the Half Angle identity:

sin (\frac{θ}{2}) = \frac{1 - cos ( θ )}{2}

Use the Double Angle identity

cos (2 θ) = 1 - 2 sin^{2} (θ)

Substitute

θ

with

\frac{θ}{2}

cos (θ) = 1 - 2 sin^{2} (\frac{θ}{2})

Switch sides

2 sin^{2} (\frac{θ}{2}) = 1 - cos (θ)

Divide both sides by

2

sin^{2} (\frac{θ}{2}) = \frac{( 1 - cos ( θ ) )}{2}

Square root both sides

Choose the root sign according to the quadrant of

\frac{θ}{2}

range [0, \frac{π}{2}] [\frac{π}{2}, π] [π, \frac{3 π}{2}] [\frac{3 π}{2}, 2 π] quadrant I II III I V sin positive positive negative negative cos positive negative negative positive

sin (\frac{θ}{2}) = \frac{( 1 - cos ( θ ) )}{2}

= \frac{1 - cos ( \frac{π}{4} )}{2}

= \frac{1 - cos ( \frac{π}{4} )}{2}

Use the following trivial identity:

cos (\frac{π}{4}) = \frac{2}{2}

cos (\frac{π}{4})

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{2}{2}

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

Simplify

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

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

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

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

Join

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

1 - \frac{2}{2}

Convert element to fraction:

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

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

Multiply the numbers:

1 \cdot 2 = 2

= \frac{2 - 2}{2}

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

Apply the fraction rule:

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

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

Multiply the numbers:

2 \cdot 2 = 4

= \frac{2 - 2}{4}

= \frac{2 - 2}{4}

Apply radical rule:

n \frac{a}{b} = \frac{n a}{n b},

assuming

a \geq 0, b \geq 0

= \frac{2 - 2}{4}

4 = 2

4

Factor the number:

4 = 2^{2}

= 2^{2}

Apply radical rule:

n a^{n} = a

2^{2} = 2

= 2

= \frac{2 - 2}{2}

= \frac{2 - 2}{2}

Rewrite using trig identities:

sin (\frac{7 π}{6}) = - \frac{1}{2}

sin (\frac{7 π}{6})

Rewrite using trig identities:

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

sin (\frac{7 π}{6})

Write

sin (\frac{7 π}{6})

sin (π + \frac{π}{6})

= sin (π + \frac{π}{6})

Use the Angle Sum identity:

sin (s + t) = sin (s) cos (t) + cos (s) sin (t)

= sin (π) cos (\frac{π}{6}) + cos (π) sin (\frac{π}{6})

= sin (π) cos (\frac{π}{6}) + cos (π) sin (\frac{π}{6})

Use the following trivial identity:

sin (π) = 0

sin (π)

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}

= 0

Use the following trivial identity:

cos (\frac{π}{6}) = \frac{3}{2}

cos (\frac{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}

Use the following trivial identity:

cos (π) = (- 1)

cos (π)

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}

= (- 1)

Use the following trivial identity:

sin (\frac{π}{6}) = \frac{1}{2}

sin (\frac{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}

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

Simplify

= - \frac{1}{2}

Use the following trivial identity:

cot (\frac{π}{3}) = \frac{3}{3}

cot (\frac{π}{3})

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

= \frac{3}{3}

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

Simplify

\frac{2 - 2}{2} - (- \frac{1}{2}) - \frac{3}{3} : \frac{1}{2} + \frac{3 - 2 + 2 - 2 3}{6}

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

Apply rule

- (- a) = a

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

Combine the fractions

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

\frac{2 - 2}{2} - \frac{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}{2} :

multiply the denominator and numerator by

3

\frac{2 - 2}{2} = \frac{2 - 2 \cdot 3}{2 \cdot 3} = \frac{2 - 2 \cdot 3}{6}

For

\frac{3}{3} :

multiply the denominator and numerator by

2

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

= \frac{2 - 2 \cdot 3}{6} - \frac{3 \cdot 2}{6}

Since the denominators are equal, combine the fractions:

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

= \frac{2 - 2 \cdot 3 - 3 \cdot 2}{6}

Factor

2 - 2 3 - 32 : 3 (3 - 2 + 2 - 2)

2 - 2 \cdot 3 - 3 \cdot 2

3 = 33

= 2 - 2 33 - 3 \cdot 2

Factor out common term

3

= 3 (2 - 2 3 - 2)

Expand

3 2 - 2 - 2 : 3 - 2 + 2 - 2

2 - 2 3 - 2

2 - 2 3 = 3 - 2 + 2

2 - 2 3

Apply radical rule:

a b = a \cdot b

3 2 - 2 = 3 (2 - 2)

= 3 (2 - 2)

Factor

2 - 2 : - (2 - 2)

2 - 2

Factor out common term

- 1

= - (2 - 2)

= - 3 (2 - 2)

Apply radical rule:

n ab = n a n b,

assuming

a \geq 0, b \geq 0

= 3 - (2 - 2)

Expand

- (2 - 2) : - 2 + 2

- (2 - 2)

Distribute parentheses

= - (2) - (- 2)

Apply minus-plus rules

- (- a) = a, - (a) = - a

= - 2 + 2

= 3 2 - 2

= 3 2 - 2 - 2

= 3 (3 2 - 2 - 2)

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

Factor

6 : 2 \cdot 3

Factor

6 = 2 \cdot 3

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

Cancel

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

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

Apply radical rule:

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

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

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

Subtract the numbers:

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

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

Apply radical rule:

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

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

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

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

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

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

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

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

Multiply by the conjugate

\frac{3}{3}

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

(3 - 2 + 2 - 2) 3 = 3 - 2 + 2 - 23

(3 - 2 + 2 - 2) 3

= 3 (3 - 2 + 2 - 2)

Apply the distributive law:

a (b - c) = ab - a c

a = 3, b = 3 - 2 + 2, c = 2

= 33 - 2 + 2 - 3 \cdot 2

= 33 - 2 + 2 - 23

Apply radical rule:

a a = a

33 = 3

= 3 - 2 + 2 - 23

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 + 2 - 2 3}{6}

= \frac{1}{2} + \frac{3 2 - 2 - 2 3}{6}

= \frac{1}{2} + \frac{3 - 2 + 2 - 2 3}{6}

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Frequently Asked Questions (FAQ)

What is the value of sin((7pi)/8)-sin((7pi)/6)-cot(pi/3) ?
The value of sin((7pi)/8)-sin((7pi)/6)-cot(pi/3) is 1/2+(3sqrt(-\sqrt{2)+2}-2sqrt(3))/6