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

The value of arccos(sin(-(5pi)/6)) is (2pi)/3

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arccos(sin(-(5pi)/6))

Solution

arccos (sin (- \frac{5 π}{6}))

Solution

\frac{2 π}{3}

Decimal Notation

120

Solution steps

arccos (sin (- \frac{5 π}{6}))

Use the following property:

sin (- x) = - sin (x)

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

= arccos (- sin (\frac{5 π}{6}))

Rewrite using trig identities:

sin (\frac{5 π}{6}) = cos (\frac{π}{3})

sin (\frac{5 π}{6})

Use the following identity:

sin (x) = cos (\frac{π}{2} - x)

= cos (\frac{π}{2} - \frac{5 π}{6})

Simplify:

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

\frac{π}{2} - \frac{5 π}{6}

Least Common Multiplier of

2, 6 : 6

2, 6

Least Common Multiplier (LCM)

Prime factorization of

2 : 2

2

2

is a prime number, therefore no factorization is possible

= 2

Prime factorization of

6 : 2 \cdot 3

6

6

divides by

26 = 3 \cdot 2

= 2 \cdot 3

2, 3

are all prime numbers, therefore no further factorization is possible

= 2 \cdot 3

Multiply each factor the greatest number of times it occurs in either

2

6

= 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{π}{2} :

multiply the denominator and numerator by

3

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

= \frac{π 3}{6} - \frac{5 π}{6}

Since the denominators are equal, combine the fractions:

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

= \frac{π 3 - 5 π}{6}

Add similar elements:

3 π - 5 π = - 2 π

= \frac{- 2 π}{6}

Apply the fraction rule:

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

= - \frac{2 π}{6}

Cancel the common factor:

2

= - \frac{π}{3}

= cos (- \frac{π}{3})

Use the following property:

cos (- x) = cos (x)

cos (- \frac{π}{3}) = cos (\frac{π}{3})

= cos (\frac{π}{3})

= arccos (- cos (\frac{π}{3}))

Rewrite using trig identities:

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

cos (\frac{π}{3})

Use the basic trigonometric identity:

cos (x) = - cos (π - x)

= - cos (π - \frac{π}{3})

Simplify:

π - \frac{π}{3} = \frac{2 π}{3}

π - \frac{π}{3}

Convert element to fraction:

π = \frac{π 3}{3}

= \frac{π 3}{3} - \frac{π}{3}

Since the denominators are equal, combine the fractions:

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

= \frac{π 3 - π}{3}

Add similar elements:

3 π - π = 2 π

= \frac{2 π}{3}

= - cos (\frac{2 π}{3})

= arccos (- (- cos (\frac{2 π}{3})))

Simplify

= arccos (cos (\frac{2 π}{3}))

Use the inverse trig property:

\frac{2 π}{3}

arccos (cos (\frac{2 π}{3}))

For

0 \leq x \leq π, a rccos (cos (x)) = x

0 \leq \frac{2 π}{3} \leq π

= \frac{2 π}{3}

= \frac{2 π}{3}

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Frequently Asked Questions (FAQ)

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