What is the value of arccos(sin(-(11pi)/4)) ?

The value of arccos(sin(-(11pi)/4)) is (3pi)/4

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arccos(sin(-(11pi)/4))

Solution

arccos (sin (- \frac{11 π}{4}))

Solution

\frac{3 π}{4}

Decimal Notation

135

Solution steps

arccos (sin (- \frac{11 π}{4}))

Use the following property:

sin (- x) = - sin (x)

sin (- \frac{11 π}{4}) = - sin (\frac{11 π}{4})

= arccos (- sin (\frac{11 π}{4}))

sin (\frac{11 π}{4}) = sin (\frac{3 π}{4})

sin (\frac{11 π}{4})

Rewrite

\frac{11 π}{4}

2 π + \frac{3 π}{4}

= sin (2 π + \frac{3 π}{4})

Apply the periodicity of

sin

sin (x + 2 π) = sin (x)

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

= sin (\frac{3 π}{4})

= arccos (- sin (\frac{3 π}{4}))

Rewrite using trig identities:

sin (\frac{3 π}{4}) = - sin (\frac{5 π}{4})

sin (\frac{3 π}{4})

Use the following identity:

sin (x) = - sin (2 π - x)

sin (x)

Use the following property:

sin (θ) = - sin (- θ)

sin (x) = - sin (- x)

= - sin (- x)

Apply the periodicity of

sin

sin (2 π + θ) = sin (θ)

- sin (- x) = - sin (2 π - x)

= - sin (2 π - x)

= - sin (2 π - \frac{3 π}{4})

Simplify:

2 π - \frac{3 π}{4} = \frac{5 π}{4}

2 π - \frac{3 π}{4}

Convert element to fraction:

2 π = \frac{2 π 4}{4}

= \frac{2 π 4}{4} - \frac{3 π}{4}

Since the denominators are equal, combine the fractions:

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

= \frac{2 π 4 - 3 π}{4}

2 π 4 - 3 π = 5 π

2 π 4 - 3 π

Multiply the numbers:

2 \cdot 4 = 8

= 8 π - 3 π

Add similar elements:

8 π - 3 π = 5 π

= 5 π

= \frac{5 π}{4}

= - sin (\frac{5 π}{4})

= arccos (- (- sin (\frac{5 π}{4})))

Simplify

= arccos (sin (\frac{5 π}{4}))

Rewrite using trig identities:

sin (\frac{5 π}{4}) = cos (\frac{3 π}{4})

sin (\frac{5 π}{4})

Use the following identity:

sin (x) = cos (\frac{π}{2} - x)

= cos (\frac{π}{2} - \frac{5 π}{4})

Simplify:

\frac{π}{2} - \frac{5 π}{4} = - \frac{3 π}{4}

\frac{π}{2} - \frac{5 π}{4}

Least Common Multiplier of

2, 4 : 4

2, 4

Least Common Multiplier (LCM)

Prime factorization of

2 : 2

2

2

is a prime number, therefore no factorization is possible

= 2

Prime factorization of

4 : 2 \cdot 2

4

4

divides by

24 = 2 \cdot 2

= 2 \cdot 2

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

2

4

= 2 \cdot 2

Multiply the numbers:

2 \cdot 2 = 4

= 4

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

4

For

\frac{π}{2} :

multiply the denominator and numerator by

2

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

= \frac{π 2}{4} - \frac{5 π}{4}

Since the denominators are equal, combine the fractions:

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

= \frac{π 2 - 5 π}{4}

Add similar elements:

2 π - 5 π = - 3 π

= \frac{- 3 π}{4}

Apply the fraction rule:

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

= - \frac{3 π}{4}

= cos (- \frac{3 π}{4})

Use the following property:

cos (- x) = cos (x)

cos (- \frac{3 π}{4}) = cos (\frac{3 π}{4})

= cos (\frac{3 π}{4})

= arccos (cos (\frac{3 π}{4}))

Use the inverse trig property:

\frac{3 π}{4}

arccos (cos (\frac{3 π}{4}))

For

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

0 \leq \frac{3 π}{4} \leq π

= \frac{3 π}{4}

= \frac{3 π}{4}

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Frequently Asked Questions (FAQ)

What is the value of arccos(sin(-(11pi)/4)) ?
The value of arccos(sin(-(11pi)/4)) is (3pi)/4