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

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

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arccos(sin(-(2pi)/3))

Solution

arccos (sin (- \frac{2 π}{3}))

Solution

\frac{5 π}{6}

Decimal Notation

150

Solution steps

arccos (sin (- \frac{2 π}{3}))

Use the following property:

sin (- x) = - sin (x)

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

= arccos (- sin (\frac{2 π}{3}))

Rewrite using trig identities:

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

sin (\frac{2 π}{3})

Use the basic trigonometric identity:

sin (x) = sin (π - x)

= sin (π - \frac{2 π}{3})

Simplify:

π - \frac{2 π}{3} = \frac{π}{3}

π - \frac{2 π}{3}

Convert element to fraction:

π = \frac{π 3}{3}

= \frac{π 3}{3} - \frac{2 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 - 2 π}{3}

Add similar elements:

3 π - 2 π = π

= \frac{π}{3}

= sin (\frac{π}{3})

= arccos (- sin (\frac{π}{3}))

Rewrite using trig identities:

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

sin (\frac{π}{3})

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{π}{3})

Simplify:

2 π - \frac{π}{3} = \frac{5 π}{3}

2 π - \frac{π}{3}

Convert element to fraction:

2 π = \frac{2 π 3}{3}

= \frac{2 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{2 π 3 - π}{3}

2 π 3 - π = 5 π

2 π 3 - π

Multiply the numbers:

2 \cdot 3 = 6

= 6 π - π

Add similar elements:

6 π - π = 5 π

= 5 π

= \frac{5 π}{3}

= - sin (\frac{5 π}{3})

= arccos (- (- sin (\frac{5 π}{3})))

Simplify

= arccos (sin (\frac{5 π}{3}))

Rewrite using trig identities:

sin (\frac{5 π}{3}) = cos (- \frac{7 π}{6})

sin (\frac{5 π}{3})

cos (\frac{π}{2} - x) = sin (x)

= cos (\frac{π}{2} - \frac{5 π}{3})

= cos (- \frac{7 π}{6})

= arccos (cos (- \frac{7 π}{6}))

Use the inverse trig property:

\frac{5 π}{6}

arccos (cos (- \frac{7 π}{6}))

For

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

Rewrite using trig identities:

cos (- \frac{7 π}{6}) = cos (\frac{5 π}{6})

cos (- \frac{7 π}{6})

cos (x + 2 π \cdot k) = cos (x)

= cos (- \frac{7 π}{6} + 1 \cdot 2 π)

= cos (\frac{5 π}{6})

= arccos (cos (\frac{5 π}{6}))

0 \leq \frac{5 π}{6} \leq π

= \frac{5 π}{6}

= \frac{5 π}{6}

Popular Examples

20cos((2pi)/3)

20 cos (\frac{2 π}{3})

(250)/(sin(45))

\frac{250}{sin ( 4 5 ^{\circ} )}

arctan((11.2)/(18.3))

arctan (\frac{11.2}{18.3})

arctan((-1.34)/(0.84))

arctan (\frac{- 1.34}{0.84})

7/(tan(50))

\frac{7}{tan ( 5 0 ^{\circ} )}

Frequently Asked Questions (FAQ)

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