What is the value of cos(arctan(-(sqrt(3))/3)+pi) ?

The value of cos(arctan(-(sqrt(3))/3)+pi) is -(sqrt(3))/2

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cos(arctan(-(sqrt(3))/3)+pi)

cos (arctan (- \frac{3}{3}) + π)

- \frac{3}{2}

Decimal Notation

- 0.86602 \dots

Solution steps

cos (arctan (- \frac{3}{3}) + π)

Use the following property:

arctan (- x) = - arctan (x)

arctan (- \frac{3}{3}) = - arctan (\frac{3}{3})

= cos (- arctan (\frac{3}{3}) + π)

Rewrite using trig identities:

cos (π) cos (arctan (\frac{3}{3})) + sin (π) sin (arctan (\frac{3}{3}))

cos (π - arctan (\frac{3}{3}))

Use the Angle Difference identity:

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

= cos (π) cos (arctan (\frac{3}{3})) + sin (π) sin (arctan (\frac{3}{3}))

= cos (π) cos (arctan (\frac{3}{3})) + sin (π) sin (arctan (\frac{3}{3}))

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)

Rewrite using trig identities:

cos (arctan (\frac{3}{3})) = \frac{3}{2}

cos (arctan (\frac{3}{3}))

Rewrite using trig identities:

cos (arctan (\frac{3}{3})) = \frac{1 + ( \frac{3}{3} ) ^{2}}{1 + ( \frac{3}{3} ) ^{2}}

Use the following identity:

cos (arctan (x)) = \frac{1 + x ^{2}}{1 + x ^{2}}

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

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

Simplify

= \frac{3}{2}

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

Rewrite using trig identities:

sin (arctan (\frac{3}{3})) = \frac{1}{2}

sin (arctan (\frac{3}{3}))

Rewrite using trig identities:

sin (arctan (\frac{3}{3})) = \frac{( \frac{3}{3} ) 1 + ( \frac{3}{3} ) ^{2}}{1 + ( \frac{3}{3} ) ^{2}}

Use the following identity:

sin (arctan (x)) = \frac{x 1 + x ^{2}}{1 + x ^{2}}

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

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

Simplify

= \frac{1}{2}

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

Simplify

= - \frac{3}{2}

tan (3 2^{\circ}) 80

5 + 4 cos (- \frac{π}{2})

cos (2 arccos (- 1))

tan (\frac{π}{7}) tan (\frac{2 π}{7}) tan (\frac{3 π}{7})

331.17 cos (5 0^{\circ})

What is the value of cos(arctan(-(sqrt(3))/3)+pi) ?
The value of cos(arctan(-(sqrt(3))/3)+pi) is -(sqrt(3))/2