What is the value of (sec(11pi))/6 (sin(pi))/6+(cos(7pi))/6 ?

The value of (sec(11pi))/6 (sin(pi))/6+(cos(7pi))/6 is -1/6

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(sec(11pi))/6 (sin(pi))/6+(cos(7pi))/6

\frac{sec ( 11 π )}{6} \frac{sin ( π )}{6} + \frac{cos ( 7 π )}{6}

- \frac{1}{6}

Decimal Notation

- 0.16666 \dots

Solution steps

\frac{sec ( 11 π )}{6} \cdot \frac{sin ( π )}{6} + \frac{cos ( 7 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}

= \frac{sec ( 11 π )}{6} \cdot \frac{0}{6} + \frac{cos ( 7 π )}{6}

= \frac{sec ( 11 π )}{6} \cdot \frac{0}{6} + \frac{cos ( 7 π )}{6}

Simplify

= \frac{cos ( 7 π )}{6}

Rewrite using trig identities:

cos (7 π) = - 1

cos (7 π)

cos (7 π) = cos (π)

cos (7 π)

Rewrite

7 π

2 π \cdot 3 + π

= cos (2 π 3 + π)

Apply the periodicity of

cos

cos (x + 2 π \cdot k) = cos (x)

cos (2 π \cdot 3 + π) = cos (π)

= cos (π)

= cos (π)

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)

= - 1

= \frac{- 1}{6}

Simplify

= - \frac{1}{6}

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

- arctan (tan (\frac{2 π}{7}))

\frac{40}{cos ( 3 0 ^{\circ} )}

arctan (\frac{- 4}{5})

- sec (\frac{11 π}{6}) tan (\frac{11 π}{6})

What is the value of (sec(11pi))/6 (sin(pi))/6+(cos(7pi))/6 ?
The value of (sec(11pi))/6 (sin(pi))/6+(cos(7pi))/6 is -1/6