What is the value of cos(-pi)+isin(-pi) ?

The value of cos(-pi)+isin(-pi) is -1

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cos(-pi)+isin(-pi)

cos (- π) + i sin (- π)

- 1

Solution steps

cos (- π) + i sin (- π)

Use the following property:

sin (- x) = - sin (x)

sin (- π) = - sin (π)

= cos (- π) + i (- sin (π))

Use the following property:

cos (- x) = cos (x)

cos (- π) = cos (π)

= cos (π) + i (- sin (π))

Simplify

= cos (π) - i sin (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)

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

= - 1 - i 0

Simplify

- 1 - i 0 : - 1

- 1 - i 0

Apply rule

0 \cdot a = 0

= - 1 - 0

Simplify

= - 1

= - 1

- \frac{π}{2} + sin (- \frac{π}{2})

csc (7 π)

cos (sin (0))

6^{2} + 8^{2} + 2 \cdot 6 \cdot 8 \cdot cos (9 0^{\circ})

tan (\frac{2 π}{3} + \frac{π}{4})