Find the standard form of the complex number shown below 5(cos7pi/6 + isin7pi/6)
To find the standard form of the complex number, we can use Euler's formula, which states that e^(iθ) = cos(θ) + isin(θ).
In this case, the given complex number is 5(cos(7π/6) + isin(7π/6)).
Let's convert the expression into Euler's form:
cos(7π/6) + isin(7π/6) = e^(i7π/6)
Now, substitute this back into the original equation:
5*cos(7π/6) + 5i*sin(7π/6) = 5*e^(i7π/6)
We can simplify this by using the fact that e^(iπ) = -1:
5*e^(i7π/6) = 5*(-1)^(7/6)
Next, simplify the exponent:
(-1)^(7/6) = (-1)^(1/6) * (-1)^(6/6)
= (-1)^(1/6) * (-1)^1
= (-1)^(1/6) * -1
Now, let's find the value of (-1)^(1/6):
Converting to polar form, -1 = e^(iπ)
Therefore, (-1)^(1/6) = (e^(iπ))^(1/6) = e^(iπ/6)
Finally, substitute this value back into the equation:
5*(-1)^(7/6) = 5*(e^(iπ/6)) * -1
= -5e^(iπ/6)
So, the standard form of the given complex number is:
-5e^(iπ/6)