The number cis 75 + cis 83 + cis91 +...+ cis 147 is expressed in the form r cis(theta), where 0 <= theta < 360.
Find theta in degrees.
I need help.
nvm its 111
one way:
cos 75 + cos 83 + .....cos 147 = a
sin 75 + sin 83 + .... sin 147 = b
tan theta = b/a
r = sqrt (a^2+b^2)
I don't understand?
calculate a above.
calculate b above.
theta= arctan (b/a)
How do I calculate a and b?
To find the value of theta in degrees, we will first simplify the expression and then convert it to the trigonometric form.
The given expression is:
cis 75 + cis 83 + cis 91 +...+ cis 147
To simplify this expression, we notice that the given terms follow a pattern. The argument of the terms increases by 8 in each successive term. Hence, we can rewrite the expression as a series:
cis 75 + cis 75*1.1 + cis 75*1.2 +...+ cis 75*1.6
Now, let's use Euler's formula to express each term in the form of cos and sin:
cis x = cos(x) + i*sin(x)
Applying Euler's formula to each term, we get:
cos(75) + i*sin(75) + cos(83) + i*sin(83) + cos(91) + i*sin(91) +...+ cos(147) + i*sin(147)
Using the property of complex numbers, we can separate the real part (cosine) and the imaginary part (sine):
(cos(75) + cos(83) + cos(91) +...+ cos(147)) + i(sin(75) + sin(83) + sin(91) +...+ sin(147))
Now, we can apply the formula for the sum of an arithmetic series, which is of the form: sum = (n/2)(first term + last term).
In our case, the first term is cos(75) and the last term is cos(147), so we have:
sum of cosines = (n/2)(cos(75) + cos(147))
The same applies to the sum of sines, where:
sum of sines = (n/2)(sin(75) + sin(147))
Since the argument values (75, 83, 91,..., 147) are an arithmetic sequence with a common difference of 8 and the first term is 75, we can determine the number of terms using the formula:
number of terms = (last term - first term) / common difference + 1
Using the values:
first term = 75
last term = 147
common difference = 8
We can calculate the number of terms:
number of terms = (147 - 75) / 8 + 1 = 10
Plugging the values into the formulas for sum of cosines and sum of sines, we get:
sum of cosines = (10/2)(cos(75) + cos(147))
sum of sines = (10/2)(sin(75) + sin(147))
Now, let's calculate the values of cos(75) and cos(147) using a calculator or trigonometric table. Similarly, calculate the values of sin(75) and sin(147).
Once we have these values, we can calculate the sum of cosines and the sum of sines. Finally, we can express the result in the form r cis(theta), where r is the magnitude and theta is the argument.
To do this, we calculate:
magnitude, r = sqrt((sum of cosines)^2 + (sum of sines)^2)
argument, theta = arctan(sum of sines / sum of cosines)
The value of theta will be between 0 and 360 degrees because theta represents the argument.