Answer:
a. cosθ = ¹/₂[e^jθ + e^(-jθ)] b. sinθ = ¹/₂[e^jθ - e^(-jθ)]
Step-by-step explanation:
a.We know that
e^jθ = cosθ + jsinθ and
e^(-jθ) = cosθ - jsinθ
Adding both equations, we have
e^jθ = cosθ + jsinθ
+
e^(-jθ) = cosθ - jsinθ
e^jθ + e^(-jθ) = cosθ + cosθ + jsinθ - jsinθ
Simplifying, we have
e^jθ + e^(-jθ) = 2cosθ
dividing through by 2 we have
cosθ = ¹/₂[e^jθ + e^(-jθ)]
b. We know that
e^jθ = cosθ + jsinθ and
e^(-jθ) = cosθ - jsinθ
Subtracting both equations, we have
e^jθ = cosθ + jsinθ
-
e^(-jθ) = cosθ - jsinθ
e^jθ + e^(-jθ) = cosθ - cosθ + jsinθ - (-jsinθ)
Simplifying, we have
e^jθ - e^(-jθ) = 2jsinθ
dividing through by 2 we have
sinθ = ¹/₂[e^jθ - e^(-jθ)]
Answer:
6.4
Step-by-step explanation:
By the Pythagorean Theorem:
Hope this helps!
Answer:
-8.95
Step-by-step explanation:
(-6) + (-4.25) = -10.25
-10.25 + 1.3 = -8.95
Answer:
$114,000
Step-by-step explanation:
Use formula
where
I = interst
P = principal
r = rate
t = time
First, find the interst for 25 years:
I = unknown
P = $190,000
r = 0.03 (3% as decimal)
t = 25
Now find the interest for 5 years:
I = unknown
P = $190,000
r = 0.03 (3% as decimal)
t = 5
The unpaid balance is