Regarding the approach to induction motor operation and how to solve the very simple problem being known some basic equations of power balance such as input power equation of IM (P=[(EV/X)sin(Angle E- Angle V)]

A 12 pole, 60 Hz synchronous machine has an internal rms voltage of 10 kV.
Its internal synchronous reactance is 10 ohms. The machine is connected to a network
with a terminal voltage of 11 kV rms. The phase angle difference between the internal
and external voltages is 30 degrees (internal minus external).
(a) Is this machine generating or consuming real power? How much in MW?
(b) What is the synchronous speed of the machine?
(c) What is the torque in Nm produced by the machine?

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Solution:
(a) From the basic electrical model of a synchronous machine, the real ...

I need check about these :
A) Solvethe following, giving all positive values of theangle between 0° and 360° to the nearest minute only.
i) cos2x-sin^2 (x/2)+3/4 = 0
ii) cos4x =sin2x
iii) sin^2 theta = cos^2 theta +3/2
iv) sin(2x-10°)= 1/2
v) sin2x-cosx = 0
b) Write equivalent equations in the form of inverse

Two vectors are given by A = -3i + 7j -4k and
B = 6i -10j + 9k. Evaluate the quantities:
a) cos^-1 [A.B/AB] ( the . between the A.B is a dot product)
b) sin^-1 [ |A x B|/AB]
c) Which gives theangle between the vectors.
I haven't a clue as to what to do with this problem. Do I use the cross products of unit vectors t

William Tell must split the apple atop his son's head from a distance of 27 meters. When he aims directly at the apple, the arrow is horizontal. At what angle must he aim the arrow to hit the apple if the arrow travels at a speed of 35m/sec?

A kite 100ft above the ground moves away horizontally at a speed of 8ft/sec. At what rate is theangle between the string andthe horizontal decreasing when 200ft of string has been let out?

Consider the electric power system shown {see attachment}. Thepower-flow solution of this system can be obtained without resorting to iterative techniques. The elements of the bus admittance matrix Ybus have been calculated as: {see attachment}.
Calculate the phase angle {see attachment} by using the real and reactive power

For the given circuit, calculate thepower in the load as a percentage of maximum power using the given formula, for a firing angle of 45 degrees.
Please refer to the attachment for the circuit andthe formula.

Please see the attached file
Q1. Prove the identity
cos3t = 4cos^3t- 3cost
Q2. Prove the identity:
(cos3a - cos7a) / (sin7a + sin3a) = tan2a
Q3. Solvetheequation, giving answer in radians in the range 0 to 2 pi
sin(2t) -1 = cos(2t)

A spherical shell starts from rest and rolls down an incline. The coefficient of static friction between the shell and incline is (Us=.2). What is the maximum angle of incline such that the shell will roll without slippage?