Identity (1b) states that a cos wave is just a sin wave shifted to the left by a phase angle of π/2 or 90°.
sin(−α) = −sin(α)
cos(−α) = cos(α)
2c. tan(−α) = −tan(α)
sin(α ± β) = sin(α)·cos(β) ± cos(α)·sin(β)
A common mistake is to believe that, for example:
sin(α + β) = sin(α)+ sin(β)This is not correct! It would only be correct if the sin curve was a straight line through the origin, instead of a wavy curve. The previous identities show the correct way to break up the sum of the angles.
sin(α + β) = sin(α)·cos(β) + cos(α)·sin(β)which we just proved, is the most important identity of this group. Any other identity in this group (and in fact almost any trigonometric identity) can be derived by modifying it to look like this one. For example:
sin((α + 90°) + β),then applying identity (3a) and then using (1b) again to clean up.
4. sin2(α) + cos2(α) = 1
sin(α ± β) = sin(α)·cos(β) ± cos(α)·sin(β)If we use the upper signs in these identities and let β equal α we get:
sin(2α) = 2·sin(α)·cos(α)
cos(2α) = cos2(α) − sin2(α)
= 1 − 2 sin2(α)
= 2 cos2(α) − 1
If we now change the name of the angle α to be A/2 then these become the so-called half angle identities:
sin(α)·sin(β) = ½[cos(α − β) − cos(α + β)]
sin(α)·cos(β) = ½[sin(α + β) + sin(α − β)]
7c. cos(α)·cos(β) = ½[cos(α + β) + cos(α − β)]
(b) Use the result to find the sum of the two sinusoidal waveforms
y1 = 3 sin(θ) and y2 = 4 cos(θ).
A = C cos(φ) and B = C sin(φ).
so the sum of the waveforms is
(b) Define the two sinusoidal waves:
Here are their graphs:
Solution part (a):
y1 + y2 = 1.0 cos(t) · sin(20 t).The factor 1.0 cos(t) in this product plays the role of a slowly varying amplitude for the rapidly varying oscillation sin(20 t). (In other words imagine a sinusoidal wave sin(20 t), but that at certain times it has a small amplitude and at other times it has a large amplitude.) The periodic constructive and destructive interference is known as the beat phenomenon in acoustics.
P = i 2 R,where P is the power produced measured in watts, i is the electric current flowing through the resistor in amperes and R is the resistance of the resistor in ohms.
i = 0.7071 amps
i = 1.0 sin(t) amps
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