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# Phasor analysis confusion in the trigonometric identities

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#### hioyo

##### Advanced Member level 4
I was learning about phasor analysis from Fundamentals of electric circuits by Alexander and Sadiku.

In chapter 9 the below trigonometric identity is given.

As per my knowledge sin is positive in the 2nd quadrant and negative in all other quadrants(ALL SILVER TEA CUPS). In the first equation Sin(wt±180) means it is lying in the 2nd quadrant and 3rd quadrant depending phase shift is +180 or -180.

When it is lying in the 2nd quadrant**(Sin(wt-180))**it should be positive and when it is lying in the 3rd quadrant **(Sin(wt+180))**should be negative.This is my understanding.

But in the book it is given as Sin(wt±180)=-sinwt .May I know where I went wrong

I don't get the meaning of "quadrant" in this question. Quadrants are used for AC problems in two regards:
1. I/V diagrams
2. complex numbers, e.g. representation of complex power
The above quoted trigonometric identies are addressing neither of this.

In other words, I don't see which conflict your are talking about. The identities are unconditionally valid.

Hi,

don´t know about phasor analysis...

But generally:
* sine should be positive from 0° to 180°, this usually is quadrant I and II (1 and 2).
* cosine is positive from 0° to 90° and 270° to 360° this is quadrant I and IV (1 and 4)

Klaus

I was learning about phasor analysis from Fundamentals of electric circuits by Alexander and Sadiku.

In chapter 9 the below trigonometric identity is given.

View attachment 175423

As per my knowledge sin is positive in the 2nd quadrant and negative in all other quadrants(ALL SILVER TEA CUPS). In the first equation Sin(wt±180) means it is lying in the 2nd quadrant and 3rd quadrant depending phase shift is +180 or -180.

When it is lying in the 2nd quadrant**(Sin(wt-180))**it should be positive and when it is lying in the 3rd quadrant **(Sin(wt+180))**should be negative.This is my understanding.

But in the book it is given as Sin(wt±180)=-sinwt .May I know where I went wrong

if you look at sine(green) then if you start after +180 degrees you get it inverted.
if you look at cos(blue) and start from 180 degrees then you it inverted
You can visualise all other cases on the same way.

I don't get the meaning of "quadrant" in this question. Quadrants are used for AC problems in two regards:
1. I/V diagrams
2. complex numbers, e.g. representation of complex power
The above quoted trigonometric identies are addressing neither of this.

In other words, I don't see which conflict your are talking about. The identities are unconditionally valid.
Please check this link. This explains the quadrant in trigonometry.

My confusion is when the angle is sin(wt-180) ,this lies in 2nd quadrant and sine is positive in 2nd quadrant.

and when it is sin(wt+180) is located in 3rd quadrant and sin is negative in this quadrant.

But in the book it is given as Sin(wt±180)=-sinwt that is it is always negative ,whether it is located in 2nd(In 2nd quadrant sin is positive) quadrant or 3rd quadrant(In 3rd quadrant sin is negative),

Hi,

sin(30°) = 0.5 (1Q)
sin(30° + 180°) = -0.5 so it is -sin(30°) .. (3Q)
sin(30° - 180°) = -0.5 so it is -sin(30°) .. (3Q)

you can do this with any other angle than 30°.

Klaus

My confusion is when the angle is sin(wt-180) ,this lies in 2nd quadrant and sine is positive in 2nd quadrant.
No. wt can have any value (=lie in any quadrant), so does wt-180.

But in the book it is given as Sin(wt±180)=-sinwt that is it is always negative ,whether it is located in 2nd(In 2nd quadrant sin is positive) quadrant or 3rd quadrant(In 3rd quadrant sin is negative),

if sin(wt) is positive, then sin(wt+/-180) is negative, or -sin(wt)
if sin(wt) is negative, then sin(wt+/-180) is positive, or -sin(wt)

it is not "always negative" it is always -1 times whatever sin(wt) was.

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