Trig Identity: Tan(x) = sin(2x) / [ cos(2x) + 1 ]

How I discovered a trig identity that I was not previously aware of using COMPLEX numbers and the Euler Identity:
exp(ix) = cos(x) + i sin(x)

The trig identity: Tan(x) = sin(2x) / [ cos(2x) + 1 ]

[Theylieohio]

I think it's Awesome that the India students learn from your channel. Yuu Trehan!!!

[georgehnatiuk]

Let " p " = rho and " s " = VSWR
If A = arcTan(s), B = pi/4, then p = Tan(A + B) = Tan [ arcTan(s) - pi/4 ] = [ s - 1 ] / [ s + 1 ]
If A = pi/4, B = arcTan(p), then s = Tan(A - B) = Tan [ pi/4 + arcTan(p) ] = [ 1 + p ] / [1 - p ]

What you wrote in your comment above is this portion of what the Trig functions reduce to algebraically:

p = [ s - 1 ] / [ s + 1 ] " p " is rho and " s " is VSWR
s = [ 1 + p ] / [1 - p ]

The reason I use the TAN function is the number of steps in making the calculations is reduced with fewer keystrokes as I stated in my original post.

The definition of VSWR (s) from rho (p) is

s = [ 1 + p ] / [1 - p ]

Let's do a calculation using an Hp RPN calculator to find VSWR from rho with as few keystrokes as possible. Start with the definition of VSWR from rho which is your calculation:

Let rho, p = .34
DISPLAY
1) .34 (3 keys) .34 <== rho
2) STO 00 (2 keys) 0.34000
3) 1 1
4) + 1.34000
5) 1 1
6) RCL 00 (2 keys) .34000
7) - .65000
8) -/- 2.03030 <== ANS VSWR

TOTAL Keystrokes = 12


Now lets use the TAN and ATAN functions in degree mode:

1) .34 (3 keys) .34 <== rho
2) ATAN (2 keys) 18.77803
3) 45 (2 keys) 45
4) + 63.77803
5) TAN 2.03030 <== ANS VSWR

TOTAL Keystrokes = 9 Using your method, the # keystrokes is increases by 3 or 33%.

I was just trying to show a more efficient computation method which requires fewer steps and keystrokes therefore easier in my opinion.

ALSO, I do not know about you but as often as I make these calculations, I STILL have to pause and think when dividing the binomials. Do I put the subtraction in the numerator or denominator?

[ s - 1 ] / [ s + 1 ] = p " p " is rho and " s " is VSWR
[ 1 + p ] / [1 - p ] = s

Using the Trig function method, this confusion is avoided for the form is the same for either calculation where you add or subtract 45 degrees (pi/4) in the argument -- there is no division. I remember it as " p from s -- SUBTRACT "

p = Tan [ arcTan(s) - pi/4 ]
s = Tan [ arcTan(p) + pi/4 ]

There really is no "Best" method. I always try to develop computation methods that are easy to remember by using mathematical identities and manipulations to get me there.

George