What is the Origin of the Trigonometric Identity?

In summary, the identity \frac{1-\cos(\beta)}{\sin(\beta)}=\frac{\sin(\frac{\beta}{2})}{\cos(\frac{\beta}{2})} is derived from the half-angle identities for sine and cosine, as well as the tangent half-angle formula. It can also be obtained by substituting x = 2θ into \frac{1 - \cos x}{\sin x} = \tan \frac{x}{2}.
  • #1
Piano man
75
0
I'm reading through a solution to a problem and at one point the following identity is used:

[tex]\frac{1-\cos(\beta)}{\sin(\beta)}=\frac{\sin(\frac{\beta}{2})}{\cos(\frac{\beta}{2})}[/tex]

I've been trying to figure out where this comes from but with haven't got it yet.
Any ideas?
 
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  • #2
Just substitute in the well known formulas

1 = cos^2 b/2 + sin^2 b/2
cos b = cos^2 b/2 - sin^2 b/2
sin b = 2 cos b/2 sin b/2
 
  • #3
Sin (a + a) = sin a cos a+ cos a sin a
=> Sin 2a = 2Sin a Cos a
=> SIN B = 2 Sin (B/2) Cos (B/2) [2a = B] . . . . (i)

Cos (a + a) = cos a cos a - sin a sin b
=> cos 2a = cos^2 (a) - sin^2 (a)
=> cos 2a = (1 - sin^2 (a) ) - sin^2 a
=> cos 2a = 1 - 2sin^2 (a)
=> 1 - cos 2a = 2s n^2 (a)
=> 1 - cos B = 2sin^2 (B/2) . . . (ii)

n0w (ii) / (i) you should do it n0w urself . . . Am en0ugh tired already
 
  • #4
You could start with the half-angle identities for sin and cos:
49a19a42dd45749f43f81d5028e5a276.png

3fea203908f8dabb66acad4ad3a3c37c.png


And since tanx = sinx/cosx, dividing them gives you [tex]\pm\sqrt{\frac{1 - cos\theta}{1 + \cos\theta}}[/tex]
which can be simplified to [tex]\frac{\sin\theta}{1 + \cos\theta} = \frac{1 - \cos\theta}{\sin\theta}[/tex]
(the ± goes away because each of the last two expressions give you the correct sign of tanθ/2)

You could also start with
5202340c6725f876ce8e6d659843bd13.png

and substitute x = 2θ → θ = x/2, then solve for tanx/2 which will end up giving you the same expressions above.
 

FAQ: What is the Origin of the Trigonometric Identity?

What is a trigonometric identity?

A trigonometric identity is an equation that is true for all values of the variables involved. These identities are used to simplify and solve trigonometric equations and expressions.

What are the basic trigonometric identities?

The three most basic trigonometric identities are the Pythagorean identities, which include sin^2x + cos^2x = 1, tan^2x + 1 = sec^2x, and cot^2x + 1 = csc^2x. These identities relate the three main trigonometric functions: sine, cosine, and tangent.

How do you prove a trigonometric identity?

To prove a trigonometric identity, you must manipulate one side of the equation using algebraic rules and trigonometric identities until it is equivalent to the other side. This process may involve using basic identities, factoring, or using reciprocal or quotient identities.

What are the sum and difference identities?

The sum and difference identities are used to express the sine, cosine, and tangent of the sum or difference of two angles in terms of the sine, cosine, and tangent of the individual angles. These identities include sin(a+b) = sin a cos b + sin b cos a, cos(a+b) = cos a cos b - sin a sin b, and tan(a+b) = (tan a + tan b) / (1 - tan a tan b).

How are trigonometric identities used in real life?

Trigonometric identities have various applications in fields such as engineering, physics, and astronomy. They are used to solve problems involving angles and distances, and to analyze and design structures such as bridges and buildings. They are also used in navigation, GPS systems, and satellite communication.

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