Trigonometric functions like sin(2x)=2sin(x)cos(x)

In summary, the conversation discusses known relations in trigonometric functions and the question of whether there are similar formulas for arcsin and arccos. The conversation also touches on the topic of inverse trigonometric functions and their relationships.
  • #1
TheDestroyer
402
1
Hi guyz, as we know we have some known relations in the trigonometric functions like

sin(2x)=2sin(x)cos(x) and sin(x/2)=1/2-1/2 cos2x

My question is are there similar formulas for arcsin and arccos?

I know those only !

arcsin x =ln(ix-sqrt(1-x^2))
arccos x =ln(-ix-sqrt(1-x^2))

I'm working in electrodynamics, and i reached an integral for the kinetic energy and the answer should be mc^2-mc^2, but I'm getting arcsin !

i knew the answer already about the energy, but i don't this i should neglect the problem of the damn inverse function,

Can anyone help and tell me some formulas?

Thanks...
 
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  • #2
You are not getting "arcsin"; you might be getting the arcsine to some argument. What argument?
 
  • #3
"sin(x/2)=1/2-1/2 cos2x"

That's not correct. Perhaps the identity you were thinking of was
sin^2(x)=1/2-1/2 cos2x.I'm not really sure what your question is but you haven't fogotten about the trig of arc_trig relationships have you? I mean like cos(arctan(x)) = 1/sqrt(1+x^2) etc.
 
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  • #4
i suppose you could take a formula like sin(2x) = 2sin(x)cos(x), set u = sinx, cosx = sqrt(1-u^2), x = arcsin(u), and apply arcsin to the previous formula,

to get say 2arcsin(u) = arcsin(2u[sqrt(1-u^2)]).

seems pointless though. i.e. any formula for sin, cos, IS a formula for arcsin, arccos.
 

1. What are trigonometric functions?

Trigonometric functions are mathematical functions that relate the angles of a right triangle to the lengths of its sides. The most commonly used trigonometric functions are sine, cosine, and tangent.

2. What is the identity sin(2x)=2sin(x)cos(x)?

This identity is known as the double angle formula for sine. It states that the sine of twice an angle (2x) is equal to twice the sine of the angle (x) multiplied by the cosine of the angle (x).

3. How is this identity derived?

This identity can be derived using the sum and difference formula for sine and the trigonometric identity cos(2x)=cos^2(x)-sin^2(x). By substituting cos(2x) in the identity with its equivalent form, and using the Pythagorean identity sin^2(x)+cos^2(x)=1, the double angle formula for sine can be derived.

4. Why is this identity useful?

This identity is useful in simplifying trigonometric expressions and solving trigonometric equations. It also allows us to express a trigonometric function in terms of a single angle, which can make calculations and graphing easier.

5. Are there other double angle formulas for trigonometric functions?

Yes, there are double angle formulas for cosine and tangent as well. They are cos(2x)=cos^2(x)-sin^2(x) and tan(2x)=2tan(x)/1-tan^2(x), respectively.

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