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x = (π/2 ± ln(2+√3)i)+2πn, where n is in the set of integers.
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Verify that Cos theta cot theta plus sin theta equals csc theta?
It's easiest to show all of the work (explanations/identities), and x represents theta. cosxcotx + sinx = cscx cosx times cosx/sinx + sinx = csc x (Quotient Identity) cosx2 /sinx + sinx = csc x (multiplied) 1-sinx2/sinx + sinx = csc x (Pythagorean Identity) 1/sinx - sinx2/sinx + sinx = csc x (seperate fraction) 1/sinx -sinx + sinx = csc x (canceled) 1/sinx = csc x (cancelled) csc x =csc x (Reciprocal Identity)
What is the integral of sinx2?
If you mean the the integral of sin(x2)dx, It can only be represented as an infinite series or a unique set of calculus functions known as the Fresnel Integrals (Pronounced Frenel). These functions, S(x) and C(x) are the integrals of sin(x2) ans cos(x2) respectively. These two integrals have some interesting properties. To find out more, go to: http://en.wikipedia.org/wiki/Fresnel_integral I hope this answers your question.
What is the integral of sin x squared?
The integral of sin x2 is one of the Fresnel Integrals. It does not have a closed form solution. However, you can calculate a series solution by integrating the Taylor series, as follows: The Taylor series expansion about x = 0 for sin x is sin x = x - (x3/3!) + (x5/5!) - (x7/7!) +/- ... Substitution of x2 for x yields sinx2 = x2 - (x6/3!) + (x10/5!) - (x14/7!) +/- ... Term-wise integration, using the power rule gives {integral}sinx2 = (x3/3) - (x7/7*3!) + (x11/11*5!) - (x15/15*7!) +/- ... This is the answer. It is the Fresnel Integral S(x). There is a similar one for the integral of cos x2, called C(x). It can be written in more compact form: S(x) = (Sum from n = 1 to infinity) of (-1)n x4n+3/(4n+3)*(2n+1)! It looks better in Sigma notation, with fractions, but if you work out the first 4 terms, you will see agreement with the result for integrating the series expansion. Here is a link to Fresnel Integral on Wikipedia: http://en.wikipedia.org/wiki/Fresnel_integral Thank you for posing this question.
What happens to the speed of a light wave when it travels from air to liquid?
The speed of light decreases when it travels from air to a denser medium such as a liquid. This change in speed is due to the difference in the refractive indices of the two mediums, which affects how light waves propagate.
How do you use snell's law to find the speed of light in jello?
Snell's law combines trigonometry and refractive indices to determine different aspects of refraction. The law is as follows: (n1)(sinX1) = (n2)(sinX2); where n1 is the refractive index of the first medium, X1 is the angle of incidence (the angle between the incident ray and the normal), n2 is the refractive index of the second medium, and X2 is the angle of refraction (the angle between the refracted ray and the normal). Setting up an experiment using jello and a laser, one can determine the index of refraction in the jello. Shine the laser at an arbitrary angle and record this angle. Then measure the refractive angle seen in the jello (this is the angle between the ray in the jello and the normal). The index of refraction for air is 1.0003. Now substitute all three values into Snell's law and solve for n2, the refractive index of jello. An index of refraction is defined as the speed of light in a vacuum divided by the speed of light in a medium. Once n2 is determine, use the following equation: n2 = c / v. Substitute n2 and the speed of light in a vacuum (which is approximately 299,792,458 meters per second), and solve for v. The value obtained will be the speed of light in jello.
