Tagged Questions

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Is it true that $|\sin^2z+\cos^2z|=1, \forall z \in\Bbb C$?

We know that equation $\sin^2z+ \cos^2z=1$ which holds $\forall z \in\Bbb R$, actually holds $\forall z \in\Bbb C$. Is it true that $|\sin^2z+\cos^2z|=1, \forall z \in\Bbb C$? Thanks in ...
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How to Find End Point, after rotation

I am having an 3D object, length of the object is 27.5 meter, rotation value is -30 degree and the rotate origin point will be one end. After rotating the object i want to find the coordinate of ...
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An inequality involving arctan of complex argument

I have the following conjecture: $$\text{Re}\left[(1+\text{i}y)\arctan\left(\frac{t}{1+\text{i}y}\right)\right] \ge \arctan(t), \qquad \forall y,t\ge0.$$ Which seems to be ...
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Integrating $\int_0^{\pi/2} \cos^a(x) \cos(bx) \ dx$

Please help me in this integral : $$\int_0^{\pi/2} \cos^a(x) \cos(bx) \ dx \quad \text{if}\; b>a>-1$$ Please help me I used everything and can't evaluate it.
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Someone told me that there is a method for integrating rational functions $R(\cos{\theta}, \sin { \theta})$ by doing contour integration of the complex function $$\frac {R \left( \frac {z + \frac1z} ... 5answers 1k views Prove that \sum\limits_{k=0}^{n-1}\dfrac{1}{\cos^2\frac{\pi k}{n}}=n^2 for odd n In old popular science magazine for school students I've seen problem Prove that \quad  \dfrac{1}{\cos^2 20^\circ} + \dfrac{1}{\cos^2 40^\circ} + \dfrac{1}{\cos^2 60^\circ} + ... 3answers 1k views Express \cos 6\theta  in terms of \cos \theta I think I'm supposed to use the chebyshev polynomials, as in$$ \cos n \theta = T_n(x) = \cos(n \arccos x)$$But no idea what now? 2answers 210 views Interpolation using trigonometric polynomials of bounded modulus Consider a grid of points T=\{t_1,\ldots,t_m\} with 0\le t_i\le 1. I would like to derive conditions on t_1,\ldots,t_m (interpolation points) under which for any sequence of complex numbers ... 1answer 147 views How to compute \int_0^\infty \frac{\sin t}{t^{s+1}} dt ? How to compute \displaystyle\int_0^\infty \frac{\sin t}{t^{s+1}}\;\text dt ? Here, the real part of the complex number s is negative and greater than -1. 1answer 88 views Why is e^{g(x)} = \pi where g(x) is holomorphic in Weierstrass factorization of sine function? Why is e^{g(x)} = \pi where g(x) is holomorphic in Weierstrass factorization of sine function? I just can't get why it's true. 3answers 166 views Incoherence using Euler's formula Using the relation \ e^{ix} = \cos(x) + i\sin(x) and substituting for \ x = \pi, we have the well-known Euler identity,  e^{i\pi} = -1. Substitute also for  x = -\pi , we have  e^{-i\pi} = ... 0answers 70 views Bernoulli generating function and cotangent May I ask for a little help in solving a problem about Bernoulli number generating function? Bernoulli number generating function is given by:$$f(z):=\begin{cases} \frac{z}{e^{z}-1} & z \in ...
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Is it true that: $$\coth^{-1}(z) = \tanh^{-1}\left(\frac{1}{z}\right), z\in \mathbb{C}$$ I used this identity: $$\coth{z} = \dfrac{-1}{\tanh{z}}$$ To obtain such a result.
how to prove $\displaystyle \frac{\sin (2n+1)\theta}{\sin \theta} = …$
How to prove $$\displaystyle \frac{\sin (2n+1)\theta}{\sin \theta} = (2n+1) \prod_{k=1}^{n}\left(1 - \frac{\sin^2 \theta}{\sin^2 \left( \frac{k\pi }{2n+1} \right ) } \right )$$ So far, I manage to ...