5
votes
4answers
497 views

How to transform the result of this integral into powers of $\sin\,x$?

I want to calculate the integral $I = \displaystyle\int{\cos ^5 x\, \mathrm dx}$. Using the variable $t = \sin x$, I get $$I = \sin x - \frac{2}{3}\sin^3 x + \frac{1}{5}\sin^5 x + C.$$ With ...
0
votes
2answers
160 views

Sum of roots of a trigonometric equation

The equation $$ \sin x + \sqrt{3} \cos x = 1,$$ where $x \in [-2\pi, 2\pi]$ has four roots $-\dfrac{3\pi}{2}$, $\dfrac{\pi}{2}$, $\dfrac{\pi}{2}$ and $\dfrac{11\pi}{6}$. Sum of roots of the given ...
17
votes
4answers
512 views

How do I introduce a new variable in a trigonometric equation?

I have the trigonometric equation \begin{equation*} \sin^8 x + 2\cos^8 x -\dfrac{1}{2}\cos^2 2x + 4\sin^2 x= 0. \end{equation*} By putting $t = \cos 2x$, I have \begin{equation*} \dfrac{3}{16} t^4+ ...
13
votes
2answers
583 views

Mathematica Sec and Csc

How can I prevent Mathematica from using the "old fashioned" functions "Sec" and "Csc"? In Germany these functions are "old fashioned" as they are not taught anymore at school.
4
votes
2answers
1k views

ArcTan expressed as a radian fraction?

This type of answer is what I'm looking for: In[58]:= ArcTan @ 1 Out[58]= π/4 This is what mathematica gives me: ...