Articles of trigonometry

What is the easiest/most efficient way to find the taylor series expansion of $e^{1-cos(x)}$ up to and including degrees of four?

So I have $$e^{1-cos(x)}$$ and want to find the taylor series expansions up to and including the fourth degree in the form of $$c_{0} \frac{x^0}{0!} + c_{1} \frac{x^1}{1!} + c_{2} \frac{x^2}{2!} + c_3 \frac{x^3}{3!} + c_{4} \frac{x^4}{4!} + HOT$$ or just $$1 + c_{1} x^1 + c_{2} x^2 + c_3 x^3 + c_{4} x^4 +HOT$$ […]

How do I solve $\int \sec^3 \theta d\theta$

Possible Duplicate: Indefinite integral of secant cubed I guess I need to learn a new technique because those I know didn’t help me here. Wolframalpha uses the reduction formula, which I haven’t been introduced to yet. So is there a way to solve this without using the reduction formula?

What is the Arctangent of Tangent?

This is probably just me misunderstanding trig properties. I remember from my trig days that $\tan(\arctan(x)) = x$ But I can’t remember if that holds the other way. Can someone help me out? Is this valid: $$\arctan(\tan(\theta)) = \theta$$

If $p =\frac{4\sin\theta \cos\theta}{\sin\theta +\cos\theta}$ Find the value of $\frac{p+2\sin\theta}{p-2\sin\theta}$

Problem : If $\displaystyle p =\frac{4\sin\theta\cos\theta}{\sin\theta +\cos\theta}$, find the value of $\displaystyle \frac{p+2\sin\theta}{p-2\sin\theta} + \frac{p+2\cos\theta}{p-2\cos\theta}$. Please help how to proceed in such problem..Thanks..

Fitting a sine function to data

I have a sequence of $n$ points $(x_i,y_i)$, for $i=1,\dots,n$. I would like to find the function, of the form $y=V\sin(x+\phi)$, which best fits the points. Which numerical method could I use? I have a slow system, with little memory, so I am searching for a fast and efficent method, even if not very accurate. […]

The value of $\cos^4\frac{\pi}{8} + \cos^4\frac{3\pi}{8}+\cos^4\frac{5\pi}{8}+\cos^4\frac{7\pi}{8}$

Problem : The value of $\cos^4\frac{\pi}{8} + \cos^4\frac{3\pi}{8}+\cos^4\frac{5\pi}{8}+\cos^4\frac{7\pi}{8}$ If this could have been like this $\cos\frac{\pi}{8} + \cos\frac{3\pi}{8}+\cos\frac{5\pi}{8}+\cos\frac{7\pi}{8}$ then we can take the terms like $(\cos\frac{3\pi}{8}+\cos\frac{5\pi}{8} ) + (\cos\frac{\pi}{8} +\cos\frac{7\pi}{8})$ and solve further , but in this case due to power of 4 I am unable to proceed please suggest… thanks..

Prove $ \frac{a\sin A+b\sin B+c\sin C}{a\cos A+b\cos B+c\cos C}=R\left(\frac{a^2+b^2+c^2}{abc}\right) $

$ \frac{a\sin A+b\sin B+c\sin C}{a\cos A+b\cos B+c\cos C}=R\left(\frac{a^2+b^2+c^2}{abc}\right) $ This is what I have so far: I know that $A + B + C = 180^\circ$, so $C = 180^\circ – (A+B)$. Plugging this in, I get that $\sin C = \sin(A+B)$ and $\cos C = -\cos(A+B)$. When I plug this back into the equation, […]

If $m \tan(\theta – \pi/6) = n \tan(\theta + 2\pi/3)$ then find $\cos 2\theta$

If $$m \tan(\theta – \pi/6) = n \tan(\theta + 2\pi/3)$$ then find $\cos 2\theta$ in terms of $m$ and $n$. What is the correct method to solve this question?

Prove $\frac{\sec{A}+\csc{A}}{\tan{A} + \cot{A}} = \sin{A} + \cos{A}$ and $\cot{A} + \frac{\sin{A}}{1 + \cos{A}} = \csc{A}$

Can anyone help me solve the following trig equations. $$\frac{\sec{A}+\csc{A}}{\tan{A} + \cot{A}} = \sin{A} + \cos{A}$$ My work thus far $$\frac{\frac{1}{\cos{A}}+\frac{1}{\sin{A}}}{\frac{\sin{A}}{\cos{A}}+\frac{\cos{A}}{\sin{A}}}$$ $$\frac{\frac{\sin{A} + \cos{A}}{\sin{A} * \cos{A}}}{\frac{\sin{A}}{\cos{A}}+\frac{\cos{A}}{\sin{A}}}$$ But how would I continue? My second question is $$\cot{A} + \frac{\sin{A}}{1 + \cos{A}} = \csc{A}$$ My work is $$\frac{\cos{A}}{\sin{A}} + \frac{\sin{A}}{1 + \cos{A}} = \csc{A}$$ I think I […]

Solving $\sin(5x) = \sin(x)$

If I have an equation: $$\sin(5x) = \sin(x)$$ In what case can I equate $$5x = x$$ Is it only when there is a multiply of $2\pi n$ on either side, where n is any integer so $$ 5x = x+2\pi n$$ Also with this method can I get every possible solution or does that […]