Articles of trigonometry

Evaluate $\mathop {\lim }\limits_{x \to 0} \left( {{1 \over {{{\sin }^2}x}} – {1 \over {{x^2}}}} \right)$

I tried l’Hospital but that will require a lot (and I mean A LOT!!!) of differentiating Is there a shortcut? $$\mathop {\lim }\limits_{x \to 0} \left( {{1 \over {{{\sin }^2}x}} – {1 \over {{x^2}}}} \right)$$ Thanks in advance

Prove this proprety of $f(x)$

I’ve asked this question before a long time ago, but I didn’t get a complete answer. This is the link to the incomplete answer: Prove the following property of $f(x)$? Let $$f(x)=|a_1\sin(x)+a_2\sin(2x)+a_3\sin(3x)+\cdots+a_n\sin(nx)|.$$ Given that $f(x)$ is less than or equal to $|\sin(x)|$ for all $x$, prove that $|a_1+a_2+\cdots+a_n|$ is less than or equal to $1$. […]

Is there a formula for sine and cosine?

I’m an Android programmer and am working on a graphing calculator. I have been looking for a formula for sine and cosine to put in there. I have a decent understanding of mathematics but can not seem to find this formula. Any help would be great, thanks.

Finding the values of $\cos \frac{n\pi}{2}$ and $\sin \frac{n\pi}{2}$.

i know that the values of $\cos n\pi=(-1)^{n}$ and $\sin n\pi=0$. Now i want to know that what is the general expressions of $\cos \frac{n\pi}{2}$ and $\sin \frac{n\pi}{2}$.

How to create an identity for $\sin \frac{x}{4}$

I am trying to understand how to create an identity for $\sin \frac{x}{4}$. There are a number of steps that I have not understood. 1) $\sin\left( \frac{x}{4}\right) = \sin\left( \frac{\frac{x}{2}}{2}\right)$ Question: How are these two equivalent? Thank you for the answer given to this part. 2) I know it has something to do with $\pm […]

Simplify: 0.3cos(4t) + 0.2cos(6t) in terms of cos( ) function.

Please help me with this, I need to compare the equation $$s(t)= 50[1+ 0.3\cos(4t) + 0.2\cos(6t)]\cos10t$$ with the given general equation $$s(t)=A[1+ u \cos(bt)]\cos(ct)$$ . I don’t know but it might be something like simplifying the form xcos(a)+ycos(b) into zcos⁡(c) form . So, if only i knew how to simplify the equation to its general […]

$\lim x \sin (1/x)$, when $x \to 0$

Here’s my solution: $$\lim x\sin (1/x) = \lim\, x \dfrac{\sin (1/x)}{x(1/x)} = \lim\, x/x = 1$$ when $x \to 0$ However on the internet I read that the solution of this equation is 0. How can this be? Where a I making a mistake?

How to find $\lim_{x\to 0} \frac{1-\cos x \sqrt{\cos 2x}}{x^2}$

By plotting $\dfrac{1-\cos x \sqrt{\cos 2x}}{x^2}$, we find that in sufficiently small domain near $x = 0$, $f(x)\to 0$ as $x\to 0$. So the limit seems to be $0$. Now I tried to evaluate it using Wolfram alpha, it gives the limit to be $\frac{3}{2}$. How can this be true. Also how can we find […]

If $\arctan(x)+\arctan(y)+\arctan(z)=\pi/2$ how to show that $xy+yz+zx=1$?

If $\arctan(x)+\arctan(y)+\arctan(z)=\pi/2$ how to show that $xy+yz+zx=1$ ?

point on line to form angle

In planning some video game camera behavior, I hit a math problem I am having some difficulty with. The camera in this case is restricted to moving along a line and chooses a location along the line to get close to important objects while keeping them in frame. So to phrase the question in a […]