# Tagged Questions

14 views

65 views

### How do you find two functions $f$ and $g$ such that $f(x) \cdot g(x)=f(x)-g(x)$?

This was inspired by this question ( Logarithms with trigonometric inequality ) I already know the answer ( $f(x)=\tan^2 x$ and $g(x)=\sin^2 x$). However I am interested in how to find this answer. ...
18 views

### Finding Inverse of Function With Two Instances of X

I need to find $f^{-1}(2)$ where $f(x) = 2 + x^2 + tan(πx/2)$ I know can substitute $f(x)$ with $y$ and swap $x$ and $y$: $$x = 2 + y^2 + tan(πy/2)$$ But I'm having trouble eliminating the tangent: ...
153 views

### Is there an everywhere-defined function that satisfies $f(x+y)=\frac{f(x)+f(y)}{1-f(x)f(y)}$

Is there a function $f:\mathbb{R}\to\mathbb{R}$ which is differentiable and satisfies the following: (1) $f(x+y)=\frac{f(x)+f(y)}{1-f(x)f(y)}$ (2) $f'(0)=1$ (1) is the functional equation for ...
633 views

### $\cos x\,$ is the only function satisfying $\,f(x)\,f(y)-f(x+y)=\sin x\,\sin y.$

Find all continuous functions $f$ which satisfy the functional equation $$f(x)\,f(y)-f(x+y)=\sin x\,\sin y,$$ for all $x,y\in\mathbb R$. I can prove that $f(n\pi)=\cos\left(n\pi\right)$ for all ...
99 views

### Formula for solving for Cx and Cy…

I'm trying to create a formula to find the third point in a triangle based on two known points and three known sides. Known Sides: $AB, BC, AC$ Known Points: $A(x, y), B(x, y)$ Unknown Points: ...
143 views

### A Functional Differential Equation

I was having a play with some trig. identities and noticed the following: $\cos{x}=\frac{\sin{2x}}{2\sin{x}}$ Now, $\cos{x} = \frac{d}{dx}\sin{x}$ so I made the following analogous differential ...
600 views

### fabric design using trigonometric functions

is there any trigonometric function or any others that involve trigonometric function, that draw cool fabric shapes or patterns? I have seen some pictures like but with trigonometric functions... ...
936 views

### About the addition formula $f(x+y) = f(x)g(y)+f(y)g(x)$

Consider the functional equation $$f(x+y) = f(x)g(y)+f(y)g(x)$$ valid for all complex $x,y$. The only solutions I know for this equation are $f(x)=0$, $f(x)=Cx$, $f(x)=C\sin(x)$ and $f(x)=C\sinh(x)$. ...
190 views

### How can prove this equation.

if $a+b=c+d=e+f=\dfrac{\pi}{3}$, $\dfrac{\sin{a}}{\sin{b}}\cdot\dfrac{\sin{c}}{\sin{d}}\cdot\dfrac{\sin{e}}{\sin{f}}=1$, Prove that: ...
172 views

### functional equations for trigonometric functions

It is well known that the following system of functional equations: $\begin{cases} f(x+y)=f(x)f(y)-g(x)g(y) \\ g(x+y)=f(x)g(y)+g(x)f(y) \end{cases}$ admit the solution $(f,g)=(\cos,\sin)$. Are there ...
I've recently learned that the cotangent satisfies the following functional equation: $$\dfrac1{f(z)}=f(z)-2f(2z)$$ (true for $f(z)\neq 0$). Can we solve this equation for real or complex ...