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• 8,932

### Showing $\tan x = \frac{\sin\alpha\sin y}{1 - \sin\alpha\cos y}$, given $\sin x = \sin \alpha\sin (x + y)$

Divide $$\sin x = \sin \alpha\sin x\cos y + \sin\alpha\cos x\sin y$$ by $\cos x$ to get $$\tan x = \sin \alpha\tan x\cos y + \sin\alpha\sin y$$ and solve for $\tan x$
• 17.7k

### The triangle $ABC$ is right-angled in $A$. Prove that the inequality $(1-\sin B)(1-\sin C)\leq \frac{{(\sqrt{2}-1)}^2}{2}$ holds.

Hint : It is a right angle triangle so, $B + C = 90^{\circ}$. So, this can be re-written as $$(1-\sin B)(1-\cos B)$$ I hope you will be proceed from to get maxima at $B=45^{\circ}$ and then the final ...

### The triangle $ABC$ is right-angled in $A$. Prove that the inequality $(1-\sin B)(1-\sin C)\leq \frac{{(\sqrt{2}-1)}^2}{2}$ holds.

As it was pointed out in other answers and comments, this amounts to determining the maximum value of $f(x)=(1-\sin x)(1-\cos x)$ for $x \in [0, \frac{\pi}{2}]$. Since $f$ is differentiable, the ...
• 20.9k
Accepted

• 11
1 vote

### Proof that $\frac{d^a}{dx^a}\sin(x) = \sin(x+\pi a/4)$ iff these two infinite series are equivalent?

Your operator is linear and homogeneous, i.e. : $$\frac{d^a}{dx^a}f(\lambda x)=\lambda^a \frac{d^af}{dx^a}(\lambda x)$$ Therefore, your claim is equivalent to proving that $\exp$ is fixed by your ...
• 2,540
1 vote
Accepted

### Explanation to how the length between 2 centers is adjusted as angle changes

TL DR Yes, for any given angle you will need to adjust the x axys. Intuition Start with the two centers aligned on the x axys. If you move the center up the new distance between the two centers will ...
• 1,824
1 vote
Accepted

• 21k
1 vote

### Is it possible to adjust the sine function to pass through specific points?

What about piecewise-defined functions? If those work for you, you can use the periodic version of interpolation by cubic splines. That would most likely satisfy the condition you want on the second ...
• 3,982

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