8 votes
Accepted

Simplifying $\frac{1-\cos^2(\alpha)}{\sin(\alpha) \cos(\alpha)}$ until there is only one trigonometric function left

You can use the fact that \begin{align} \frac{1-\cos^2(\alpha)}{\sin(\alpha)\cos(\alpha)}&=\frac{\sin^2(\alpha)}{\sin(\alpha)\cos(\alpha)}\\ &=\frac{\sin(\alpha)}{\cos(\alpha)}\\ &=\tan(\...
  • 1,699
6 votes

Simplifying $\frac{1-\cos^2(\alpha)}{\sin(\alpha) \cos(\alpha)}$ until there is only one trigonometric function left

You must know the Pythagorean trigonometric identity $$ \sin^2(\alpha)+\cos^2(\alpha)=1\implies 1-\cos^2(\alpha)=\sin^2(\alpha). $$ From there, use the definition of $\tan(\alpha)$ that you already ...
  • 738
4 votes
Accepted

Can we construct a sine approximating function $f$ with $f'(n\pi) = 0$ and infinitely differentiable?

Consider the functions $$ h_a(x) = \frac{a^2 x^3}{1+a^2 x^2} \, . $$ For each $a > 0$ is $h_a$ strictly increasing, infinitely differentiable, and has a triple zero at $x=0$. Also $$ |h_a(x) - x| ...
  • 92.8k
4 votes

Simplifying $\frac{1-\cos^2(\alpha)}{\sin(\alpha) \cos(\alpha)}$ until there is only one trigonometric function left

The other answers give the right calculations, but they miss an important point. The function $f(\alpha)=\frac{(1-\cos^2 \alpha)}{(\sin \alpha \cos \alpha)}$ is defined when: $$\sin \alpha \cos \alpha ...
  • 1,787
4 votes
Accepted

Question inside $\int \tan^2 x dx$

This is a phenomenon that happens with all indefinite integrals. The original integral $$\int \tan^2 x\:dx$$ has singularities at $\frac{\pi}{2} + \pi k$ for $k\in\Bbb{Z}$, therefore, no, the original ...
  • 26.8k
4 votes

In $\triangle ABC$, if $AC=4$ and $BC=5$, and $\cos(A-B)=\frac{7}{8}$, find $\cos(C)$

Here's my approach: 1.) First, we draw a line $AD$ from $A$ such that $\angle ABD=\angle DAB=\beta$, this means that $\angle DAC=\alpha-\beta$ ($\angle BAC=\alpha$). This implies that $AD=BD=x$. Now, ...
  • 2,102
3 votes
Accepted

Why does cancelling by $\sin x$ when solving $4\tan x = 5\sin x$ for $0\leq x < 2\pi $ miss solutions?

For the same reason why solving $x = x^2$ by dividing by $x$ would lose solutions. If you divide both sides by something, you implicitly give rise to two cases. For $x=x^2$, if you divide by $x$, then ...
  • 34.1k
3 votes
Accepted

Determine relative position of 3 large (equal) circles and 1 smaller circle within a minimum enclosing circle

For Case 1, using Descartes' Kissing Circles Theorem, we can determine the maximum possible value of $r_2$. From the $r_1$ circles forming an equilateral triangle, it follows that the enclosing ...
3 votes

Proving that, if $\cos2\alpha$ is irrational, then $\sin\alpha$ and $\tan\alpha$ are irrational

I am not sure why the previous answer got downvoted, but contradiction is the easiest way forward. You can also use contraposition, the arguments are pretty much the same. Start by assuming ...
  • 26.3k
2 votes

$A(z_1)$, $B(z_2)$ and $C(z_3)$ be vertices of ABC s/t $|z_1|=|z_2|=|z_3|=1$, $z_1+z_2\cos\alpha+z_3\sin\alpha=0$ then find $\bar z_2z_3+z_2\bar z_3$

$\left|z_2 \cos \alpha + z_3 \sin \alpha \right|^2 = 1 \Rightarrow Re \left(z_2 \overline{z_3}\right) =0 \Rightarrow z_2 \overline{z_3}= \pm i$ Use the determinant form of the area to get max area as $...
  • 3,295
2 votes

Seeking elegant proof of $\sum_{cyc}\cot(x-y)\cot(y-z)=1$

Given any three numbers $\;x,y,z\;$ define $$ X := e^{ix},\; Y := e^{iy},\; := e^{iz}\quad \text{ and }\quad a := X^2,\; b := Y^2,\; c := Z^2. $$ Then, by definition of cotangent $\; \cot(x-y) = i(a+b)...
  • 31.1k
2 votes
Accepted

Show that there exists θ in (13π/6, 7π/2) such that tan(2θ-5π)tan(3θ+4π) = 2/3

It is asked about $\theta$ such that $6.806784\lt\theta\lt10.995574$ and $\tan(2\theta-5\pi)\tan(3x+4\pi)=\dfrac23$. First we have $\tan(2x-5\pi)\tan(3\theta+4\pi)=\tan(2x)\tan(3x)$ and because $\tan(...
  • 26.7k
2 votes

Reduce a linear combination of sin and cos to a single trigonometric function

Let $\rho:=\sqrt{a^2+b^2}$. Then there exists a $\vartheta$ such that $a=\rho\cos\vartheta$ and $b=\rho\sin\vartheta$. Now, $$a\sin x+b\cos x=\rho\cos\vartheta\sin x+\rho\sin\vartheta\cos x=\rho\sin(x+...
  • 400
2 votes
Accepted

Reduce a linear combination of sin and cos to a single trigonometric function

Let's work “backwards”. We want an expression of the form $$c \sin(x + d)$$ $$= c \cos(d) \sin(x) + c \sin(d) \cos(x)$$ This needs to be equal to $a \sin(x) + b\cos(x)$, giving us the system of ...
  • 11.1k
2 votes

Need help with $\int_0^\pi\arctan^2\left(\frac{\sin x}{2+\cos x}\right)dx$

$$\begin{align*} I &= 2\int_0^\infty\arctan^2\left(\frac{2t}{3+t^2}\right)\,\frac{dt}{1+t^2} \\[1ex] &= 2\sqrt3\int_0^\infty\arctan^2\left(\frac{2\sqrt3 t}{3+3t^2}\right)\,\frac{dt}{1+3t^2} \...
  • 12.4k
2 votes

Proving that, if $\cos2\alpha$ is irrational, then $\sin\alpha$ and $\tan\alpha$ are irrational

$\cos 2\alpha = 1 - 2\sin^2 \alpha$ If $\cos 2\alpha$ is irrational then $\frac {1-\cos 2\alpha}{2} = \sin^2 \alpha$ is irrational. A rational number raised to an integer power is rational. $\sin \...
  • 7,871
2 votes
Accepted

Is this u-substitution valid?

Of course squaring still yields a true identity, but as your comment regarding absolute value suggests (and as Brian Moehring made explicit in the comments) $\cos x = \sqrt{1 - u^2}$ cannot hold on ...
2 votes
Accepted

In $\triangle ABC$, if $AC=4$ and $BC=5$, and $\cos(A-B)=\frac{7}{8}$, find $\cos(C)$

If the Law of cosines works the Law of sines probably works. From $\frac{\sin A}{5}=\frac{\sin B}{4}=k$ and $\cos A\cos B+\sin A\sin B=\cos(A-B)$ we get the equation $$\sqrt{1-25k^2}\sqrt{1-16k^2}+20k^...
  • 3,649
2 votes

Proving $\,\sin^272^\circ - \sin^260^\circ = \frac18\left(\sqrt5 - 1\right)$

Now $$\sin^2 72°=\cos^2(90°-72°)=\cos^2(18°)=\frac{1+\cos36°}{2}=\frac{1+\frac{\sqrt 5+1}{4}}{2}$$ $$=\frac{5+\sqrt 5}{8}$$ Thus $$\sin^2 60°=\frac 34$$ Hence $$\color{orange}{\sin^272^\circ - \sin^...
  • 5,563
2 votes

Proving $\,\sin^272^\circ - \sin^260^\circ = \frac18\left(\sqrt5 - 1\right)$

$$\sin^2t=\frac{1-\cos(2t)}2\quad\text{ and }\quad\cos(s°)=-\cos((180-s)°)$$ hence $$\begin{align}\sin^2(72°)-\sin^2(60)&=\frac{\cos(120)-\cos(144°)}2\\ &=\frac{\cos(36°)-\cos(60°)}2\\ &=\...
  • 9,189
1 vote
Accepted

Finding the equation of a hyperbola given the foci and a tangent line

As the image above clearly shows, it is a property of hyperbola that the tangent line at any point $T$ on the hyperbola, makes equal angles with the line segments $TF_1$ and $TF_2$. This as remarked ...
1 vote
Accepted

Express $\tan\alpha-i$ in the form $r(\cos\theta+i\sin\theta)$

To answer the last question first, the tangent function has period $\pi,$ and is undefined at odd multiples of $\frac{\pi}{2}.$ Thus, in order to specify the behavior of $\tan\alpha-i,$ it suffices to ...
1 vote

Show that $\sin(x)$ is differentiable with $\sin(x)$ as series

$$\sum_{n=0}^{\infty}\left(\frac{x^{2n+1}-x_0^{2n+1}}{x-x_0}\right)\frac{(-1)^n}{(2n+1)!}= \sum_{n=0}^{\infty}\big({x^{2n}+x^{2n-1}x_0^{}+\ldots+xx_0^{2n-1}+x_0^{2n}}\big)\frac{(-1)^n}{(2n+1)!}$$ but $...
1 vote
Accepted

You are racing around a circular track $C$...

What you have done thus far is correct. We can use the identity $$\sin^2\left(\frac{\theta}{2}\right) = \frac{1 - \cos\theta}{2}$$ If $0 \leq \theta < 2\pi$, then $\sin\left(\frac{\theta}{2}\right) ...
1 vote
Accepted

Doing $\sin\to-\cos$ with CDF causes trouble (cumulative distribution function)

Let's try another example: Probability density function $f(x) = B\sin\left(5x-\frac\pi2\right)$, domain $\left[0, \frac\pi{10}\right]$. Making the substitution $\sin(5x-\pi/2) = -\cos(5x)$, we find $$ ...
  • 88.8k
1 vote

Why does cancelling by $\sin x$ when solving $4\tan x = 5\sin x$ for $0\leq x < 2\pi $ miss solutions?

We can’t cancel terms on both sides without knowing whether it is zero. Just like solving $x^2=x \Leftrightarrow x^2-x=0 \Leftrightarrow x(x-1)=0 \Leftrightarrow x=0 \textrm{ or }x=1, \tag*{} $ we ...
  • 8,630
1 vote

Prove and generalize $\cos\frac\pi9\cos\frac{2\pi}9\cos\frac{3\pi}9\cos\frac{4\pi}9=\frac1{16}$

Multiplying the given product by $2^4 \sin \frac{\pi}{9}$ and using the double-angle formula $\sin (2x)=2\sin x\cos x$ reduces the product one by one yields $$ \begin{aligned} \because \quad & 2^4 ...
  • 8,630
1 vote

Show that there exists θ in (13π/6, 7π/2) such that tan(2θ-5π)tan(3θ+4π) = 2/3

A first step is to plot the function $$f=\tan(2x-5\pi)\tan(3x+4\pi)-2/3$$ and to observe that there are 5 poles induced by the tan-functions in the interval $13\pi/6,7\pi/2$ and that the function is ...
1 vote

Reduce a linear combination of sin and cos to a single trigonometric function

Hint You want to express $a\sin x+b\cos x$ in terms of a single sine (or cosine). Since the periodicity does not change, let $ a\sin x+b\cos x = A\sin (x+k)$ with $A,k$ to be determined.

Only top scored, non community-wiki answers of a minimum length are eligible