Articles of trigonometry

Calculating the shortest possible distance between points

Question: Given the points $A(3,3)$, $B(0,1)$ and $C(x,0)$ where $0 < x < 3$, $AC$ is the distance between $A$ and $C$ and $BC$ is the distance between $B$ and $C$. What is x for the distance $AC + BC$ to be minimal? What have I done? I defined the function $AC + BC$ as: […]

Show $4 \cos^2{\frac{\pi}{5}} – 2 \cos{\frac{\pi}{5}} -1 = 0$

Show $$4 \cos^2{\frac{\pi}{5}} – 2 \cos{\frac{\pi}{5}} -1 = 0$$ The hint says “note $\sin{\frac{3\pi}{5}} = \sin{\frac{2\pi}{5}}$” and “use double/triple angle or otherwise” So I have $$4 \cos^2{\frac{\pi}{5}} – 2 (2 \cos^2{\frac{\pi}{10}} – 1) – 1$$ $$4 \cos^2{\frac{\pi}{5}} – 4 \cos^2{\frac{\pi}{10}} +1$$ Now what? Theres $\frac{\pi}{5}$ and $\frac{\pi}{10}$ and I havent used the tip on $\sin$ […]

Trig equation help please

I am trying to solve $\sqrt{3}\tan\theta=2\sin\theta$ on the interval $[-\pi,\pi]$. $$\sqrt{3}\tan\theta=2\sin\theta \Rightarrow \sqrt{3}=\frac{2\sin\theta}{\tan\theta}$$ $$\Rightarrow \sqrt{3}=2\sin\theta \cdot \frac{\cos\theta}{\sin\theta} \Rightarrow 3 = 4\cos^2\theta$$ I get $\displaystyle \cos \theta = \pm{\frac{\sqrt{3}}{2}}$; the cosine of $30^{\circ}$ and $150^{\circ}$ so arrived at the solutions $\displaystyle -\frac{5}{6}\pi,-\frac{1}{6}\pi,\frac{1}{6}\pi,\frac{5}{6}\pi$. Looking in the back of the book (and checking with Wolfram), the answer is […]

How to find the maximum and minimum value of $2^{\sin x}+2^{\cos x}$

My try: Let $y$ = $2^{\sin x}+2^{\cos x}$ Applying AM GM inequality I get $y$ $> 2.2^{(\sin x+\cos x)/2}$. Now, the highest value of R.H.S is $2^{\frac{\left(2+\sqrt{2}\right)}{2}}$. Should this mean that $y$ is always greater than $2^{\frac{ \left ( 2+\sqrt{2}\right ) }{2}}$? But this is not true (we can see in the graph). Calculus method: […]

Are there any natural occurrences of taking a trig function of a trig function?

In devising challenging exercises for my students, I am tempted to give them something like $\cos(3\sin(4))$, but then I get to wondering whether such a calculation would ever be encountered in practice. Since radians are dimensionless, as are values returned by trig functions, there is no mathematical barrier to this happening, but I was wondering […]

Proving $\int_0^{\pi } f(x) \, \mathrm{d}x = n\pi$

I’ve been asked to show $$ \displaystyle \int_{0}^{\pi} \dfrac{2(1+\cos x) – \cos((n-1)x) – \cos((n+1)x) – 2\cos nx}{1-\cos 2x} \ dx = n\pi $$ The integrand simplifies nicely to $$\frac{\cos nx – 1}{\cos x – 1}$$ but I’m not sure how to proceed from here, I’ve tried using a $x \mapsto \pi – x$ sub, but […]

Prove Bernoulli Function is Constant on Streamline

I have an incompressible, inviscid fluid, under the influence of gravity, with a velocity potential: $$ \mathbf{u} = (-\cos(x)\sin(y), \sin(x)\cos(y), 0) $$ Using Euler’s equations, $$ \mathbf{u} \cdot (\mathbf{\nabla} \cdot \mathbf{u}) = \frac{-1}{\rho} \mathbf{\nabla}p – g\mathbf{\hat{z}} $$ that is, $$ \frac{\partial p}{\partial x} = \frac{\rho}{2} \sin(2x) $$ $$ \frac{\partial p}{\partial y} = \frac{\rho}{2} \sin(2y) $$ […]

Find intersection of two lines given subtended angle

I’m tinkering with a bit of graphics software. I want to be able to nudge the value of a single angle in a polygon, then redraw the new polygon. The intersection point of the two lines creating the angle will change, and their lengths will change, but I can’t work out how to calculate the […]

Reference request: Analytic study of the trigonometric functions

I’m looking for a source (or sources) which develop a complete theory of the trigonometric functions with no reference to circle geometry. That is, it is purely analytic. The starting point could be (for example) $$\arcsin x := \int_0^x \frac{dt}{\sqrt{1-t^2}} $$ or alternatively defining $\sin$ and $\cos$ as solutions to differential equations.

Scale position points in a circle.Look like normal scaling

I have an Art degree, no math involved, so sometimes when doing 3D graphics and envisioning problems, it’s hard to search for solutions over the internet since I don’t have good pointers for search terms. I’m sure this is a trivial problem with a proper name/solution. Basically I just want to grab P and scale […]