Articles of trigonometry

Find area bounded by two unequal chords and an arc in a disc

Math people: This question is a generalization of the one I posed at . Below is an image of a unit circle with center $O$. $\theta_1, \theta_2 \in (0, \pi)$ and $\gamma \in (0, \min(\theta_1,\theta_2))$. $\theta_1 = \angle ROS$, $\theta_2 = \angle POQ$, and $\gamma = \angle ROQ$. I want to find the area […]

$\cos^n x-\sin^n x=1$

For $0 < x < 2\pi$ and positive even $n$, the only solution for $\cos^n x-\sin^n x=1$ is $\pi$. The argument is simple as $0\le\cos^n x, \sin^n x\le1$ and hence $\cos^n x-\sin^n x=1$ iff $\cos^n x=1$ and $\sin^n x=0$. My question is that any nice argument to show the following statement? ‘For $0 < x […]

$a,b,c$ are positive reals and distinct with $a^2+b^2 -ab=c^2$. Prove $(a-c)(b-c)<0$

This question already has an answer here: If $a,b,c$ are positive integers, with $a^2+b^2-ab=c^2$ prove that $(a-b)(b-c)\le0$. 6 answers

Prove that $\arctan\left(\frac{2x}{1-x^2}\right)=2\arctan{x}$ for all $|x|<1$, directly from the integral definition of $\arctan$

I would like to show that for $A(x) = \int_{0}^{x}\frac{1}{1+t^2}dt$, we have $A\left(\frac{2x}{1-x^2}\right)=2A(x)$, for all $|x|<1$. My idea is to start with either $2\int_0^x\frac{1}{1+t^2}dt$ or $\int_0^{2x/(1-x^2)}\frac{1}{1+t^2}dt$, and try to transform one into the other by change of variables. (It would make more sense for the moment if we did not do any trigonometric substitutions, since […]

Find the side length of triangle ABC;

Let ABC be an equilateral triangle, and let P be a point in the interior of the triangle. Given that PA = 3, PB = 4, and PC = 5, find the side length of ABC. Relatively simple problem I think, but I can’t quite get the right way to solve this. Please nothing fancy […]

Solving this equation $10\sin^2θ−4\sinθ−5=0$ for $0 ≤ θ<360°$

The first part of the question asks me to square both sides of the equation: $$3 \cos θ=2 − \sin θ$$ So that I can obtain and solve the quadratic: $$10\sin^2θ−4\sinθ−5=0 \;\;\text{for}\;\; 0 ≤ θ<360°$$ solutions obtained in the interval are: $69.2°, 110.8°, 212.3°, 327.7°$ However the second part of this question stumps me, it […]

If $\sin\theta+\sin\phi=a$ and $\cos\theta+ \cos\phi=b$, then find $\tan \frac{\theta-\phi}2$.

I’m trying to solve this problem: If $\sin\theta+\sin\phi=a$ and $\cos\theta+ \cos\phi=b$, then find $\tan \dfrac{\theta-\phi}2$. So seeing $\dfrac{\theta-\phi}2$ in the argument of the tangent function, I first thought of converting the left-hand sides of the givens to products which gave me: $$2\sin\frac{\theta+\phi}2\cos\frac{\theta-\phi}2=a\quad,\quad2\cos\frac{\theta+\phi}2\cos\frac{\theta-\phi}2=b$$ But then, on dividing the two equations (assuming $b\ne0$), I just get the […]

Finding roots of $\sin(x)=\sin(ax)$ without resorting to complex analysis

If you were given an equation $\sin(x)=\sin(ax)$ (say $a$ is a natural number), how would you go about finding all the roots on $[0,2\pi)$ without delving into complex numbers? From a simple geometric analysis it is obvious that solving for $\pi-x=ax$ would yield 4 solutions, and $x=0$ and $x=\pi$ are another 2 obvious solutions. From […]

How do we know Taylor's Series works with complex numbers?

Euler famously used the Taylor’s Series of $\exp$: $$\exp (x)=\sum_{n=0}^\infty \frac{x^n}{n!}$$ and made the substitution $x=i\theta$ to find $$\exp(i\theta) = \cos (\theta)+i\sin (\theta)$$ How do we know that Taylor’s series even hold for complex numbers? How can we justify the substitution of a complex number into the series?

Rotate a point in circle about an angle

How should I rotate a point $(x,y)$ to location $(a,b)$ on the co-ordinate by any angle?