Articles of trigonometry

find parameter for maximize area

suppose that we have Cartesian coordinate system.and suppose that we have three point which depend on parameter $t$,where t belongs to $(0,1)$;points are $A(cos(3-t),sin(3-t))$ $B(cos(t),sin(t))$ $C(-cos(t),-sin(t))$ goal: find $t$ for which area of triangle $ABC$ is maximum first of all,i was thinking that we could find length of each side of triangles,for example $BC=2$ but […]

What is inverse tangent?

I recently started thinking about what inverse tangent is. It is obvious that the definition of tangent is $\frac{\sin x}{\cos x}$, however, what is inverse tangent? I first thought $\tan^{-1} x = \frac{\sin^{-1} x}{\cos^{-1} x}$, but it didn’t seem right when I graph it out. One interesting I find is in most of the programming […]

Graph the following function: $y = \sin \frac{1}{2}\left(x-\frac{\pi}{6}\right)$

I am working my way through Trigonometry by Gelfand and Saul. I am trying to work out the following question on p 183: Graph the following function: $$y = \sin \frac{1}{2}\left(x-\frac{\pi}{6}\right)$$ I would like to check my understanding is correct. Firstly the authors talk about the importance of ensuring the equation is in the standard […]

Definition of $\operatorname{arcsec}(x)$

My calculus book gives for the derivative of $\operatorname{arcsec}(x)$: $$\frac{d}{dx} \sec^{-1}x=\frac{1}{|x|\sqrt{x^2-1}}$$ Then it continues to state that: “Some authors prefer to define $\sec^{-1}$ as the inverse of the restriction of $\sec(x)$ to the separated intervals $[0,\pi/2]$ and $[\pi,3\pi/2]$ because this prevents the absolute value from appearing in the formula for the derivative” Question: what does […]

The limit of the function $(1-x^2)/\sin(\pi x)$ as $x\to 1$

What is $${\lim_{x \to 1}} \frac{1-x^2}{\sin(\pi x)} \text{ ?} $$ I got it as $0$ but answer in the book as $2/ \pi$. Can you guys tell me what’s wrong?

Equation $2\cos(x)-3\tan(x)=0$

I solved this equation $2\cos(x)-3\tan(x)=0$ and I got, $\frac{1}{2}=\sin(x)$ and $-2=\sin(x)$. For the first solution I got $\arcsin(1/2)=x, 30°=x$, but second is invalid because the domain of arcsin can be only between $-1$ and $1$. Right??? Thanks. EDIT: My question: is the solution valid for $\arcsin(-2)$, because the domain of $\arcsin()$ is $-1<x<1$

How can I solve $\sin(x)=\sin(2x)$?

Solve $\sin(x)=\sin(2x)$ How do I solve this equation for $x$ without a calculator? It seems pretty simple but I’m not sure how to do it.

Inverse trig question?

Hello everyone I have a question about trig. How would I solve the following. $$\tan\left(2\arcsin(4/5)+\arccos(12/13)\right)=\frac{253}{204}$$ Please help.

Have you seen this golden ratio construction before? Three squares (or just two) and circle. Geogebra gives PHI or 1.6180.. exactly

Note this golden ratio construction has been dramatically updated here with numerous golden harmonies: A Golden Ratio Symphony! Why so many golden ratios in a relatively simple golden ratio construction with square and circle? Have you seen the attached golden ratio construction before? Three squares (or just two) and circle. For the ratio of segment […]

A trigonometric inequality: $\cos(\theta) + \sin(\theta) > 0$

How can I find $\theta$ such that $$\cos(\theta)+\sin(\theta)>0$$