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 […]

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 […]

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 […]

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 […]

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?

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 $-1<x<1$

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.

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.

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 […]

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

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