Articles of trigonometry

Superelliptical Trig Functions

The six normal trigonometric functions relate to the unit circle and the coordinates of the endpoints of the radii making some angle $\theta$ with the x-axis (sine represents the y-coordinate, cosine the x-coordinate, and tangent the ratio of the two). Does there exist a analogous set of functions for the unit superellipse with equation $$x^4+y^4=1?$$ […]

Show that $|\sin{a}-\sin{b}| \le |a-b| $ for all $a$ and $b$

I’ve recently been going over the mean value and intermediate value theorems, however I’m not sure where to start on this.

If y = asinx + bcosx + C then find maxima and minima for y

I was able to solve it till y = √(a^2 + b^2) sin(α + x) + C but I don’t know how to find maxima and minima from here. If C = 0 then maxima & minima = amplitude of the sine curve but when C is non-zero then? I need help from here onwards.

How can $r$ be negative when dealing with polar coordinates?

If by definition $r=\sqrt{x^2 + y^2}$, then why do we allow $r$ to be negative? Relatedly, I do not understand the last section of this conversation discussing points being represented by multiple $\theta$: Student: So a single point could have many different values? Mentor: Correct! The values for $r$ can be given as positive and […]

Find $r$ knowing that $r=\frac{60}{\sin^{-1}\frac{60}{r}}$

I’m trying to find the value of $r$ knowing that: $$r=\frac{60}{\sin^{-1}\frac{60}{r}}$$ I’m not really sure how to approach finding the solution. Can anyone help me out? I’ve spent well over an hour on the problem to get to this point, and now I’m stuck. Thanks!

A cosine function has maximum value of 14 and a minmum value of 4, a period of 7, and a phase shift of 12.

A cosine function has a maximum value of 14 and a minimum value of 4, a period of 7, and a phase shift of 12. Write an equation representing this cosine function… Could someone tell me if I’am write and if I’am wrong explain why and a solution my answer $y\,\,\, = \,\,\,14\cos 2\pi {{\left( […]

Notation for powers of trigonometric functions

Possible Duplicate: $\arcsin$ written as $\sin^{-1}(x)$ When I learnt the trig identity $\sin^2\theta + \cos^2\theta \equiv 1$, I learnt that $\sin^2\theta = (\sin\theta)^2$. So why isn’t $\sin^{-1}\theta = (\sin\theta)^{-1} = \dfrac{1}{\sin\theta}$? Because $\csc\theta = \dfrac{1}{\sin\theta} $, but $\csc\theta \ne \sin^{-1}\theta$ How can these two same notations, just with different numbers, mean different things?

amplitude of sine wave with multiple frequencies

I’m having some troubles determining the amplitude/magnitude of the following equation. $$ A\cos(2\omega t+\beta_1)+B\cos(3\omega t+\beta_2)+C\cos(5\omega t+\beta_3) $$ Since each part is at a different frequency, i cannot sum the magnitudes of each part. I have also thought about using variations of the double/triple angle formulae and some basic trigonometric identities, so that I can write […]

From $\tan(1/A) = \tan(1/B) + \tan(1/C)$ to $A + B + C = ABC$

In this recent question, the equation $$\tan\left(\frac{1}{A}\right) = \tan\left(\frac{1}{B}\right) + \tan\left(\frac{1}{C}\right)$$ is said to imply $$A + B + C = ABC$$ without any stated constraints. Where does this come from, and is it even correct? I must be misunderstanding the context here, because even with integers like $B = 4$ and $C = 5$, […]

How to find $\lim_{x\to 0}\frac{\tan 3x}{\tan 5x}$?

I am asked to find the following limit: $$ \lim_{x \to 0} \frac{\tan 3x}{\tan 5x}$$ My problem is in simplifying the function. I followed two different approaches to solve the problem. But both seems incorrect. Apprach 1) Since $\tan \theta = \frac{sin \theta}{cos \theta}$ and $\cot \theta = \frac{\cos \theta}{\sin \theta}$, we have: $$\frac{\tan 3x}{\tan […]