Mathematics Stack Exchange
Mathematics Stack Exchange

Questions tagged [trigonometry]

Questions about trigonometric functions (both geometric and circular), relationships between lengths and angles in triangles and other topics relating to measuring triangles.

Trigonometry is a branch of mathematics that studies relationships involving lengths and angles of triangles.

Trigonometry is most simply associated with planar right-angle triangles. The applicability to non-right-angle triangles exists, but, since any non-right-angle triangle (on a flat plane) can be bisected to create two right-angle triangles, most problems can be reduced to calculations on right-angle triangles. Thus the majority of applications relate to right-angle triangles.

One exception to this is spherical trigonometry, the study of triangles on spheres, surfaces of constant positive curvature, in elliptic geometry. Trigonometry on surfaces of negative curvature is part of hyperbolic geometry.

Trigonometric identities are equalities that involve trigonometric functions and are true for every single value of the occurring variables. Geometrically, these are identities involving certain functions of one or more angles.

See Wikipedia's list of trigonometric identities.

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Solutions of $\tan(2x+3) = -1/2$

this is a silly question but here I go. I was solving a question that required evaluating the derivative of an equation, which would result in finding a local min and a local max. The two points are in Quadrant 2 and 3,respectively. The original…
Gtexx
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Testing $\sin\theta$ and $\cos\theta$ without referring to the trigonometric functions

This is very much not my area so apologies if this is an obvious no. Suppose values have been calculated for $\sin\theta$ and $\cos\theta$. Is it possible to test their correctness, without referring to the trigonometric functions, imaginary…
Max
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Why am I getting the correct value for $\sin\left(2\tan^{-1}\frac{4}{3}\right)$ even though the usage of the formula is incorrect?

The expression: $$\sin\left(2\tan^{-1}\left(\frac{4}{3}\right)\right)$$ Way 1: If I punch the above expression in my calculator, I get $\frac{24}{25}$. Way…
tryingtobeastoic
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Number of solution of ${\left( {\sin x - 1} \right)^3} + {\left( {\cos x - 1} \right)^3} + {\sin ^3}x = {\left( {2\sin x + \cos x - 2} \right)^3}$

Number of solution of the equation ${\left( {\sin x - 1} \right)^3} + {\left( {\cos x - 1} \right)^3} + {\sin ^3}x = {\left( {2\sin x + \cos x - 2} \right)^3}$ in the interval $[0,2\pi]$ is equal to_____ My approach is as follow $a = \sin x - 1;b =…
Samar Imam Zaidi
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Finding the least positive integer such that $\sum \cos{\theta_{i}} \leq \gamma$

Let $n \in \mathbb{N}$. How to find the least positive integer $\gamma$ such that $$ \sum\limits_{i=1}^{n} \cos{\theta_i} \leq \gamma$$ provided $$\prod\limits_{i=1}^{n} \tan{\theta_i} = 2^{\frac{n}{2}}$$ for any $\theta_i \in (0,\frac{\pi}{2})$,…
anonymous
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How do I calculate the size of padded envelope needed for a given size box?

When you put a box of dimensions (W x H X D) in a padded envelope of dimensions (X x Y), what is the mathematics? The padded envelope also has to have a flap of length (F), it also has welded seams (S). There also needs to be some extra gap so the…
Mathew
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Is there an elementary way to understand this trigonometric identity?

Let $\{k_r\}$ be the $N$ solutions between $0$ and $\pi$ of $\cos(N k) = h$, where $\lvert h \rvert <1$. I have come across the following identity in a physics research problem: \begin{align} \Bigg(\prod_{r=1}^N 2 \sin(k_r) \prod_{1\leq r
Nick Jones
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How to simplify $\sqrt{\tan^2 x + \cot^2x }$?

How to simplify : $$\sqrt{\tan ^2 x + \cot ^2x }$$ the option are : (i) $ \tan x \cdot \sin x$ (ii) $\sin x \cdot \cos x $ (iii) $ \sec x \cdot \csc x $ (iv) $ \frac{1}{\tan x - \cot x}$ (v) $ \csc^2 x - \sec ^2 x$ My approach : Since…
valkedin
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Superposition of two cosine terms

This is an example problem from my textbook. I read through its solution, but was confused by the following step. $$x=\frac{\omega^{2}A\cos(\omega t-\delta)}{\sqrt{(\omega_{0}^{2}-\omega^{2})^{2}+\omega^{2}\gamma^{2}}}+A\cos\omega t$$ The textbook…
Sirou Ewei
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How to use double angle identities to find $\sin x$ and $\cos x$ from $\sin 2x $?

If $\sin 2x =\frac{5}{13}$ and $0^\circ < x < 45^\circ$, find $\sin x$ and $\cos x$. The answers should be $\frac{\sqrt{26}}{26}$ and $\frac{5\sqrt{26}}{26}$ Ideas The idea is to use double angle identities. One such identity is $\sin 2x=2\sin…
Mike
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Proving $\frac{2\sin x+\sin 2x}{2\sin x-\sin 2x}=\csc^2x+2\csc x \cot x+\cot^2x$

Prove $$\dfrac{2\sin x+\sin 2x}{2\sin x-\sin 2x}=\csc^2x+2\csc x \cot x+\cot^2x$$ Proving right hand side to left hand side: $$\begin{align}\csc^2x+2\csc x \cot x+\cot^2x &= \frac{1}{\sin^2x}+\dfrac{2\cos x}{\sin^2x}+\dfrac{\cos^2x}{\sin^2x}…
Joe
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Proving $\frac{\cos A - \cos B}{\sin A + \sin B} = \frac{\sin B - \sin A}{\cos A + \cos B}$

Prove $$\dfrac{\cos A - \cos B}{\sin A + \sin B} = \dfrac{\sin B - \sin A}{\cos A + \cos B}$$ I tried as shown below and am not sure how to do it. Your help is appreciated. Thanks. Proving from left hand side: $$\dfrac{\cos A}{\sin A + \sin B} -…
Joe
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Is my process of proving this problem correct when $2x$ & $3x$ are not acute angles?

Problem: Prove $$\cot^{-1}(\tan 2x)+\cot^{-1}(-\tan 3x)=x$$ My proof: $$\begin{align}\text{L.H.S}&=\cot^{-1}(\tan 2x)+\cot^{-1}(-\tan3x) \\ &=\cot^{-1}(\cot…
tryingtobeastoic
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Is this statement related to trigonometry valid to write?

Is this statement related to trigonometry valid to write? $$\sin\left(\frac{\pi}{6}-2x\right)=\sin4x$$ $$\implies \frac{\pi}{6}-2x=\pm 2n\pi+4x \tag{i}$$ where $n$ is any whole number. Can I write (i)?
tryingtobeastoic
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Identity $\arctan{\frac{1-\beta}{2\sqrt{\beta}}}=\arcsin{\frac{1-\beta}{1+\beta}}$

In the book, Control Systems Engineering - frequency design, the author used the equality $$\phi_{max}=\arctan{\frac{1-\beta}{2\sqrt{\beta}}}=\arcsin{\frac{1-\beta}{1+\beta}}$$ Is this some famous identity? Am I seriously missing out since I've…
Yami
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