Questions tagged [trigonometry]

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

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Trigonometric problem in solving a PDE

I'm self-studying partial differential equations with course material from 2018 and I have example solutions to the exercises. I have tried to arrive at the example solution for this PDE already a few ...
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Interpolation of a 2D segment using its projection

Consider the following diagram: The blue segment is projected on the orange projection screen from a specific point of view. The projection is shown at the bottom of the image. About the blue segment ...
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How does $\frac{-(3-\sqrt{3})}{(3+\sqrt{3})}$ become $\frac{1-\sqrt{3}}{1+\sqrt{3}}$?

How can I change $\dfrac{-(3-\sqrt{3})}{(3+\sqrt{3})}$ to $\dfrac{1-\sqrt{3}}{1+\sqrt{3}}$? Background: I tried solving $\tan(345°)$ with the trigonometric angle sum/difference identity. I used $\tan(...
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How do i prove this $\frac{\sin^2(20)}{\sin(140)\sin(80)} = \frac{\cos(80)}{\cos(20)}$

How do you prove $$\frac{\sin^2(20)}{\sin(140)\sin(80)} = \frac{\cos(80)}{\cos(20)}$$ Any help or hints will be appreciated. I have been stuck for quite some time! Thank you in advance. EDIT: After ...
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Solve equation for all solutions $\tan(3x)+1=0$

I am working on proving the $\tan(3x) = -1 $ for all solutions. The big thing in the problem that is new to me is the substitution that has to be done. After the substitution I am not sure how the ...
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How to use fraction rules for dividing

Hi I am very confused with fractions such as $\dfrac {\dfrac{a}{b}}{c}$ can this be simplified to $\dfrac {a}{b} \cdot \dfrac {1}{c}$? so for example I was using the definition of a derivative to find ...
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Determine $a$ such that $3x\cdot\cos x - ax = 0$ has () solutions...

An equation is given $3x\cdot\cos x - ax = 0$, for $x$ in $[0,2\pi]$. I am asked to determine $a$ such that the equation has two and three solutions. It seems that for only one solution $a$ is in the ...
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What is a distance function of circles outgoing from the center while shifting at rates proportional to the distance to center towards the +Y axis?

A continuation of What is a distance function of circles outgoing from the center while shifting at constant rates towards the positive Y axis?, so please first look at that question before proceeding ...
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3 answers
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What is a distance function of circles outgoing from the center while shifting at constant rates towards the positive Y axis?

I have the following problem: I want a distance equation that depending on the X and Y Coordinates will give me a distance to the circle center. The circle center is drifting towards a certain ...
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Is $\sin(\pi/2-\arctan(y/x))/\sin(2\arctan(y/x))=\frac{1}{2\sin(\arctan(y/x))}$ an identity?

So while deriving a formula I got to a point where I got these expressions as a result: $$\frac{\sin(\pi/2-\arctan(y/x))}{\sin(2\arctan(y/x))}$$ $$\frac{1}{(2\sin(\arctan(y/x)))}$$ They are ...
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The points A $(0,0)$, B$(\cos(\alpha),\sin(\alpha))$ and C $(\cos(\beta),\sin(\beta))$ are the vertices of a right angled triangle.

$(0,0)$, B$(\cos(\alpha),\sin(\alpha))$ and C $(\cos(\beta),\sin(\beta))$ are the vertices of a right angled triangle. Derive a relation between $\alpha$ and $\beta$." /> I tried using the slope ...
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Find the length of AP such that $\theta$ maximum.

Find the length of AP such that $\theta$ maximum. First, I think to construct a function length of AP with variable $\theta$ ($AP(\theta)$). Next, we find maximum $\theta$ by equation $\dfrac{d AP(\...
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Determine Scalene Trapezoid with three sides and an angle adjacent to unknown side

Ran into this problem recently: I have a scalene trapezoid with parallel bases $b_1$ and $b_2$, and legs $l_1$ and $l_2$. Both base side lengths are known, but only one leg is known. In addition, one ...
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Deriving addition from various other operations

problem I need to derive addition and/or subtraction from a limited set of mathematical operations: limitations I can do arithmetic with constant values e.g. x * 2 ...
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Determining the input angle from the output of a trigonometric function

I am attempting to answer a set of questions where I need to find all possible values of $x$, in the range $0 < x < 2\pi$, as a fraction of $\pi$, in a question such as: $$\cos x = 1$$ For this ...
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Is it true that $\cos(\cos(1)) > \sin(\cos(1))$?

Let $\cos(1)$ be $\theta$. Then $\cos(\cos(1)) = \cos(\theta)$ and $\sin(\cos(1)) = \sin(\theta)$. We know that both $\cos(\theta)$ and $\sin(\theta)$ lies between $-1$ and $1$. What to do after this??...
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Angle dependencies between equilateral and right-angled triangle

Given an equilateral triangle $\triangle ABC$ and a right-angled triangle $\triangle ABD$ where $\angle ADB$ is the right angle and, therefore, the hypotenuse $AB$ is shared with the $\triangle ABC$. ...
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Generalizing the Pythagorean trig identity $\sin^2{\theta}+\cos^2{\theta}=1$

The following is a generalization of the half-angle formulas presented at Nabla - Applications of Trigonometry for a triangle. Generalization. Let $a$, $b$, $c$, $d$ be the sides of a general convex ...
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Are goniometric functions only defined for oriented angles?

My textbook defines sine and cosine functions of only oriented angles. For example cosine of an oriented angle is the signed abscissa / the radius of the circle. I've got two questions: Every time we ...
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how can you understand trigonometry from its origin? [closed]

I've been looking at Toomer's translation of the Almagest and in it, he provides a section on how chords were formed from Euclidean geometry but I find that it's impossible to learn Euclidean geometry ...
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Radically different answers for $\frac{\mathrm d}{\mathrm dx}\left(\arccos\frac{\sqrt{1 - x^3} - \sqrt{1 + x^3}}{2}\right)$

Find the derivative with respect to $x$ of $$\cos^{-1}\left(\frac{\sqrt{1 - x^3} - \sqrt{1 + x^3}}{2}\right).$$ Here's my work: Substituting $x^3 = \cos(2\theta):$ $$\begin{aligned}\cos^{-1}\left(\...
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4 votes
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$\sin(25°)+\cos(115°)$?

What is the value of $\sin(25°)+\cos(115°)$? Using $\cos(90°+\theta)=-\sin(\theta)$, we get, $$\sin(25°)+\cos(115°)=\sin(25°)-\sin(25°)=0$$ But when I searched the same on Google, it showed $-0....
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-1 votes
2 answers
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Is the statement true: $\cos x\cos 2x\cos 4x=1/4\cos 3x$? [closed]

How to show that $\cos x\cos 2x\cos 4x=1/4\cos 3x$? I have tried by $\cos x\cos 2x\cos 4x=\frac{1}{2}\cos x[2\cos 2x\cos 4x]$. $=\frac{1}{2}\cos x[\cos 6x+\cos 2x]$. $=\frac{1}{4}[2\cos x\cos 6x+2\...
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prove that $\arctan\frac{\cos x-\sin x}{\cos x+\sin x}=\frac{\pi}{4}-x$, where $0<x<\pi$

I tried to solve this $$\begin{align} \arctan\frac{\cos x-\sin x}{\cos x+\sin x}&=\arctan\frac{1-\tan x}{1+\tan x}\\&=\arctan\frac{\tan\frac{\pi}{4}-\tan x}{1+\tan\frac{\pi}{4}\tan x}\\&=\...
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1 vote
1 answer
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Tangent Space View Direction based factor value remap

I'm trying to setup a mask similar to what Fresnel produces. Unfortunately Fresnel gives pretty bad results at grazing angles so I ended up using this : ...
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$\sin(\pi/2-\pi/x)=\cos(\pi/x)$?

I have suspicion which is that $\sin(\pi/2-\pi/x)=\cos(\pi/x)$ is an identity. I visualized both functions in Geogebra and it looks like they are in fact identical. However I am unable to prove that ...
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1 answer
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Phase Angle of a complex fraction

I'm having a confusion with a problem given, any help will be appreciated. For example it is given a transfer function, $G(s)= \frac{(s+20)}{(s+1)(s+100)}$ Substitute $j\omega$ to get the frequency ...
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2 votes
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Proving rank deficiency of a matrix whose elements are given by trigonometric functions

I want to show that a specific $(N^2+1)\times 3N$ matrix ($N\geq 3$) is rank deficient, specifically that it has rank $3N-1$. Ideally, I would like to show that this is the case for all $N\geq 3$ but ...
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Simplifying atan(x)-atan(y)

From this question post and this youtube video "proof", I am convince that for $x>0$ and $y>0$, the following holds true: $$ \tan^{-1}(x)-\tan^{-1}(y)=\tan^{-1}\left(\frac{x-y}{1+xy}\...
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-3 votes
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How to rewrite this trigonometric identity (Euler Formula to the n-th power)? [closed]

Is $$\exp(i\cdot na)=e^{i\cdot na} =((\cos(a)+i \sin(a))^n$$ equal to $$\cos(na)+i \sin(na)?$$ or is it $$n \cos(a)+i n \sin(a)$$
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Why is the Arccosine of $30$ degrees undefined?

Recently, while working on Trigonometry, a problem came up in which I was asked to evaluate the value of $\cos(\arccos(30^\circ))$, and I stated that the value of this function was $30$ degrees (...
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Robotics kinematics calculating roll of a universal joint

I'm making a hexapod robot that uses universal joints for the connections. I'm simulating it in 3D so I need to be able to calculate the angles for each joint to properly visualize it. I've figured ...
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0 answers
42 views

How to evaluate cos(120), sin(240), tan(-60), cot(300), tan(330, cos(-60), sin(-150), cos(-120), tan(330), cos(-60), and sin(-150) by hand? [closed]

HELP I neeed to solve this for an exam. I thing the solutions involve trigonometric ratios and relations, but I'm not sure. Could anyone give me some help, please?
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1 answer
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With rotation, is it computationally more efficient to use quaternions?

When coding rotation in 3D graphics/games, is it computationally more efficient to use quaternions in all circumstances? Say I'm not rotating an airplane or something prone to gimbal lock. Maybe I ...
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2 votes
1 answer
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General addition theorem for $cas(x):=sin(x) + cos(x)$ | Summation form

I like addition theorems in trigonometry and recently YouTuber Dr Barker posted the video "My New Favourite Trig Function" playing around with following: Define $$ \operatorname{cas}(x) := \...
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1 vote
1 answer
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Find the side length of a square with line segments of length 1, 2, and 3 extending from each corner and intersecting at their tips

I know trigonometry should be involved in this somehow but am stuck at where to construct the triangles.
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-5 votes
1 answer
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If $c^4+a^4-2c^2(a^2+b^2)+a^2b^2+b^4=0$, then prove that $C=60^\circ$ or $120^\circ$ [closed]

If $$c^4+a^4-2c^2(a^2+b^2)+a^2b^2+b^4=0$$ then prove that $C=60^\circ$ or $120^\circ$.
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1 vote
1 answer
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Prove: out of all triangles with a given angle and side, the triangle with the biggest perimeter will always be an isosceles triangle [duplicate]

While studying for a test I came across this question. Before approaching I had already found that if there's a triangle $ABC$ with the angle $\angle BAC = \alpha$ (in radians) and the opposite side $...
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$\theta$ is real if and only if $e^{i\theta}$ is in circle

Let us define $e^z = \displaystyle \sum_{k=0}^{\infty} \frac{z^n}{n!},$ where $z$ is a complex number. I want to show that, $\theta$ is real if and only if $e^{i\theta}$ is in circle. That is, I want ...
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-1 votes
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Does a 90 degree angle have a adjacent side for SIN COS and TAN calculations?

Right triangle diagram I know you can calculate sin from angle A; 5/5 would equal 1, what about COS and Tan? All the examples i found online would reference the calculations with angles B and C, but ...
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1 vote
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Grover's Algorithm Trigonometric doubt

The Grover's operator $G$ in Quantum Computing, has the following effect $$ G|\Phi\rangle=(4\sin^2\Delta -1)|\Phi\rangle-2\sin\Delta|z\rangle $$ where $|\Phi\rangle=\frac{1}{\sqrt{2^n}}(|x_1\rangle+|...
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1 vote
1 answer
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Triangle identity with two large sides and 1 small side

I have been tasked at work to take over a task from a former colleague and I cant wrap my head around the trig proof in his notes I can prove it works when I use some real numbers, but cant get how to ...
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2 answers
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Finding number of solutions to $\sin(x)=x/10$ using an algebraic method.

I am trying to find the number of solutions of the equation $\sin(x)={x/10}$. While I know about the graphical method of doing this, I want to know if there are any quicker and/or algebraic method to ...
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1 vote
2 answers
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How to find the scaling factor of a rotated rectangle circumscribing another rectangle of same size?

Suppose rectangle 1 with length $l$, and width $w$, which has a center $C$ (where the diagonals intersect), rectangle 2 with same length $l$, width $w$, and center $C$, but rotated $\theta$ radians ...
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2 votes
1 answer
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What will be the value of $ 2\cos^{2}\theta - 1 $ , if $ \cos^{4}\theta - \sin^{4}\theta = \frac{2}{3}$

The question was What will be the value of $ 2\cos^{2}\theta - 1 $, if $ \cos^{4}\theta - \sin^{4}\theta = \frac{2}{3}$ Here is my working: $\cos^{4}\theta - \sin^{4}\theta = \frac{2}{3} $ $(\cos^{2}\...
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1 vote
2 answers
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How to solve $\sin(x) = \pm a$ for $a \not = 0$?

I was solving the below equation: $\left|\sqrt{2\sin^2x + 18 \cos^2x} - \sqrt{2\cos^2x + 18 \sin^2x} \right| = 1$ for $x \in [0, 2\pi]$. My attempt: $$\begin{align}&\left|\sqrt{2\sin^2x + 18 \...
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2 votes
2 answers
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Solve the trig system of equations

This is a repost, I'm new here and I did a terrible job explaining the question last time. I'm trying to make an IK system, the rig basicly works in 2d with the bones shown below. (Rig in Unreal) The ...
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1 answer
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Geometric interpretation of direction cosines

I have a task to: From the definition of the direction cosines, it is easy to see that the sum of the squares of the direction cosines is 1. For the special case of $\mathbb R^3$, draw a sketch and ...
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8 votes
2 answers
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Getting two different answers on differentiating $\cos^{-1}(\frac{3x+4\sqrt{1-x^2}}{5})$

Question given in my book asks to find $\frac{dy}{dx} $ from the following equation.$$y=\cos^{-1}\left(\frac{3x+4\sqrt{1-x^2}}{5}\right)$$ My Attempt: Starting with substitutions, Putting $\frac35=\...
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1 answer
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$\sin^2 + \cos^2 = 1$

$\alpha +\beta =\frac{3\pi}4$ $\sin(\alpha +\beta)= \frac{\sqrt{2}}2$ Why then $\cos(\alpha+\beta)= -\frac{\sqrt{2}}2$ if $\sin^2(x) + \cos^2(x) = 1$ I keep getting answer on calculator sqrt(2)/2 Can ...
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