# 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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### 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 ...
1 vote
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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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1 vote
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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??...
1 vote
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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 ...
1 vote
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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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### 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 vote
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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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### 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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### 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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### 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 $120^\circ$.
1 vote
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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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### 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 ...
1 vote
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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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### $\sin^2 + \cos^2 = 1$

$\alpha +\beta =\frac{3\pi}4$ $\sin(\alpha +\beta)= \frac{\sqrt{2}}2$ 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 ...