# 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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### is tan(π/2 - x)=cot(x) because tan in the 2nd area is "negative" and also cot(-x) equals to -cot(x)?

trying to figure out if that negative/positive calculation is true. and so at the end there will be two negative and the cot(x) will be positive. is that so? thank you.
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### Proof without words for the sum and product of three tangents

How to get the $\tan\beta \tan\gamma$ in the third graph? Proof without words for the sum and product of three tangents The image is original from MAA https://www.maa.org/sites/default/files/...
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### Find a triangle's vertices given sides, angles, and inscribing circle

I have a triangle inscribed in a circle of radius 1 centered on the origin. The angles A B C and opposite sides a b c of the triangle are given (but variable). Given the coordinates of one vertex (0,-...
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### Prove 2D rotation by $\theta$ is a linear transformation using trig identities for $r\cos(\alpha) + s\cos(\beta)$ and $r\sin(\alpha) + s\sin(\beta)$

Theorem The rotation by $\theta$ function $T:\mathbb{R}^2 \to \mathbb{R}^2$ defined for $\vec{x}%=\begin{bmatrix}x_1 \\ x_2\end{bmatrix} =\begin{bmatrix}r\cos(\alpha) \\ r\sin(\alpha)\end{bmatrix}$...
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### exponential equation for the distance between two points a distance away from the function [0,1,...,9]x

Given the element list: b = [0,1,...,9] Then what is the exponential equation in relation to distance from two points with a value of d from the origin of the functions: $(b_n)x$ and $(b_n-1)x$. To ...
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### Help with determining constant values for water depth measurements using trigonometric functions

I am currently working on a problem related to water depth measurements and I am seeking some help with it. The problem is as follows: Water depth at a dock is measured during the first 16 hours of a ...
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### Find the sides of an AAA triangle inscribed in a circle with known radius

I have all of the angles for a triangle. I want to know the length of each of the sides such that the three vertices will all lie on a circle with a known diameter. How do I do it? (The application is ...
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### How to find minimum degrees required to get X distance from the far left side of a circle?

I'm trying to figure out how far each degree of a circle is, assuming the far left side of a circle is x=0. To put it another way I need to figure out how far each degree of the circle takes it from ...
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### How to prove $(\sec^2(a)+\tan^2(a))(\csc^2(a)+\cot^2(a))=1+2\sec^2(a)\cdot\csc^2(a)$ [closed]

How can I prove that $$\left(\sec^2(a)+\tan^2(a)\right)\left(\csc^2(a)+\cot^2(a)\right)=1+2\sec^2(a)\cdot\csc^2(a)?$$
1 vote
Let $D\in BC$ in $\triangle ABC$, so that $AD$ is the bisector of $\angle BAC$. Let $r$ be the radius of the circle inscribed in $\triangle ABC$ and $R$ the radius of the circumscribed circle of $\... • 447 0 votes 1 answer 24 views ### Finding angle and radius of arc with one known and one partially known point Given two points$p1$and$p2$on a circle of radius$R$and knowing the following: Arc length$Lp1$(which is simply the origin at 0,0) The$y$coordinate of$p2$(but not the$x$coordinate) Is ... • 103 2 votes 1 answer 63 views ### Prove that the sum of division of two squared sin function is 1 When I numerically compute the below summation it is always$1$. How can I prove this?$N$is an integer number and$k$is an integer number between$0$to$N-1$and$x$is real number between$0$and ... • 331 -2 votes 0 answers 20 views ### local Coordinate transformation to global coordinates Reference image Please help me to prove; -2 votes 2 answers 76 views ### What is the perímeter of the triangle ABC? I am froze in this question after some hours. It's a homework (without a detailed solution in the book) that I am trying to do. I am graduated in high school, but there is some years I didn't study ... -2 votes 1 answer 44 views ### How does$\cos^{-1} (1/\sqrt2) = \pi/4$? I thought$\cos(\pi/4) = \sqrt2/2$How does$\cos^{-1} (1/\sqrt2) = \pi/4$? I thought$\cos(\pi/4) = \sqrt2/2$I am awful at formatting, thank you ahead of time. -3 votes 0 answers 23 views ### Can you tell me the reason behind this equation , written below [closed] -cot(x+pi/3)=tan(pi over 2 plus x plus pi over 3) -4 votes 1 answer 41 views ### Trigonometric identity$(\csc \theta - \cot \theta )^2 = \frac{{1 - \cos \theta }}{{1 + \cos \theta }}$[closed] I was doing some problems related to trigonometry but I couldn't figure out how to do this sum. I really wanted the help to figure it out. Thanks very much in advance. $$(\csc \theta - \cot \theta )^... • 13 0 votes 0 answers 18 views ### We have to find maximum value of the expression. [closed] enter image description here I have tried to solve this problem by taking derivative on both the sides but solving algebra and trigonometry simultaneously doesn’t provide a neat solution. Plotting the ... 1 vote 0 answers 30 views ### Understanding the trigonometry behind a pole leaning on a wall I am copying and pasting this question I asked in the physics stack exchange. The reason I am doing so is because the question was automatically closed for being too "homework-like" and not ... 0 votes 3 answers 64 views ### Find the solutions of x such that \arcsin x=2\arctan x Find the solutions of x such that \arcsin x=2\arctan x. My solution goes like this: We have, \arcsin x=2\arctan x. Now, 2\arctan x=\arctan(\frac{2x}{{1-x^2}})=\arcsin(\frac{2x}{1+x^2}). Now, ... • 1,809 4 votes 2 answers 174 views ### Proving \sum_{k=1}^{2n-1}\frac{\sin(\frac{\pi k^2}{2n})}{\sin(\frac{\pi k}{2n})}=n I wander on the internet and found this problem (from Quora) this link The problem is proving the identity:$$\sum_{k=1}^{2n-1}\frac{\sin\left(\frac{\pi k^2}{2n}\right)}{\sin\left(\frac{\pi k}{2n}\... • 1,221 -1 votes 0 answers 86 views ### How can I briefly but concisely explain the unit circle to my daughter? [closed] My daughter is currently in Geometry, but she has already worked some with beginner concepts of Trigonometry. She understands SOHCAHTOA. The unit circle came up in conversation, and I attempted to ... 1 vote 1 answer 17 views ### Solid angle on a sphere of the intersection of multiple hemispheres Consider the following scenario where I have a unit sphere cut by$N$halfplanes that all contain the center of the sphere, thus forming several hemispheres. How can I calculate the solid angle ... -3 votes 0 answers 18 views ### trigonometric identity series [closed] enter image description here I am looking at complex analysis and I need to prove this series using complex algebra recording the identity 1+z^1+z^2+z^3+...+z^n=(1-z^(n+1))/( 1-z) 0 votes 2 answers 40 views ### Inscribed triangle in a circle from perpendicular chords I'm trying to solve the following problem but have hit a wall. Let's assume we have a circle. Said circle has a chord from point$A$to point$B\$. At one of the points where the chord crosses the ...
My problem consists as follows. Say I have a sinusoidal function whose argument is a cubic polynomial: $$f(t) = \sin(at^3+bt^2+ct+d) \quad a, b, c, d\in \mathbb{R}, \quad t\in[T_1, T_2]$$ I want to ...