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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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20 views

Calculate X, Y, Z of the 4 points in the 3D space.

I want to find the $x, y, z$ coordinates for $4$ points in a 3D space. Point $A$ is my origin $(X, Y, Z = 0,0,0)$ and other points $B, C, D$ are with reference to point A. I know all six distances ...
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2answers
36 views

Evaluating trig limit $\lim_{x\to 0}\frac{\sqrt{1-\cos(x^2)}}{1-\cos(x)}$

Evaluate: $$\lim_{x\to 0}\dfrac{\sqrt{1-\cos(x^2)}}{1-\cos(x)}$$ I have tried to simplify the expression using the identity $1-\cos(x) = 2 \sin^2 (x/2)$, but I have still failed to remove the ...
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0answers
24 views

Sine wave with different slope on each side

Upfront note that English is not my first language, and my mathematical terminology might not translate very well (although google has helped). (TL;DR at the bottom) I want to describe the animation ...
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0answers
29 views

Length of a segment intersecting a square [on hold]

I have the following schema and I need to find the lengths of the colored segments. From some calculations and help, I discovered that it should be something like $$LB = L\left(\frac{1}{\cos 45^\...
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2answers
41 views

Given a triangle with coners $A,B,C$, Show $\sin(A)+\sin(B)+\sin(C) \leq \frac{3\sqrt{3}}{2}$ [duplicate]

Given a triangle with vertices $A,B,C$, Show $\sin(A)+\sin(B)+\sin(C) \leq \frac{3 \sqrt{3}}{2}$. Here is a proof using Jensen inequality: $\sin(x)$ is concave from $0$ to $\pi$. hence $\frac{\sin(A)...
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1answer
53 views

Too many solutions to trig equation

When dealing with the equation $2\sin2x + \cos2x = 1$, I found that there is two ways of tackling the problem, both yielding different solutions. The first and most obvious way to tackle the problem ...
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3answers
37 views

How to evaluate $\int \sin^2 x \ dx$ [duplicate]

The fact that sin is squared is really throwing me off, can't seem to relate it to any standard integrals.
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0answers
34 views

Is there a way of reducing $4\sin(a)\sin(b)-\sin(a+c)\sin(b)-\sin(a)\sin(b+d)$ using the gonionmetric identities?

I'm currently working on a problem and stumbled on this equation that I'm unable to reduce using the standard goniometric identities. Perhaps the formula is impossible to reduce but any help is ...
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2answers
35 views

Trig and Triangle Math Club Question: Finding Side Length

I recently had a math club competition, and I was unsure of how to approach one of the problems on the test: In $\triangle ABC$, $\ \ \ \ \ \ \ \ \cos(2A-B) + \sin(A + B) = 2$ $\ \ \ \ \ \ \ \ \...
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13 views

What does it mean for a function to be semi-monotonic?

I mostly understand monotonic functions as described by wikipedia. However, I do not understand what it means for a function to be semi-monotonic as described in the java math class. This page helped ...
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48 views

Proving $\cos\frac{9}{11}\pi + \cos\frac{7}{11}\pi + \cos\frac{5}{11}\pi + \cos\frac{3}{11}\pi + \cos\frac{1}{11}\pi = \frac{1}{2}$

Why is: $$\cos\frac{9}{11}\pi + \cos\frac{7}{11}\pi + \cos\frac{5}{11}\pi + \cos\frac{3}{11}\pi + \cos\frac{1}{11}\pi = \frac{1}{2}$$
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1answer
40 views

How do I determine the sign of “a” in a trig function?

So tomorrow I have a quiz regarding trigonometric functions. The following is a practice question for that exam: How do I determine "a" without inserting t or y values into the equation? I would do ...
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0answers
43 views

Is it possible to prove the following inequalities to be true?

I'm trying to prove the following inequalities are true, but they're a little too complex, is it even possible to prove that they are true? I suspect they are, but unless there is some simple method ...
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1answer
16 views

Modulus of tangent of complex number

I need to find real, imaginary parts of $\tan(x+yi)$ and the modulus of it. I have: $$\operatorname{Re}(\tan(x+yi))={\frac{\sin2x}{\cos2x+\cosh2x}}$$ and $$\operatorname{Im}(\tan(x+yi))={\frac{\...
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0answers
41 views

Find $\sin x$ if $\cos x=\tan y$, $\cos y=\tan z$, $\cos z=\tan x$

If $\cos x=\tan y$, $\cos y=\tan z$, $\cos z=\tan x$, then find the value of $\sin x$ My reference says $\sin x=2\sin 18^\circ$, but how do I approach the problem ? My Attempt $$ \sin x=\tan x.\cos ...
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0answers
41 views

How to apply the cosine rule here?

ABC is an equilateral triangle. D is a point on BC and AD is produced to E such that $\angle EAC= \angle EBC.$ Find the length of AE given that BE=5 and CE= 12
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1answer
60 views

Shaded Area under square inscribed in a Circle.

Check this Question please I have tried solving this question by first finding the Area of circle and then area of square (via diagonal method). and then subtracted Its value from the total area But ...
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2answers
19 views

Convert polar coordinates to specific angle range

I shamefully admit that my trig-skill have rusted. I have a point on the uniform circle by $\sin(\alpha) = x$ and $\cos(\alpha) = y$ coordinate. For example: $\alpha = 0 \to (0,1)^T$ The angle is ...
3
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1answer
37 views

In this visual proof for the law of cosines, why are the products of subsegments of two intersecting chords equal?

The first line of the visual proof below states that $$(2a\cos\theta-b)b=(a-c)(c+a)$$ I understand the line segments represented by each part of the equation, but what makes the equation true? In ...
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0answers
31 views

Inverse Trig Identities

I've been attempting to interpolate between two trig functions for a piece of software I'm writing, but feel I've finally reached the end of my mathematical knowledge. $$\frac{\pi x-\sin^{-1}\left(\...
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0answers
28 views

find $k^2$ if $\left| a\sin^2 \theta+b\sin \theta \cos \theta+c\cos^2 \theta-\frac{(a+c)}{2} \right|=\frac{k}{2}$

If $ \left| a\sin^2\theta+b\sin\theta\cos\theta+c\cos^2\theta-\dfrac{(a+c)}{2} \right|=\dfrac{k}{2}$, then find $k^2.$ My Attempt \begin{align} \pm k &= 2a\sin^2 \theta+2b\sin \theta \cos \...
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3answers
56 views

Is there a simple way to calculate $\sin \frac{3\pi}{10}-\sin \frac{\pi}{10}$? [duplicate]

I know how to find the exact value of $\sin \frac{\pi}{10}$ using double and triple angle formulas and the fact that $\frac{5\pi}{10}=\frac{\pi}{2}$ but it maybe too complicated for high school ...
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3answers
26 views

Trigonometry Addition and Subtraction Formula

My professor showed us how to solve $\cos(\theta - X)$ where $\cos(\theta) = \frac{3}{5}$ and is in Quadrant IV, and $\tan(X) = -\sqrt{3}$ and is in Quadrant II. Since, $\cos(A-B) = \cos(A)\cos(B)+\...
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3answers
52 views

challenge trig question - no calculator

The challenge trignometry question is: simplify $sin (80^\circ) + sin (40^\circ) $ using trignometric identities. All the angle values are in degrees. This is what I did: Let $a=40^\circ$. So we ...
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2answers
31 views

Height of a Triangle, and a Progression of Triangles

I am researching some algorithms and it turns out that the following figure I made can model what is happening in a "step". I am not a mathematician, so I was having a hard time with this one. The ...
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1answer
29 views

atan2 and Cardan angles - Problem with sign

I'd like to get the three Cardan angles starting from the below matrix and by using the function atan2. The problem is that I get two conflicting results. where $- \pi/2 \le \beta \le \pi/2$. If I ...
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4answers
59 views

What is the range of $f(x)= \lvert \cos(x) - \sin(x) \lvert$

I know that $ 0 \leq \lvert \cos(x) - \sin(x) \lvert $ but I am not sure how to proceed. Thanks for the help in advance!!
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1answer
26 views

Trigonometric inequalities with substitution

Using inequality $\tan \frac{x}{2} > \frac{x}{2}$ prove that $\sin x > x- \frac{x^3}{4}$ I tried with substitution $\tan \frac{x}{2} = t$ $\sin x = \frac{2t}{t^2+1}$ $t>\frac{x}{2}$ $2t&...
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2answers
25 views

system of equations involving inverse function

The number of ordered pairs $(x,y)$ which satisfy the system of equations $(\cos^{-1} x)^2+\sin^{-1}(y)=1$ and $\cos^{-1}(x)+(\sin^{-1}y)^2=1$ are Try: Adding these two equations. we ...
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1answer
46 views

Solving $\sin(-\theta)=0.35 $. Is $\sin$ postive or negative? Where are the angles located? [on hold]

My question: $$\sin(-\theta)=0.35 \qquad\text{range: } 0<\theta<360$$ Is $sin$ positive or negative in this case? and where would the angles locate at? Thank you!
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Finding $a$ and $b$ such that $2\cos 2\pi a t \sin 2 \pi b t$ is $0$ at even $t$ and $\pm 2$ at odd $t$ [on hold]

Trying to solve this wave function. I appreciate anyone who chooses to help. The problem I'm trying to solve says: A wave with function $g(t) = 2\cos(2\pi a t)\sin(2\pi b t)$ is graphed with $a&...
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1answer
38 views

Hello, Trying to solve a complex trig problem

I am trying to understand and solve this problem. If anyone is willing to help out I appreciate you! :) Problem: Choose a value of w between 0.01 and 0.05 and again plot the graph of $f(t) = \sin(2\pi ...
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4answers
58 views

Why does this method fail for finding the Fourier series for $\cos\left(\frac{x}{2}\right)$ on the interval $-\pi \lt x \lt \pi$?

This question is strongly related to this question (that does not have an answer). From "Riley, Hobson and Bence - Mathematical methods for physics and engineering", Section 12.5 - "Non-periodic ...
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1answer
66 views

Limit of $\frac{\tan{(\sin{(x)}})}{\sin{(\tan{(x)}}}$ when x approaches 0

How would one approach finding this limit without using Taylor's series? $$\lim_{x \to 0} \frac{\tan{(\sin{(x)}})}{\sin{(\tan{(x)}})}$$
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1answer
10 views

show that if $\beta$ is an acute angle and that $\tan\beta=2\sqrt2$ then $\cos\beta=\frac{1}{3}$

show that if $\beta$ is an acute angle and that $\tan\beta=2\sqrt2$ then $\cos\beta=\frac{1}{3}$ This question question has stumped me as I see no obvious way to go with it. This is all I've done so ...
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4answers
67 views

Proof that $\left|\arctan (x)-\frac{π}{4}-\frac{(x-1)}{2}\right| \leq \frac{(x-1)^2}{2}$

I'm trying to prove that, for every $x \geq 1$: $$\left|\arctan (x)-\frac{π}{4}-\frac{(x-1)}{2}\right| \leq \frac{(x-1)^2}{2}.$$ I could do it graphically on $\Bbb R$, but how to make a formal ...
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0answers
19 views

Algbraic proof of compound periodicity of a function

While studying trigonometric functions I keep encountering exercises of this sort: $f\left(x\right)=\frac{\tan\left(x\right)}{1+\sin\left(x\right)}$. Show that the periodicity of $f(x)$ is $2\pi $. ...
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1answer
84 views

Evaluate $\int _{0}^{{\pi}/{3}} (\ln ({\sin(x)}/{\sin (x+{\pi}/{3})}))^2dx$ [duplicate]

Evaluate $$I=\int _{0}^{\frac{\pi}{3}} \left(\ln \left(\frac{\sin x}{\sin \left(x+\frac{\pi}{3}\right)}\right)\right)^2 dx$$ My try: we have $$I=\int_{0}^{\frac{\pi}{3}} \ln ^2(\sin x)dx+\ln ^2 \...
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1answer
15 views

How to calculate sides and degrees of triangle

So I understand that we can derive the degrees from a triangle with $$TAN^{-1}\left(\frac{AB}{CB}\right)= A^o (degree)$$ But this only applies on a Right Triangle. In the next example I want to get ...
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0answers
19 views

Simplify trigonometric expression of hyperbolic functions

I have $\cos^2x\cosh^2y - \sin^2x\sinh^2y$. I saw it written simplified as $\cosh^2 y - \sin^2 x$. But I don't get how to get it. My attempts were to write $\cosh^2y -1$ instead of $\sinh^2y$ but ...
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0answers
31 views

Sine/Sinc simplification

I'm trying to get the inverse Fourier transform of a raised cosine filter. The frequency domain equation of the filter is: $$ \begin{equation} H(f)= \begin{cases} 1 &|...
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2answers
34 views

$\cos^8x.\sec^6y,\frac12,\sin^8x.\csc^6y$ in AP if $\cos^4x.\sec^2y,\frac12,\sin^4x.\csc^2y$ in A.P

If $\cos^4x.\sec^2y,\dfrac{1}{2},\sin^4x.\csc^2y$ are in A.P, then prove that $\cos^8x.\sec^6y,\dfrac{1}{2},\sin^8x.\csc^6y$ in AP. My Attempt $$ \cos^4x.\sec^2y+\sin^4.x\csc^2y=\frac{\cos^4x}{\cos^...
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4answers
89 views

Show that $\det\left[\begin{smallmatrix}1&\cos a&\cos b\\ \cos a&1&\cos(a+b) \\ \cos b&\cos(a+b)&1 \end{smallmatrix}\right]=0$ [on hold]

I am unable to show - without expanding, by using determinant properties - that $$\det\begin{bmatrix} 1 &\cos a &\cos b\\ \cos a &...
2
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1answer
51 views

Prove $\tan\frac\pi{16}+2\tan\frac\pi8+4=\cot\frac{\pi}{16}$

Prove that $\tan\dfrac{\pi}{16}+2\tan\dfrac{\pi}{8}+4=\cot\dfrac{\pi}{16}$ My Attempt \begin{align} &\tan\dfrac{\pi}{16}+2\tan\dfrac{\pi}{8}+4=\dfrac{1}{\cot\dfrac{\pi}{16}}+\dfrac{2}{\cot\dfrac{\...
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1answer
48 views

Can't resolve a simple equation

I am stuck with an equation I've been trying to solve for a while in different ways. Tried searching on the internet about the properties of arccosh (or cosh-1) and cosh but still couldn't find the ...
0
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3answers
42 views

Approximating $\cos(47^{\circ})$

Given that $\cos(45^{\circ}) = \frac{\sqrt{2}}{2}$, what would $\cos(47^{\circ})$ be. Using differential approximation, I get $\cos(47^{\circ})$ is about $\cos\left( \frac{45\pi}{180}\right)-2\sin\...
2
votes
0answers
40 views

Find the area a this shape inscribed in the unit circle

I'm having real troubles solving this one geometry problem. I've attempted to draw it as best as I could. Shape $[ABCD]$ is partly inside the Unit Circle. The only other information states that tan$\...
-3
votes
1answer
47 views

Why $\cos a\cdot \cos b+\sin a\cdot \sin b = \cos (b-a)$ [duplicate]

I do not get why $$\cos a\cdot \cos b+\sin a\cdot \sin b = \cos (b-a)$$ Thank you for your help
0
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2answers
44 views

How can I find the limit $\lim_{x\to \frac12}\frac{4x^2 - 1}{\arcsin(1 - 2x)} $ [closed]

How can I find the limit $$\lim_{\left(x\to 1/2\right)}\ \frac{4x^2 - 1}{\arcsin(1 - 2x)}\quad ? $$
0
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0answers
26 views

Trig Identity; $\cot(20^\circ)\cot(40^\circ)\cot(60^\circ)\cot(80^\circ)=\frac13$ [duplicate]

How to solve $\cot(20^\circ)\cot(40^\circ)\cot(60^\circ)\cot(80^\circ)=\frac13$? I tried by converting it to sin and cos but got stucked. Please help!