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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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3answers
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How do you solve the trigonometric equation $\sin(x)+x=9$?

How do you solve the trigonometric equation $\sin(x)+x=9$? More generally, how do you solve equations with both trigs and 'x's without graphing? And maybe I only want real number answers.
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1answer
14 views

Linear Algebra - Linear Combination and Perpendicular to Triangle

I am working on my maths homework and encounter the following question which I have no clue to answer: Let A = (1, 1, 2), B = (-3, 1, 4), C = (-1, -1, 0) be points in space. Q1: Find all values x ...
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5answers
33 views

Proving the identity $(\tan^2(x)+1)(\cos^2(-x)-1)=-\tan^2(x)$

Proving the trigonometric identity $(\tan{^2x}+1)(\cos{^2(-x)}-1)=-\tan{^2x}$ has been quite the challenge. I have so far attempted using simply the basic trigonometric identities based on the ...
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0answers
23 views

Dot and Cross Products [on hold]

A mechanic applies a force of 42 Newtons straight down to a ratchet that is 0.59 meters long. What is the magnitude of the torque when the handle makes a 38° angle above the horizontal?
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1answer
33 views

If $g'(0)>0$ write this expression $\lim\limits_{x \to 0} \frac{\sin(f(x))}{\sin(g(x))}$ using $f(0),f'(0)$ and $g(0)$.

If $g'(0)>0$ write this expression $\lim\limits_{x \to 0} \frac{\sin(f(x))}{\sin(g(x))}$ using $f(0),f'(0)$ and $g(0)$. This came up in my Analysis 1 exam, and i couldn't do it.
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1answer
18 views

Degree between vector and point

I have a vector and a point $(x, y)$. The vector starts from $(0, 0)$ and goes to $(x_1, y_1)$. $x$, $y$, $x_1$, $y_1$ are known. How can I get the degree that vector should rotate clockwise to face ...
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2answers
46 views

Does the trigonometric identity $\cos^2(\theta)+\sin^2(\theta)=1$ apply even when $\theta$ is not in radians or degrees but simply a fraction?

I have been trying to solve this question but have so far been unable to do so as the question does not seem to be "cohesive throughout". Here is my reasoning: The question is: given that $\cos A=−3/...
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2answers
24 views

How may I simplify the following trigonomic expression;

How may I simplfy for the general equation showing the values of theta that satisfy the following; $ \{\theta \in \mathbb R \mid 2\sin^4(\theta) = \cos^2(\theta) \}$
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0answers
25 views

Fitting a sinusoid through three arbitrary points

Let's say there are three arbitrary points on the x-y plane. Does there always exist a function of the form y = A.sin(Bx+C) that satisfies all the three points for real A, B and C? P.S. No pair among ...
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2answers
23 views

Similarity between graphs of sin and tan inverse

Why is it that the graphs of tan inverse and sin in the interval $$\left[-\frac \pi 2 , \frac \pi 2\right]$$ are so similar. Is it just some coincidence or something deeper?
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2answers
37 views

Can I input negative angles into the cosine half-angle formula?

So the cosine half-angle formula says: Now, we know that co-terminal angles have equal cosines. Consider that $\cos (7\pi/4)$ = $\cos(-\pi/4)$. However, if you apply the half angle formula to $(7\pi/...
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0answers
12 views

Figuring out positions of some points given other known points and angles between the known and unknown points

So the data in question is a set of points p which all have known positions in space, a set of points q which all have unknown ...
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1answer
18 views

Circle with rotating line: Locate section on a tangent with known velocity in the section

I have a hard time phrasing this in the title but let me try to explain. You all probably know the demonstration graphics on the unit circle for trigonomic functions (look here for an example). Now I ...
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3answers
39 views

The lengths of the sides of a triangle are $\sin\alpha$, $\cos\alpha$ and $\sqrt{(1+\sin\alpha\cos\alpha)}$…

The lengths of the sides of a triangle are $\sin\alpha$, $\cos\alpha$ and $\sqrt{(1+\sin\alpha\cos\alpha)}$, where $0^o < \alpha < 90^o$. The measure of its greatest angle is....... What I have ...
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1answer
16 views

Find out the Coordinate of watch tower by coordinate geometry [on hold]

A watch tower will be built equidistant from both camp A and Camp B and near to Camp C. Find the coordinates of the watch tower. Camp A coordinate:(-600,200) ; Camp B coordinate: (300,500); Camp C ...
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1answer
33 views

How can I write y in terms of x for the equation given below? [on hold]

Here is the equation: $x=\frac{1}{c}\cdot \text{tanh}^{-1}(y)-b\text{i}\cdot \text{tanh}^{-1}(b\text{i}\cdot y)$ $b$ and $c$ constants and $\text{i}$ for imaginary.
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0answers
27 views

A Tricky Trigonometric Problem [on hold]

A radar dish is observed from 3 points A,B and C in a straight horizontal line. The tangents of the elevations are in the ratios : 6:3:2. Show that the distance of the dish from the point A is $sqrt{...
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3answers
174 views

Side length of a quadrilateral incribed on a circle

I've been doing math for 10 years now, yet every so often I get stumped by a "basic" high school question. This is one of those times. Here's the question: Part a is easy; we apply the cosine rule ...
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0answers
15 views

Chord of contact in polar coordinate

Can you please help me to derive the equation of chord of contact for a circle in polar coordinate? I have found the equation in cartesian coordinate but I cannot map that into the polar coordinate. ...
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1answer
31 views

What are the polar coordinates of $(2\sqrt3, 2)$?

My answer to this is $(4,\frac{π}6)$. But a calculator said that $(-4,\frac{7π}6)$ is also an answer, and there are infinitely many solutions. Is that correct?
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2answers
52 views

Solve tan(x)+cos(x)=1/2

Is it possible (not numerically) to find the $x$ such as: $$ tan(x)+cos(x)=1/2 $$ ? All my tries finishes in a 4 degree polynomial. By example, calling c = cos(x): $$ \frac{\sqrt{1-c^2}}{c}+c=\...
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0answers
27 views

Finding an inverse Laplace transform of an integral that involves the $\max\left\{0,\dots\right\}$ function

I'm trying to tackle a complicated real world (electronics) question. In order to get the last part of the proof I need to find the following 'difficult' integral: $$\mathcal{L}_\text{s}^{-1}\left[\...
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5answers
58 views

Minimum value of $\cos x+\sin x$ for $0 \le x \le 1$

What will be the minimum value of $$\cos x+\sin x$$ for $0\le x \le 1$? The answer is $1$. I tried finding it's minima, but there is none for critical point. Which other approach shall I try?
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2answers
35 views

How does this factorise?

I was completing a question in a booklet and marked it, but I don't understand how this expression: $$-2\sin 2x + 2\cos x = 0$$ Turns into this: $-2(\sin x\cos x + \sin x\cos x) + 2\cos x = 0$ ...
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2answers
21 views

Single variable calculus : Maximum rate of change : Trig functions

I have a calculus question which i will display here as an image: I am interested to understand part (b) of this question. I actually got the answer, but i feel i need more to understand how to ...
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4answers
97 views

What is $ \sin(x)+\sin(x−π)+\sin(x+π) $?

So I have this trig question: $ \sin(x)+\sin(x−π)+\sin(x+π) = $ _____ The answer is $- \sin(x)$ I can't figure out how to solve it. Any help?
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1answer
15 views

Calculating the 4th coordinate of a tetrahedron based on three others

I'm doing experiments where we have microscopic regular tetrahedrons moving around in a solution. We can detect and track the location of the 4 corners of the tetrahedron using some clever microscopy. ...
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2answers
26 views

Evaluate $\int \frac{ 2\exp\left((-\tan^2(t))/a^2\right) }{\cos^3(t)a^2}dt$ using substitution.

Evaluate : $\displaystyle\int \frac{ 2\exp\left((-\tan^2(t))/a^2\right) }{\cos^3(t)a^2}dt$ The hint was to use $x=\cos(t)$ and the fact that $\int f'(x)e^f(x)dx=e^f(x)$. Since $$x=\cos(t)\;\text{...
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0answers
22 views

Find the function given its entire graph, complete domain, full codomain and exact parts of its graph as other functions

I need to find a continuous function with its domain in the closed continuous interval $ [0-\frac{\pi}{2}] $. Its complete range or codomain is within closed continuous interval $ [1-\sqrt2] $. I also ...
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3answers
69 views

Nested trig functions (incl. inverse trig functions)

Edit: Although this problem has received a kind answer, I would still appreciate more comprehensive explanation. I am still rather confused. This problem has confused me a bit: The standard method ...
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2answers
63 views

Find the number of points in the region $x^{2}+y^{2}\leq4$ satisfying the equation $\tan^4x+\cot^4x+1=3\sin^2y$.

Given, $x^{2}+y^{2}\leq4$ $\tan^4x+\cot^4x+1=3\sin^2y$. It's a past problem of an UG entrance.I tried it solving using the graphical method,but couldn't.And also using trigonometric ...
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1answer
35 views

How to solve this trigonometric equation?…

How to find the value of x? $$\tan x=\frac{-x}{\sqrt{(2(3\pi/4)^2-x^2)}}$$
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1answer
32 views

Finding the particular solution to a trigonometric differential equation? [on hold]

This problem that I'm linking is the first in a long list of homework problems where we find the particular, complementary, and total solutions to differential equations. I'm familiar with how to find ...
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2answers
196 views

$\cos(\theta-\phi)=\frac{2ab}{a^2+b^2}$ where $a=\sin(\theta)+\cos(\phi)$ and $b=\cos(\theta)+\sin(\phi)$

I'm really stuck trying to answer this question and have spent endless hours doing so. If $a=\sin(\theta)+\cos(\phi)$ and $b=\cos(\theta)+\sin(\phi)$, prove that $\cos(\theta-\phi)=\frac{2ab}{a^2+b^2}...
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1answer
40 views

Understanding Trig equations

I understand why $\sin(x)=\frac{\sqrt{3}}{2}$ has two answers, $\frac{\pi}{3}$ and $\frac{2\pi}{3}$ but I don't understand why $\tan(x)=1$ only has one solution (according to my book and other places ...
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4answers
65 views

Methodology for Verifying Trigonometric Identities

Lets say I have an equation like: $$\frac{\sin^2x+\cos^2x}{\cos^2x\sec^2x}=1$$ Our teacher said you have to verify the equality by simplifying the left hand side or the right hand side without moving(...
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3answers
97 views

What is wrong with this proof of $3\arcsin x$?

We know that \begin{align*} (1)2\arcsin x&= \arcsin(2x\sqrt{1-x^2})\\ (2) \arcsin x + \arcsin y &= \arcsin[x\sqrt{1-y^2}+y\sqrt{1-x^2}]\\ (3) 3\arcsin x &= \arcsin x + 2\arcsin x \end{...
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1answer
61 views

$y-xy' =2(x+yy'), y(1) =1 $ is..

Question: $y-xy' =2(x+yy'), y(1) =1 $ is.. Firstly, I do not even know what the question is asking for. Secondly, Why does this question put $y(1)=1$? Isn't it obvious or is there any special ...
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3answers
190 views

Integrate $\int\frac{\sin^{-1} (x)}{(1-x^2)^{3/4}} \,\mathrm d x$

Integrate $$\int\frac{\sin^{-1} (x)}{(1-x^2)^{\frac{3}{4}}} \,\mathrm d x$$ I have followed some steps from here, but am not able to solve this question. Any help would be appreciated. Update: After ...
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4answers
56 views

How is the integral $2/π\int_{0}^{π} x^2\cos(nx) dx = \frac{4(-1)^n}{n^2}$?

I thought it would be this : $$2/π\int_{0}^{π} x^2\cos(nx) dx = 2/π\int_{0}^{π} x^2(-1)^n = 2/π(-1)^n\int_{0}^{π} x^2=\frac{2}{π(-1)^n}\biggl[\frac{x^3}{3}\biggr]_0^π =\frac{2(-1)^n}{3π^3}. $$ But it ...
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1answer
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Writing y value of Curtate Trochoid in the function of x?

The parametric equations of a trochoid are $x = Rt-d\sin(t)$ $y = R-d\cos(t)$ For $d < R$, there should be only one corresponding y value for every $x$ value. So can we express this equation as ...
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1answer
34 views

Triangles' relation to Tangent

I saw an equation on my book which replaced $\tan (B/2)$ with $\sqrt{ s(s - b) /(s - a) (s - c) }$ How are they related?
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0answers
23 views

Why convert Quaternion to Euler Angle

I've recently played with the IMU filter in MATLAB. When using their examples, they always plot the rotations by stating something alike this: ...
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1answer
43 views

In solving an inverse trigonometric equation is it sufficient to check for one case for the existence of a solution?

This has been bugging me for quite a while. $\arctan x + \arctan y$ is defined as $$f(x) = \begin{cases}\arctan\left(\dfrac{x+y}{1-xy}\right), &xy < 1 \\[1.5ex] \pi + \arctan\left(\dfrac{...
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3answers
31 views

If $\cos(n\pi)= (-1)^n$, then why is $-\frac{4}{n}\cos(n\pi) = \frac{4}{n}(-1)^{n+1}$?

If $\cos(n\pi)= (-1)^n$, then why is $-\frac{4}{n}\cos(n\pi) = \frac{4}{n}(-1)^{n+1}$? I lectures has this written down in one of the solutions to an exercise however I'm not sure how he got $\frac{4}...
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1answer
60 views

How was the system of $degrees$ devised? [on hold]

We are all familiar with the equivalence relationship between radians and degrees, $$1^c =\big(\frac{180}{\pi}\big)^o$$ I was wondering what else degrees are equivalent to. What is the basis of ...
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3answers
39 views

Show that the expressions $\sin^{-1}(\frac{1}{\sqrt{x}})$ and $\frac{1}{\sqrt{x}}$ are the same for big values

How can you show that the expressions $\sin^{-1}(\frac{1}{\sqrt{x}})$ and $\frac{1}{\sqrt{x}}$ for big values are the same? The opposite side of a triangle is given with $1/\sqrt(x)$, the angle ...
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1answer
32 views

How do you graph Sin of a double angle? [closed]

How does the double angle affect the sine graph, does it compress / stretch the graph, etc? eg. sin(2θ) = π/4
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3answers
68 views

Find the number of roots of the equation, $x^3 + x^2 +2x +\sin x = 0$ in $[-2\pi , 2\pi]$.

Find the number of roots of the equation, $$x^3 + x^2 +2x +\sin x = 0$$ in $[-2\pi , 2\pi]$. What I have tried: $$x^3 + x^2 +2x = -\sin x$$ $$x^2 +x +2 = \frac{-\sin x }{x}$$ $$(x + \frac{1}{2})^...
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0answers
26 views

Triangulation with 3 known points and time is involved?

I am completely stuck on how to visualize this problem, let alone code it. Understanding the mathematics behind it could help me out a lot. Thanks! Let's suppose that the unknown point is actually a ...