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

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Find the tan A if the triangle is inside the square?

ABCD is a square. The problem asks for me to find $\tan(\angle QAP)$ if I am given the fact that $CP = CQ = \frac{AB}{4}$. This is what I have so far: I drew a line from $Q$ to $P$ to make ...
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1answer
42 views

Check my work on a problem involving Law of Cosines?

The problem is this: Jane walks North for 3 miles, then turns $45^\circ$ to the right. After that, she walks another 4 miles. How many miles will Jane be from her starting point? Give your answer ...
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68 views

Prove that $\frac{1}{90} \sum_{n=1}^{90} 2n \cdot \sin((2n)^\circ) = \cot (1^{\circ})$

Show that $$\frac{(2\sin(2^\circ)) + (4\sin(4^\circ))+ (6\sin(6^\circ)) + \ldots +(180\sin(180^\circ))}{90} = \cot(1^\circ).$$ I used a lot of steps, and typing it all down on here would take me an ...
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1answer
80 views

simplify cos 1 degree + cos 3 degree +…+cos 43 degree?

I am currently working on a problem and reduced part of the equations down to $\cos(1^\circ)+\cos(3^\circ)+.....+\cos(39^\circ)+\cos(41^\circ)+\cos(43^\circ)$ How can I calculate this easily using ...
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1answer
27 views

check my work on this problem: given tan(2x), find sin x + cos x?

$\tan 2x = - 24/7$ $90^\circ < x < 180^\circ$. Find the value of $\sin x+\cos x$. What I have so far: $\tan(2x) = -\frac{24}{7} \Rightarrow \frac{2\tan(x)}{1-\tan^{2}x} = -\frac{24}{7}$. ...
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2answers
73 views

Prove that $\cos(2a) + \cos(2b) + \cos(2c) \geq -\frac{3}{2}$ for angles of a triangle

Let the three internal angles of a triangle are $a,b,c$. Prove that $$\cos(2a) + \cos(2b) + \cos(2c) \geq -\frac{3}{2}.$$ I'm looking for an elementary, geometric proof. So avoid derivatives and ...
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1answer
32 views

Determine the range of f(x)=(sinx)/x

I am having trouble understanding the solution to this question. ''Determine the range of the following function: $f(x)$ = $(1$ $if$ $x=0)$ or (${\sin x\over x}$ if $x$$\neq$$0$) where the domain ...
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2answers
37 views

How to decompose this matrix exponential?

I would some help with the steps to decompose the below matrix exponential. $\exp\left[ \zeta \left ( \begin{matrix} -\cos(x) & i \sin(x) \\ -i \sin(x) & \cos(x) \end{matrix} \right ) ...
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24 views

System of Trigonometric Equations

Could someone please help me with the following system of equations
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2answers
155 views

How can I improve my explanation of why the ratio $\pi=\frac{C}{d}$ holds for all circles?

I'm trying to informally explain why $\pi$ holds for all circles. I would like to know if there is anything pertinent that I can add, or that is wrong with this explanation. It's an explanation, not ...
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4answers
67 views

Trigonometric function problem

Given: $f(x)=2\arctan(x) +\arcsin(2x/(1+x^2))$ prove that for every $x \ge 1, f(x)=\pi.$ any idea how to approach this question? thanks
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28 views

Singularities and trigonometric functions

For $f(z)=tan(z)/z$, I have found the singularities to be $z=0, z=\pi/2+2k\pi, z=3\pi/2+2k\pi$. k is an integer. I am trying to find the removable singularities. I have shown z=0 is a removable ...
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2answers
56 views

Simple Trigonometric Equation

If $\cos 2\theta=(\sqrt2+1)(\cos\theta-1/\sqrt2)$, then what is the value of $\theta$ ? I don't know how to solve this question. can someone tell me the required steps to solve this question.
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1answer
28 views

Find the magnitude of the $x$ and $y$ components to the nearest whole number?

Find the magnitude of the $x$ and $y$ components of $V$ to the nearest whole number: $||V||=27$ and $V$ has a direction of $60°$ $||V||=12$ and $V$ has a direction of $107°$ Please help.
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0answers
35 views

Will it clear the fence? Projectile motion

I'm probably being very dense, but I'm having a lot of trouble with this. The top of a vertical tower is 20m above ground level. When a ball is thrown horizontally from the top of this tower, it ...
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2answers
82 views

How do we make sense of angles which take irrational measures such as $\sqrt 2 ^\circ$?

If you were asked to draw such an angle how would you do so? Would you take it to a limit? Can the degree measure take the value of all real numbers?
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2answers
41 views

Integrate via substitution and derivation rule

i have to solve this integral $$\int_{-r}^{+r}\int_{-\sqrt{r^2-x^2}}^{+\sqrt{r^2-x^2}} \sqrt{1-\frac{x^2+y^2}{x^2+y^2-r^2}} \operatorname d y \operatorname d x$$ with substitution and then the ...
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2answers
39 views

Why is this trig function never undefined

$$f(x) = \frac{\left(\cos x\space +\space 0.5\right)}{\left(1\space +\space 0.5\cos x\right)^2}$$ Looking at the graph I know that the function is never undefined, but how would I show this or ...
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2answers
67 views

Atan2 Faster Approximation

I am using atan2(y, x) for finding the polar angle from the x-axis and a vector which contains the point (x,y) for converting Cartesian coordinates to polar coordinates. But, in my program which will ...
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4answers
133 views

Number of iterations to reach cosine's fixed point

I was messing around with my calculator the other day when I saw something interesting happen. Whenever I repetitively took the cosine of any number, it always ended up on a particular number ...
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1answer
51 views

Calculate Angle between Two Intersecting Line Segments

Need some help/direction, haven't had trig in several decades. On a 2 dimensional grid, I have two line segments. The first line Segment always starts at the origin (0,0), and extends to (1,0) along ...
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3answers
149 views

Is there a reason of $\cos(11x)+\sin(11(x+1))\approx 0$

Is there a reason of $$\cos(11x)+\sin(11(x+1))\approx 0$$
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4answers
93 views

Why are angles in “degrees” dimensionless?

Angles measured in radians are considered to be dimensionless because the radian measure of angles is defined as the ratio of two lengths $\theta=\frac{s}{r}$ (where $s$ is some arc measuring ...
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1answer
39 views

Solve for r where $\tan^{-1}\frac{60}{r} = \sin^{-1}\frac{60}{r - 10}$?

I'm trying to find the value of r where $$\tan^{-1}\frac{60}{r} = \sin^{-1}\frac{60}{r - 10}$$ It's taken me a few hours to get to this point (my trig skills are pretty bad), and I'm not sure where ...
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1answer
51 views

Help Figuring out Period of Trig Function

I'm working on a javascript project for hobby that I'm trying to get some neat effects working in. The issue is its been a long time since I was in trig and I'm having a hard time grasping exactly how ...
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1answer
56 views

Find $r$ knowing that $r=\frac{60}{\sin^{-1}\frac{60}{r}}$

I'm trying to find the value of $r$ knowing that: $$r=\frac{60}{\sin^{-1}\frac{60}{r}}$$ I'm not really sure how to approach finding the solution. Can anyone help me out? I've spent well over an ...
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1answer
209 views

Exact arctangent of product of tangents

Calculate $x$, if $$\tan(x)=\tan9\tan69\tan33$$ (Using sexagesimal degrees) Since $\tan3x=\tan(60-x)\tan x \tan(60+x)$: \begin{align*} \tan27&=\tan69\tan9\tan51\\ ...
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0answers
19 views

Proving that sum of (possibly) phase shifted sinusoids of frequency $w$ gives a sine wave with that frequency

I've been trying to get a proof of it WITHOUT using Euler's formula. The only thing I found was this link. The proof itself is quite long for a theorem it proves, I got a feeling it could be done in a ...
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1answer
86 views

Evaluate $\int \frac{\mathrm dx}{1+\cos^2 x}$

$$\int \frac{1}{1+\cos ^2x} \,\mathrm dx$$ I have to integrate the expression above: I tried with substitutions $\cos x=t$ and $1+(\cos x)^2=t$, but those didn't work, and I couldn't find any useful ...
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2answers
419 views

Can both $x$ and $\sin(x)$ be rational at the same time?

Except, of course, trivial $x=0$ case ($\sin0=0$); $x$ is measured in radians. The question turned out to be more complicated than it seemed to me at the first sight. All I came up with, that posed ...
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1answer
21 views

How to solve this integral with trigonometric functions?

How can I compute this integral manually? $\int_{1}^{t} sin2(t-\tau) cos2\tau d\tau$ I've tried some substitutions, trigonometric manipulations, but still cannot reach a reasonable next step. Any ...
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2answers
33 views

Squeeze Theorem Question

My question is from this Video In the last example He says that $$\lim_{x \to 0} x^2 \cos(\frac{1}{x^2}) = 0$$ Squeeze Theorem: $$g(x) \leq f(x) \leq h(x)$$ Given: $$-1 \leq \cos(x) \leq 1$$ he ...
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2answers
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Having trouble with application

We are doing application problems in trigonometry and I am having trouble drawing a sketch from the words provided The question is: A captain knows that his ship is due south of a lighthouse. His ...
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2answers
31 views

Confusion to use formula $l = r\theta$

I have been teaching my brother some trignometry. There is a formula as arc length of circumference of a circle. The basic formula is $$l = r\theta.$$ But sometimes for length they use $l = 2r$ and ...
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0answers
32 views

Infinity and the complex infinity

What's the difference between infinity and the complex infinity, and why is $\tan 90^{\circ}$, according to Wolfram Alpha, equal to the complex infinity and not undefined? Please see the following ...
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2answers
69 views

Simulating simultaneous rotation of an object about a fixed origin given limited resources.

Sorry if the title is a bit cryptic. It's the best I could come up with. First of all, this question is related to another question I posted here, but that question wasn't posed correctly and ended ...
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2answers
51 views

Trigonometric inequality $\frac{\cos x -\tan^2(x/2)}{e^{1/(1+\cos x)}}>0$

How can I solve the following inequality? $$\frac{\cos x -\tan^2(x/2)}{e^{1/(1+\cos x)}}>0$$
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2answers
47 views

Describing a point trigonometrically

Let us show that the transformation that reflects a point through a line through the origin is linear. This is the transformation that takes a point on one side of the line and moves it ...
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2answers
59 views

How to evaluate $\cos(22^\circ)\cos(38^\circ) - \sin(22^\circ)\sin(38^\circ)$?

How does one evaluate this? Does this generalize to $\cos(x)\cos(y) - \sin(x)\sin(y)$?
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1answer
26 views

Representing the x-component of a circular arc.

This question stemmed from my attempts to brush up on the physics of electricity and magnetism after being away from school a long while (specifically a problem related to a uniform charge across a ...
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1answer
123 views

Is there any identity for $\sum_{k=0}^{n-1}\tan(x+ka) $??

I found this series $$ \sum_{k=0}^{n-1}\tan\left(\theta+\frac{k\pi}{n}\right)=−n\cot\left(\frac{n\pi}{2}+n\theta\right) $$ but it's not what I need.
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1answer
55 views

Limit of a trigonometric sequence [duplicate]

For an arbitrary $x_{0}$ in $\left(\, 0,\pi\,\right)$ we define $x_{n + 1}=\sin\left(\, x_{n}\,\right)$. Using the limit of the sequence as $n$ tends to infinity we're supposed to find the limit of ...
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2answers
78 views

Help on proving a trigonometric identity involving cot and half angles

Prove: $\cot\frac{x+y}{2}=-\left(\frac{\sin x-\sin y}{\cos x-\cos y}\right)$. My original idea was to do this: $\cot\frac{x+y}{2}$ = $\frac{\cos\frac{x+y}{2}}{\sin\frac{x+y}{2}}$, then substitute in ...
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1answer
15 views

calculate angle for equilibrium

please see the image below for an understanding of the question i have found the tension in the string to be 57.5N and the acceleration of the system to be -1.7m/s^2 when the angle is 30deg ...
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2answers
89 views

Solving $\tan x-\tan(2x)=2\sqrt{3}$

$$\tan x-\tan(2x)=2\sqrt{3}$$ TRY #1 $$\begin{align*} \tan x-\tan(2x)=2\sqrt{3}&\implies\tan x=2\sqrt{3}+\tan{2x}\\ &\implies \tan^2x=\tan^2(2 x)+4 \sqrt{3} \tan(2 x)+12\\ ...
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2answers
23 views

Find the complementary and supplementary angles of $48^{\circ}21'12''$.

Find the complementary and supplementary angles of $48^{\circ}21'12''$. Please help me I do not know how to do addition and subtraction in DMS system. Thanks
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2answers
31 views

How to find the phase shift of this cos graph?

This graph is supposed to be of form $a cos(bx+c)+d$. I'm pretty sure that $a$, the amplitude, is $|2|$ and $b$, the period ($\frac{2\pi}{b}$), is $\frac{2}{3}$ (though some confirmation would be ...
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1answer
57 views

Rotating an object correctly when you can only rotate world axis.

This question may be useful to some people, but it is not posed correctly for my particular situation, please see: Simulating simultaneous rotation of an object about a fixed origin given limited ...
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How to solve sum of sines and cosines system of equations?

I have a set of equations to solve which in the following form: $ \cos(t_1 + t_2 + t_3 + t_4) + \sin(t_1 + t_2 - t_3 + t_4) + \cos(t_1 - t_4 + t_3 - t_5) + \sin(t_1 - t_2 + t_3 - t_5) + \cos(t_1 + ...