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

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68 views

$\sin 2x - \tan 2x = -\sin 2x\tan 2x$ trigonometric identity proof

I need to prove $$\sin 2x - \tan 2x = -\sin 2x\tan 2x$$ I tried simplifying $$ \sin 2x = 2\sin x\cos x;\quad \tan 2x = \frac{2\tan x}{1-\tan^2x}. $$ But it's so long and complicated that I ...
1
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2answers
54 views

Proving $\cos A \cdot \cos 2 A \cdot \cos 4 A \cdots \cos 2^{n-1} A = \frac{\sin 2^n A}{2^n \sin A}$

Just a bit of background on the question: When proving: $$\cos\frac{\pi}{15}\cdot \cos\frac{2\pi}{15} \cdot \cos\frac{3\pi}{15}\cdot \cos\frac{4\pi}{15} \cdot \cos\frac{5\pi}{15} \cdot \cos\frac{6\pi}...
1
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2answers
55 views

Find all solutions of $\left[\ln(\sin^{-1}(e^x))\right]^5=\ln(\sin^{-1}(e^x))$

The question is: Find all solutions of $\left[\ln(\sin^{-1}(e^x))\right]^5=\ln(\sin^{-1}(e^x))$, where $x$ is real. Give the solutions in exact form. What I have done $$\left[\ln(\sin^{-...
2
votes
4answers
76 views

$\arctan x=\frac{1}{2}i[\ln(1-ix)-\ln(1+ix)]$

In wikipedia it says, $$\arctan x=\frac{1}{2}i[\ln(1-ix)-\ln(1+ix)]$$ I want to now why is this true and what does a logarithm of a complex number even mean. I'm guessing that if I use the Taylor ...
2
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0answers
70 views

Can you solve a trig equation with a variable both inside a trig function and outside one?

I have the equation: $$d=\frac{t}{2}-\frac{sin(t)}{4}$$ I'm completely failing at how to get this in terms of $t$ I only care about it for values of $0<t<2\pi $ I've seen the graph so I know ...
0
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1answer
9 views

How is the graph of $cot(x)$ valid for negative values of $x$?

In the graph, consider a point between $\frac{-\pi}{2}$ and $-\pi$. We know that $cot(x) = \frac{cos(x)}{sin(x)}$. For negative values of $x$, i.e., $cos(-x)$ is always positive and $sin(-x)$ is ...
0
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3answers
74 views

Solve the equation on the interval $0 \le \theta\le2\pi $

I have this Final Math Exam Review, for Math Analysis/Trig = Pre-calculus. So I stumbled upon my review and this section arose where it told me to Solve the equation on the interval $0 \le \theta ...
0
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2answers
51 views

A limit question trigonometric function

Is there a solution to this question? I can't solve it. $$\lim_{n\to \infty} \frac{\sin(n^2x)}{n}$$
2
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1answer
65 views

How can the graph of $\cos(\sin x)$ be plotted?

Can anyone please discuss the procedure for drawing and analyzing graphs like $\cos(\sin x)$ and $\sin(\cos x)$?
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0answers
30 views

Trigonometry problem with complementary angles

$\tan(n.90 + (-1)^{n }.45)$ Find the integral value of the above expression My try I wrote it as ${\sin\over \cos}$ Then I used casework ie. $n = 2k+1$ or $2k$ But my case work leads to the ...
7
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3answers
226 views

Limit of the solutions of a trigonometric equation

Consider the equation \begin{equation} 2\cos(\sqrt{\lambda} \pi) \sin(\sqrt{\lambda} \frac{\pi}{n}) + \sin(\sqrt{\lambda} \pi) \cos(\sqrt{\lambda} \frac{\pi}{n})=0 \end{equation} for $n \in \mathbb{...
0
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1answer
26 views

How do I find angles in the interval, $0^\circ$ $\leq$ $\theta$ $\leq$ $360^\circ$, for $\cot \theta= -\frac{1}{\sqrt3}$ without using a calculator?

If I were to sketch the triangle for this problem on the cartesian plane, it would be in the fourth quadrant (since the side opposite to $\theta$ is negative). From the values of the opposite and ...
5
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3answers
134 views

Calculate $\int_0^{1/10}\sum_{k=0}^9 \frac{1}{\sqrt{1+(x+\frac{k}{10})^2}}dx$

How can we evaluate the following integral: $$\int_0^{1/10}\sum_{k=0}^9 \frac{1}{\sqrt{1+(x+\frac{k}{10})^2}}dx$$ I know basically how to calculate by using the substitution $x=\tan{\theta}...
0
votes
1answer
21 views

lines of bearing bounding a circle

diagram Given a circle of radius R, that is D units from the origin, I would like to find a method for finding the two lines of bearing from the origin that are tangent to to the circle. This is not ...
0
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1answer
35 views

Using Integration Techniques, how do you manipulate specific trig identities?

I've been working on this problem for the last couple hours, and I simply cannot figure out how to solve it. I've scoured the internet, checked the answer using symbolab and wolfram alpha, and yet I ...
1
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1answer
32 views

The convergence of an infinite radical involving $\cos(\alpha/3)$

By using the triple angle formula for the cosine, $\cos 3\alpha$, we get the cubic equation $ 4x^3-3x = \cos \alpha $. Now, by expressing $ x $ as $ x = \frac{1}{2}\sqrt{3+\frac{\cos \alpha}{x}}$ ...
0
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4answers
44 views

How would you show $\cos^2(x)\sin^3(x)=\frac{1}{16}(2\sin(x)+\sin(3x)-\sin(5x))$

$$\cos^2(x)\sin^3(x)=\frac{1}{16}(2\sin(x)+\sin(3x)-\sin(5x))$$ This is my try $$\cos(x)\sin(x) \cos(x) \sin^2(x))$$ $$\frac{1}{4}\sin(2x)\cos(x)\sin^2(x)$$ $$\frac 1 4\sin(2x) \frac 1 4\sin(2x)\...
13
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3answers
642 views

How would you find the exact roots of this equation?

My friend asked me what the roots of $y=x^3+x^2-2x-1$ was. I didn't really know and when I graphed it, it had no integer solutions. So I asked him what the answer was, and he said that the $3$ roots ...
3
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3answers
119 views

Evaluate $\int_{1/10}^{1/2}\left(\frac{\sin{x}-\sin{3x}+\sin{5x}}{\cos{x}+\cos{3x}+\cos{5x}}\right)^2dx$

How can I evaluate the following integral? $$\int_{1/10}^{1/2}\left(\frac{\sin{x}-\sin{3x}+\sin{5x}}{\cos{x}+\cos{3x}+\cos{5x}}\right)^2dx$$ I notice that the numerator and the denominator are ...
0
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1answer
38 views

Working out length of side of triangle?

I'm taking mooculus course from https://mooculus.osu.edu/exercises/linearTriangles1 and am given following problem : What is the intuition of the hint : 'length of DA = abscissa of D minus abscissa ...
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1answer
48 views

Proving ${\pi\over 2}=2\tan^{-1}\left({1\over A}\right)+\tan^{-1}\left({1\over B}\right)$

Let $A=2^{2^{-x}}$ and $B=2^{2^{-x}+1}(1+2^{2^{-1}})(1+2^{2^{-2}})\cdots(1+2^{2^{-x+1}})$ Showing $x\ge2$ $${\pi\over 2}=2\tan^{-1}\left({1\over A}\right)+\tan^{-1}\left({1\over B}\right)\tag1$$ ...
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0answers
31 views

How to solve the general case for a projectile that collides with any angled wall without vectors or dot products, just x and y components?

There are several formulas out there for figuring out the reflected angle of a projectile that collides with a stationary wall at any angle using vectors and dot products, I'm trying to solve it using ...
2
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2answers
67 views

Prove that $\sin x=[1+\sin x]+[1-\cos x]$ has no solution in $x\in \Bbb R$

Prove that $$\sin x=[1+\sin x]+[1-\cos x]$$ has no solution for $x\in \Bbb R$ where $[x]=\lfloor x\rfloor$ $$$$I reduced the equation into $$\sin x=2+[\sin x]+[-\cos x]$$ From here, I plotted ...
0
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2answers
85 views

$(\sin^{-1} x)+ (\cos^{-1} x)^3$

How do I find the least and maximum value of $(\sin^{-1} x)+ (\cos^{-1} x)^3$ ? I have tried the formula $(a+b)^3=a^3 + b^3 +3ab(a+b)$ , but seem to reach nowhere near ?
3
votes
2answers
63 views

Homeomorphism from $S^1\backslash(0,1)$ to $\mathbb{R}$

I am trying to derive a bijection between $S^1\backslash{(0,1)}$ and the real line, but I am stuck on using the most obvious way Let the top point of the circle be $(0,1)$, and the blue line hits ...
1
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2answers
33 views

Solving $\sin(ka/2)\cos(ka/2) = -\sin(ka)\cos(ka)$.

This is a basic high-school trigonometry problem, but it's been so long since I dealt with any of the trigonometric identities I am completely lost on how to solve it (I really should keep up with my ...
1
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3answers
117 views

How to integrate this type of fractions

Q: How to integrate this type of fractions $$\int^{\frac{\pi}{2}}_{\frac{\pi}{3}} \frac{\sin\frac{\theta}{2}}{1+\sin\frac{\theta}{2}} d\theta $$ What should I do here? I don't even know where to ...
8
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5answers
233 views

Compute $\int_0^{\pi/2}\frac{\cos{x}}{2-\sin{2x}}dx$

How can I evaluate the following integral? $$I=\int_0^{\pi/2}\frac{\cos{x}}{2-\sin{2x}}dx$$ I tried it with Wolfram Alpha, it gave me a numerical solution: $0.785398$. Although I immediately ...
3
votes
4answers
72 views

Why is the inverse tangent function not equivalent to the reciprocal of the tangent function?

I know that $$ {\tan}^2\theta = {\tan}\theta \cdot {\tan}\theta $$ So I guess the superscript on a trigonometric function is just like a normal superscript: $$ {\tan}^x\theta = {({\tan}\theta)}^{x} ...
2
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1answer
45 views

Trigonometry- Triangles

Let $A,B,C$ by the angles of a triangle. Then how to prove that $$a^3cos(B-C) + b^3cos(C-A) +c^3cos(A-B) = 3abc$$ I divided both sides by $abc$ and then tried to open the cosine function but ...
0
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1answer
29 views

Find the value of k for a given limit

If f(x)={ sin(2k - 3)x/4x x < 0 tan(3k - 4)x/2x x > 0 and lim x tending to zero f(x) exists ,then the value of k is given by ({ peicewise function) My work ...
0
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1answer
12 views

Continuous function to describe a shifted Gaussian curve

We have a set of numerical data that strictly follows Gaussian function (e.g. Fig. 1). Suppose if we shift the left half of the Gaussian curve to to its right end (see Fig. 2), trend lines based on ...
1
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5answers
74 views

Can someone show me how to simplify $\sin\left(\arccos\left(\frac{x}{x+1}\right)\right)$

I am needing help learning how to simplify the following equation: $$\sin\left(\arccos\left(\frac{x}{x+1}\right)\right)$$ Also any steps on how to get the answer would be greatly appreciated!
0
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4answers
85 views

Proof that $\frac{2\theta}{\pi} < \sin \theta < \theta$

The following inequality hold: if $\theta $ is in radians and $ 0 < \theta <\pi/2$, then $\frac{2\theta}{\pi} < \sin \theta < \theta$ How can be proved this inequality?
4
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6answers
137 views

Deriving power series for $\sin x$ without using Taylor's Theorem or $\exp z$

Starting with defining $(\cos t, \sin t)$ from the unit circle, is it possible to derive the power series for $\sin(t)$: $$\sin t = t - \frac{t^3}{3!} + \frac{t^5}{5!} - \dots$$ Note: I will be ...
2
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0answers
45 views

Shortest expression possible in terms of other variables

Given, $$b^2\cos^2\theta +a^2\cos^2\theta =a^2\cos^2\beta+a^2\sin^2\beta\cdot \sin^2\left(\cos^{-1}\left(\sqrt{\cos^2\theta +\frac{b^2}{a^2}\sin^2\theta }\right)\right)$$ what could be the shortest ...
2
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2answers
151 views

A very little known approximation for the lesser angle of a triangle

In the article A note on an Approximation in Trigonometry is proved a very interesting approximattion to the lesser angle of a triangle (in degrees): $ (1) A \approx \frac{344\Delta}{2s(s-a)+bc}$ ...
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2answers
40 views

Can we convert polar to rectangular when we are given $(1,\theta )$ where $ r=1$ and $0\le \theta <2\pi $?

Can we convert polar to rectangular when we are given $(1,\theta )$ where $ r=1$ and $0\le \theta <2\pi $?
0
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1answer
31 views

Trigonometric problem regarding a tower

The angle of elevation of a tower, $CD$, from a point $A$ due East of the tower is 45°. From a point $B$ due south of $A$, the angle of elevation is 30°. The distance from $A$ and $B$ is 400 metres. ...
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4answers
92 views

Is $\arctan(y/x) > xy/(x^{2}+y^{2})$ true for positive $x,y$?

I am trying to prove the following: $$\arctan\frac{y}{x}>\frac{xy}{x^{2}+y^{2}},\quad\forall x,y>0$$ Is the statement true, and if so how do you prove it?
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4answers
30 views

value of an inverse trignometric expression

How can we find the value of $ 3\sin(\frac12\arccos\frac19)+ 4\cos(\frac12\arccos\frac18)$ ? Substituting A = $\arccos\frac19$ My approach to this question.. I tried to use the formula $\cos A = \...
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0answers
30 views

Evaluate $x^2\left(qy-rz\right)+y^2\left(rz-px\right)+z^2\left(px-qy\right)$, where variables are sines and cosines of $\alpha+2k\pi/3$

Let $x$, $y$, $z$ be the sines of $\alpha$, $\beta$, $\gamma$, and let $p$, $q$, $r$ be the respective cosines, where the angles are in Arithmetic Progression with common difference $\frac{2\pi}{3}$. ...
1
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4answers
80 views

Prove that for all $a > 0$: $\int_0^{\pi/2}e^{-a\cos x}\cos(a\sin x)dx = \frac{\pi}{2} - \int_0^a\frac{\sin x}{x}dx$

Prove that for all $a > 0$: $$\int\limits_0^{\pi/2}e^{-a\cos x}\cos(a\sin x)dx = \cfrac{\pi}{2} - \int\limits_0^a\cfrac{\sin x}{x}dx$$ I have no idea how to solve it. But the task looks very ...
1
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0answers
24 views

When to substitute with trigo-/hyperbolic-function [closed]

I try to figure out, what indicators could look like to decide if substitution with trigonometrical function or substitution with hyperbolic function works/ works better to integrate a function. Is ...
0
votes
1answer
40 views

Converting parametric function into cartesian

I am trying to convert the parametric function $x(t) = a\cdot(t - \sin(t)) + b\cdot\cos\left(\frac{t}{2}\right)$ $y(t) = a\cdot\cos(t) + b\cdot\sin\left(\frac{t}{2}\right)$ into a cartesian form. ...
0
votes
1answer
45 views

Intersection of a line with a Sine function

A sine function is given as $f(x) = 40 + 25.\sin(\pi(x-1.5))$ domain $[0,6]$. I am asked to find a solution for $f(x) < 30$. I know that this is easily solvable with a solver such as in Excel but I ...
1
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3answers
59 views

Taking the limit $\lim_{x \to \infty} x \tan(1/x)$ without L'Hopital's Rule

I have to evaluate the following without L'Hopital's rule $$\lim_{x\to\infty} x\tan(1/x)$$ I can simplify this to be $$\lim_{x\to\infty} x\sin(1/x)$$ because $$\lim_{x\to\infty} \cos(1/x) = 1$$ ...
2
votes
1answer
58 views

Solution of $1+\frac{\cos 2x}{\sin x}+\tan x \geq 30$

Solve the given inequality: $$1+\frac{\cos 2x}{\sin x}+\tan x \geq 30$$ I am trying to convert L.H.S. in terms of one trigonometric ratio but it is not happening here. Could someone suggest some ...
1
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2answers
47 views

Confused About Equation Rearrangment

Full disclosure; I have a very very basic understanding of higher math. I apologize if this is stupid / has been answered. I don't really know how to even format my question. I have a formula used in ...
1
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2answers
67 views

Evaluating the indefinite integral $\int\frac{x^2}{\sqrt{4-x^2}}dx$

I think everything I have done is kosher, but unless I am missing an identity it is a different answer than the online quiz and wolfram alpha give. I tried to use the trig substitution $$ x=2\sin(\...