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

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16 views

How to find the period of $\cos(|\sin x|-|\cos x|)$?

My book did provide a rule as: If $f_1(x),f_2(x)$ are periodic functions with periods $T_1, T_2$ respectively, then we have $h(x)= f_1(x) + f_2(x)$ has period, as $\bullet$ LCM of $\{T_1, ...
4
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1answer
28 views

Integrating $\frac{x^3}{(81-x^2)^2}$

I've been trying to figure out this integral for an hour or so now, but keep failing. I can't figure out where I go wrong: $$I = \int \frac{x^3}{(81-x^2)^2} dx$$ Let $x = 9sin\theta \implies dx = 9 ...
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0answers
10 views

Trig substitution using reference triangles

Suppose we are doing a trig substitution and make some substition $x = a \sin \theta \equiv \sin \theta = \frac{x}{a}$ where the domain of x is $|x| \le a$ Then from the reference triangle we can ...
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0answers
27 views

Integrating trig substitution triangle equivalence

When we integrate certain integrals, such as $$\int \frac{x^2}{\sqrt{16-x^2}} dx$$ We can make a substitution like $x = 4 \sin \theta$ Then we can simplify the above integral to the following: $$8 ...
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2answers
67 views

How to find the period of $\cos(\cos\theta)$?

How to find the period of $f(\theta)=\cos(\cos\theta)$? For this, I've taken the easiest approach: Let $T$ be the least positive value for which the function is positive. Then $$f(\theta)= ...
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0answers
18 views

Polynomial function for arctan(tanx) [on hold]

What is the Equivalent polynomial function for arctan(tanx), arccos(cosx), arcsin(sinx)?
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1answer
19 views

Multiplicity in Solutions to Trig Function Equations

This is a very simple problem, but I can't figure out where I am going wrong! Say you have the following: $a \sin\theta + b \cos\theta = c. \tag{1}$ Now, this for example can be rewritten using: $R ...
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1answer
13 views

Determining North-South Line Via Non-digital Watch Method: Discussion on Background Theory

Read this recently (page 9). States that if you point the current hour hand at the sun, then the angle bisection between it and an imaginary line running through the 12 hour position will point south. ...
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3answers
31 views

Prove that $\tan x < \frac{4}{\pi}x,\forall x\in \left( 0;\frac{\pi}{4} \right)$

Prove that $$\tan x < \frac{4}{\pi}x,\forall x\in \left( 0;\frac{\pi}{4} \right)$$ I have known the solution that uses convex function. But I'd like another solution don't use it. :D
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0answers
11 views

What do I need to know before Trig?

I finished a summer course that covered Math 1 Honors and I want to enter Trigonometry/Algebra 2 this year. What do I need to know before entering this class (I haven't taken Geometry)?!
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1answer
17 views

Solve the following Trigonometric Equation

I am not sure what to do with this; $-\csc^2x + (\sqrt 2)\csc x \cot x = 0 \text{ between} (0, \pi)$ Do I convert to sine and cosine and then add the identities together?
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0answers
14 views

simplifying distance equation

The optimal angle for throwing a ball from a cliff is $$\sin \theta = \frac{1}{(2+ \beta)^{1/2}}$$ the original distance equation is $$ x = \cos \theta (\sin \theta + (\sin^{2} \theta + ...
3
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2answers
42 views

Why $\tan x>\sin x$ in this question?

The question asks me to prove the identity $\tan ^2x-\sin ^2x=\tan^2 x \sin^2 x$ and use this result to explain why $\tan x>\sin x$ for $0<x<90$ I've proved the identity and I can't explain ...
1
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1answer
19 views

Rewriting a trig function into a sum of exponential functions

Rewrite the function $2 + 4\sin(\pi t + \frac{\pi}{6})$ into a sum of exponential functions. By that I mean using Euler's formula $\sin(x) = \dfrac{e^{i\pi x} - e^{-i\pi x}}{2i}$. If it wasn't for ...
5
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1answer
32 views

Determining if a sum of trig function is periodic

$$2 + \sin(2\pi\cdot t) + 3\cos(3\pi\cdot t) - 5\sin(7t-\frac{\pi}{4})$$ Is there any manual, easy, way of knowing such a function is not periodic? I'd love to know if there's any method which ...
8
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4answers
658 views

Simple Trig Equations - Why is it Wrong to Cancel Trig Terms?

In the following problem, I first did it using a cancellation of $sin^2\theta$, working shown below, which gave the wrong answer. Having looked at the question again, I saw it could be solved by ...
1
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1answer
22 views

Finding the distance from a mirror

My friend sent me this problem, and my efforts to solve it have thus far been frustrated. I need some insight! Joe is 6 feet tall, and standing in front of a mirror that is at eye level, and is 3 ...
0
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2answers
13 views

how to find the other find circular functions using trigonometric identities. [on hold]

How can I find the other five circular functions of this problem using the trigonometric identities? $\tan x = -\frac{1}{2}$, $x$ is in Quadrant 2
2
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1answer
68 views

How do I simplify $\tan(\theta)\sin^2(\theta)$?

I am trying to simplify this trigonometric expression, so I can solve for theta. $$\tan(\theta)\sin^2(\theta) = k(q^2)/[4(L^2)mg]$$ I can't seem to simplify it down to one instance of theta despite ...
0
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0answers
27 views

Expected value of product of sinusoids

In the book Adaptive Signal Processing by Widrow, an equation (2.20) on page 23 is presented without proof as: $$E \left[ x_k x_{k-n} \right] = \frac{1}{N} \sum_{k=1} ^{N} \sin\left(\frac{2 ...
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3answers
216 views

Finding the definite integral of a trigonometric expression

Find the integral of $$ \int_0^{\frac{\pi}{2}}{{\sqrt{\sin(2\theta)}} \cdot \sin(\theta)d\theta}$$ I got $$I=\int_0^\frac{\pi}{4}{\sqrt{\sin(2\theta)} \cdot (\sin(\theta)+\cos(\theta))d\theta}$$ But, ...
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11answers
6k views

Why do both sine and cosine exist?

Cosine is just a change in the argument of sine, and vice versa. $$\sin(x+\pi/2)=\cos(x)$$ $$\cos(x-\pi/2)=\sin(x)$$ So why do we have both of them? Do they both exist simply for convenience in ...
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0answers
18 views

How are closed form solutions for eigenvalues constructed from sines and cosines?

For example a size N 3-point finite difference scheme has eigenvalues $\lambda_j = 2+cos(j\pi/(N+1))$. How is this determined? I know the Gershgorin circle theorem, but this is not what I am looking ...
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1answer
32 views

Trigonometric Problems [on hold]

$$X=A\sin(wt+L)\\ Y=B\sin(vt+Q)$$ Prove that, $$\frac{X^2}{A^2}+\frac{Y^2}{B^2}-2\left(\frac{XY}{AB}\right)\cos(L-Q)=\sin(L-Q)$$
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2answers
37 views

How do u prove this? [on hold]

How to solve? It is asked to prove $$ LHS=RHS $$ Please Which identity should I use and how to know which to use when $$ \cos (2A/1)-\sin (2A) = \cot(π/4-A) $$
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1answer
36 views

Prove that $\tan5 \theta = \frac {5\tan \theta -10 \tan ^3 \theta +\tan ^5 \theta} {1-10\tan ^2 \theta +5\tan ^4 \theta}$

As the title suggests, what is required to prove is that $$\tan5 \theta = \frac {5\tan \theta -10 \tan ^3 \theta +\tan ^5 \theta} {1-10\tan ^2 \theta +5\tan ^4 \theta}$$ I was looking back through my ...
0
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0answers
12 views

Single-statement Continuous Periodic function without trigonometry and complex numbers

Superseding the question Periodic function without trigonometry and complex numbers , I am now asking: Can I get a single-statement continuous periodic function without using trigonometric functions ...
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0answers
26 views

How to find the trigonometric functions? [on hold]

Knowing that $\tan a = -\frac 1 2$ and that $a$ is in Quadrant 2, find the other five circular functions ($\sin, \cos, \cot, \sec, \csc$) using the identities in trigonometry. Please help, thank you.
2
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2answers
58 views

Strange trigonometric proof.

I was trying to find out how to prove $$ \sin(A-\arcsin(0.3 \ \sin \ A)) \ \cdot \ \sin(A+\arcsin(0.3 \ \sin \ A)) \ = \ 0.91 \ \sin^2 \ A \ \ . $$ When I put this equation into my calculator both ...
2
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0answers
40 views

Is this expression for $x\pmod n$ interesting; nontrivial?

For example, we would get several interesting results if we had a formula for $x\pmod n$ that was uniformly convergent, however, according to Wikipedia (Floor and Ceiling Functions) these formulas do ...
0
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2answers
65 views

Why is Euler's formula a definition?

Even though there are proofs for Euler's formula for complex exponentials (see wikipedia for instance), it is mentioned as a "definition" in most textbooks. Why is that? My understanding is that a ...
6
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2answers
121 views

Find the limit analytically when the sine functions have square roots?

Find the limit analytically of the following: $\lim \limits_{x \to 0} \frac {\sin(\sqrt{2x})}{\sin(\sqrt{5x})} $ The closes thing we learned in class about this was that $\sin(x)$ over $x$ will ...
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2answers
38 views

Solving an equation with $\sin(x)$ in the exponent: $2^{\sin(x)} \cdot \cos(x) + 1 = 1$

Hi I need help with a trig problem: I have $2^{\sin(x)} \cdot \cos(x) + 1$, and I need this to equal $1$ between $x = -3$ and $3$. I keep going in circles with substitution, etc. Any help would be ...
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2answers
42 views

Does the identity ${|\cosh z|}^2={\cos}^2x+{\sinh}^2y$ given in my text hold?

In my text book I saw that ${|\cosh z|}^2={\cos}^2x+{\sinh}^2y$ But when I tried deriving it myself I got this: $${|\cosh z|}^2={\cos}^2y+{\sinh}^2x$$ See my working below: $$\cosh ...
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2answers
18 views

Impossible solutions in trigonometric equations

I'm trying to solve $\sin{4v} + \cos{4v} = 0$ I get 4 equations which I can solve for the solutions, including these 2: $4v_1 = \frac{\pi}{2} + 4v_1 + 2\pi n$ $4v_2 = -\frac{\pi}{2} + 4v_1 + 2\pi ...
0
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2answers
33 views

Find the solution set of the equation $5.(\frac{1}{25})^{\sin^2x}+4.5^{\cos2x}=25^{\frac{\sin2x}{2}}$

Problem : Find the solution set of the equation $5.(\frac{1}{25})^{\sin^2x}+4.5^{\cos2x}=25^{\frac{\sin2x}{2}}$ where $x \in [0,2\pi]$ My approach : ...
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5answers
54 views

cos(4v) + cos(v) = 0

I am given the following equation: $$\cos 4v + \cos v = 0$$ My attempt: $$\cos4v = -\cos v$$ $$\cos4v = \cos(\pi \pm v)$$ $$4v = \pm \pi \pm v + 2\pi n$$ $$4v_1 = \pi + v_1 + 2\pi n$$ ...
5
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4answers
98 views

Showing that $\left (\frac{\sin x}{x} \right )^3\geq \cos^{2}x$

Show that $$\left (\frac{\sin x}{x} \right )^3\geq \cos^{2}x,\forall x\in \left ( 0;\frac\pi2 \right )$$ Firstly, I had use the differentiation of $f(x)=\left (\frac{\sin x}{x} \right )^3- ...
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0answers
28 views

Epsilon-Delta Limit for Trigonometric Function

I'm studying an Epsilon-Delta proof for a trigonometric function: $$\lim_{x \to 1/9} \sin(x) = \sin(1/9)$$ This is the procedure from my (Italian) book: $$−\epsilon < \sin(x) − \sin(1/9) < ...
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0answers
28 views
+50

Prove that $\sin\theta_1.\sin\theta_2.\sin\theta_3=\frac{r^2_1}{16R^2}$

If $2\theta_1,2\theta_2,2\theta_3$ are the angles subtended by the circle escribed to the side $a$(opposite to vertex $A$) of a triangle at the centers of the inscribed triangle and the other two ...
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1answer
24 views

Finding cubed roots of complex number

Is this correct? $a^3 =r^3e^{i3\theta}= 5\sqrt{5}e^{i\arctan(11/2)}$ $$\implies r=\sqrt{5}, 3\theta = \arctan(11/2)+2\pi n,n\in\Bbb Z$$ $$\theta = \frac{\arctan(11/2)+2\pi n}{3}$$ $$\theta = ...
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2answers
17 views

$4\sin^2\frac{\theta}{2}.S=(n+1)\sin n\theta-n \sin (n+1)\theta$, and $4\sin^2\frac{\theta}{2}.C=-1+(n+1)\cos n\theta-n \cos (n+1)\theta$

If $S\equiv \sin\theta+2\sin2\theta+3\sin3\theta+......+n\sin n\theta$ and $C\equiv \cos\theta+2\cos2\theta+3\cos3\theta+......+n\cos n\theta$,prove that $4\sin^2\frac{\theta}{2}.S=(n+1)\sin ...
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0answers
17 views

can you illustrate this problem please? [on hold]

2 forest rangers observed a camp fire in the directions S60W and S66E from their stations. If the 2nd ranger was 2.76 miles due west of the 1st, which is the closer to the fire and how much closer is ...
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1answer
38 views

ASTC: Finding exact values of trigonometric functions

Our teacher showed us this really dodgy way of finding exact values by drawing up the 4 ASTC (all stations to central diagram) quadrants and making a right angle to the x axis. So how would I do a ...
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4answers
76 views

Prove $\sin^2(\theta)+\cos^4(\theta)=\cos^2(\theta)+\sin^4(\theta)$

Prove $\sin^2(\theta)+\cos^4(\theta)=\cos^2(\theta)+\sin^4(\theta)$. I only know how to solve using factoring and the basic trig identities, I do not know reduction or anything of the sort, please ...
3
votes
1answer
43 views

The Rhombohedron

I am trying to model a rhombohedron (using Blender) as a first pass to building Dürer's solid so I am trying to calculate the (x,y,z) values for a given side length 'a' and angle 'theta' (starting ...
4
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0answers
58 views

Prove $\cos(\sin x)>\sin(\cos x)$ [duplicate]

Prove that $\cos( \sin x)>\sin(\cos x), \forall x\in\mathbb{R}$. I have thought that we should consider their difference and show it is positive for all x, so: Let $$A=\cos\sin x-\sin\cos ...
-3
votes
1answer
18 views

Find the parameter a of function $y = 2\sin(\frac{\pi}{4}x+a)$ [on hold]

Find the parameter a of the function $y = 2\sin(\frac{\pi}{4}x+a)$ so that the corresponding trigonometric function would be even, and the value at point $x = 0$ positive. What is the fundamental ...
0
votes
0answers
25 views

Get coordinates to rotate a path around a circle JS (d3.js)

I'm trying to use the formula from this question Calculating the coordinates of a point on a circles circumference from the radius, an origin and the arc between the points to rotate a line around 180 ...
1
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2answers
63 views

trying to solve $\sqrt{\cos(x)-2\cos(2x)}+\sqrt{2}\cos(2x)=0$

The equation is $$\sqrt{\cos(x)-2\cos(2x)}+\sqrt{2}\cos(2x)=0$$ The system is $$ \begin{cases} \cos(x)-2\cos(2x)=2\cos^2(2x) \\ -\sqrt{2}\cos(2x)\ge 0 \iff \cos(2x)\le 0 \end{cases} $$ The ...