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

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11 views

Finding the root mean square of a sum of trig functions

$$v(t) = 3 - 2\sin(t) + 8\sin^2(t)$$ To find the rms of this function, I first figured out that the period $T = 2\pi$. I then set up the equation: $V = \sqrt{\frac{1}{T}\int^T_0v^2(t)\,dt}$ ...
0
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1answer
24 views

Is my explanation correct regarding Maximum value of Sine function over $\Bbb C$?

Question: What is the maximum value of sine function taking domain as $\Bbb C$? My answer is: The maximum value is not defined. Explanation: Since the range of sine function is $\Bbb C$ and $\Bbb C$ ...
1
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3answers
28 views

Show that $\csc^n\frac{A}{2}+\csc^n\frac{B}{2}+\csc^n\frac{C}{2}$ has the minimum value $3.2^n$

Show that in a $\Delta ABC$, $\sin\frac{A}{2}\leq\frac{a}{b+c}$ Hence or otherwise show that $\csc^n\frac{A}{2}+\csc^n\frac{B}{2}+\csc^n\frac{C}{2}$ has the minimum value $3.2^n$ for all $n\geq1$. ...
5
votes
3answers
38 views

Prove that $3x-x^3<\frac2{\sin2x}$

Prove that $$3x-x^3<\frac2{\sin2x},\forall x\in\left(0,\frac\pi2\right)$$ I have tried by proving that $$3x-x^3<\frac9{5\pi}x+\frac32<\frac2{\sin2x},\forall ...
2
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2answers
24 views

If the circumcenter of the triangle $ABC$ is on the incircle of the triangle,then prove that $\cos A+\cos B+\cos C=\sqrt2$

If the circumcenter of the triangle $ABC$ is on the incircle of the triangle,then prove that $\cos A+\cos B+\cos C=\sqrt2$ How should i attempt this question?I thought over it hard but could not ...
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1answer
19 views

Which is the justification for this indefinite integral relation?

Why is the following indefinite integral equation correct: $$ \int \frac{\cot(x)}{\sin^2(x)} dx= -\frac{1}{2}\cot^2(x) $$ What are the necessary steps?
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2answers
30 views

Inverse functions: what is the difference between $\tan^{-1}(x)$ and $\tan(x)^{-1}$?

I’ve never really been taught about inverse functions, and I figured this is a pretty simple question, but I couldn’t find any explanation in my math textbook about this. What is the difference ...
2
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4answers
131 views

How to solve $2 \tan x / (1 - (\tan x)^2) = (\sin 2x)^2$?

$$\frac {2\tan {x}}{1-(\tan {x})^2} = (\sin {2x})^2$$ I tried a lot but I get nowhere
2
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0answers
27 views

Geometry/Trigonometry Determine angle in a Triangle

Triangle ABC is isosceles with BC as base, AB=AC and Angle A=20 degrees. Points D and E lie on sides AB and AC respectively, such that D lies between A and B, E lies between A and C, angle BCD=50 ...
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0answers
26 views

Trignometry, bearings yacht race question

In a yacht race each yacht has to sail around a set of 4 buoys, and then return to the start line in order to finish. We will assume that the buoys are just points and the start line is also a ...
2
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0answers
30 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
36 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 ...
3
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0answers
14 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 ...
0
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3answers
39 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 ...
1
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2answers
72 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)= ...
0
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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
21 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
14 views

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

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
37 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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1answer
20 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
16 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 + ...
4
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2answers
60 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
20 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
votes
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
votes
4answers
665 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
votes
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
69 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 ...
6
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3answers
217 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, ...
55
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11answers
7k 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
33 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 $$ \cos (2A/1)-\sin (2A) = \cot(π/4-A) $$
1
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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 ...
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2answers
67 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 ...
0
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2answers
39 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 ...
3
votes
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 ...
0
votes
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 ...
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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
votes
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- ...
0
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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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votes
0answers
29 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 ...
1
vote
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 = ...