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

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3
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2answers
127 views

Formula obtained by using Trignometric approximation for a triangle with a very small side

I am reading a paper on the force between hooft polyakov monopoles, but I am completely baffled by one of the 'elementary trignometric' equation they have got using an approximation. Consider a ...
0
votes
2answers
2k views

How to calculate the displacement between points?

I'm having trouble finding the right formula for displacement between two points. I'm working on a program that will place a digital ruler and allow the user to trace their finger on the edge of the ...
3
votes
5answers
281 views

Solve for $x$; $\tan x+\sec x=2\cos x;-\infty\lt x\lt\infty$

Solve for $x$; $\tan x+\sec x=2\cos x;-\infty\lt x\lt\infty$ $$\tan x+\sec x=2\cos x$$ $$\left(\dfrac{\sin x}{\cos x}\right)+\left(\dfrac{1}{\cos x}\right)=2\cos x$$ $$\left(\dfrac{\sin ...
4
votes
2answers
171 views

Solve for $x$; $\cos^2x-\sin^2x=\sin x; -\pi\lt x\leq\pi$

Solve for $x$; $\cos^2x-\sin^2x=\sin x; -\pi\lt x\leq\pi$ $$\cos^2x-\sin^2x=\sin$$ Edit $$1-\sin^2x-\sin^2x=\sin x$$ $$2\sin^2 x+\sin x-1=0$$ $\sin x=a$ $$2a^2+a-1=0$$ $$(a+1)(2a-1)=0$$ ...
5
votes
3answers
92 views

Prove $\frac{\sin A \cos A}{\cos^2 A - \sin^2 A} = \frac{\tan A}{1-\tan^2 A}$

How would I simplify this difficult trigonometric identity: $$\frac{\sin A \cos A}{\cos^2 A - \sin^2 A} = \frac{\tan A}{1-\tan^2 A}.$$ I am not exactly sure what to do. I simplified the right side ...
4
votes
3answers
249 views

$3\sin^2x=\cos^2x;$ $ 0\leq x\leq 2\pi$ Solve for $x$

$3\sin^2x=\cos^2x;$ $0\leq x\leq 2\pi$ Solve for $x$: I honestly have no idea how to start this. Considering I'm going to get a number, I am clueless. I have learned about $\sin$ and $\cos$ but I ...
3
votes
3answers
201 views

Prove $\frac{\sec{A}+\csc{A}}{\tan{A} + \cot{A}} = \sin{A} + \cos{A}$ and $\cot{A} + \frac{\sin{A}}{1 + \cos{A}} = \csc{A}$

Can anyone help me solve the following trig equations. $$\frac{\sec{A}+\csc{A}}{\tan{A} + \cot{A}} = \sin{A} + \cos{A}$$ My work thus far ...
4
votes
1answer
246 views

Solve $\ddot\theta +k\sin(2\theta)=0$ given initial value and constraints

How is it possible to deduce from the equation $$\ddot\theta +k\sin(2\theta)=0$$ where $\theta=\theta(t)$ and $\tan(\theta)={b(t)\over a(t)}$, $k$ is constant, and $a(0)=a_0$, $a(t)^2+ b(t)^2=a_0^2$. ...
2
votes
1answer
75 views

$k$ in trigonometric equality $\sin(a) =\sin(b)$

On a test there is the question: "Solve for $x$ on the interval $[-\pi,\pi]$ where $\sin(2x) = \cos(3x)$ I know that: $\cos(x) = \sin(\frac12\pi - x)$ So you can rewrite the equation to: ...
3
votes
3answers
2k views

Is there any way to find a angle of a complex number without a calculator?

Transforming the complex number $z=-\sqrt{3}+3i$ into polar form will bring me to the problem to solve this two equations to find the angle $\phi$: $\cos{\phi}=\frac{\Re z}{|z|}$ and ...
0
votes
5answers
177 views

Verify trigonometry equation $\frac{\sin A+\tan A}{\cot A+\csc A}=\sin A \tan A$

Sorry for asking so many of these type of questions. How would I verify the following trigonometry identity: $$\frac{\sin A+\tan A}{\cot A+\csc A}=\sin A \tan A.$$ My work is $$\frac{\sin A + ...
1
vote
2answers
72 views

Verify trigonometry equation $\frac{\sin(A)}{\sin(A) + \cos(A)}=\frac{\sec(A)}{\sec(A)+\cos(A)}$

How would I verify the following trig equation? $$\frac{\sin(A)}{\sin(A) + \cos(A)}=\frac{\sec(A)}{\sec(A)+\cos(A)}$$ My work so far is to write the RHS as $$\frac{1/\cos(A)}{1/\cos(A) + \cos(A)}$$ ...
3
votes
5answers
300 views

Verify trigonometry equation $\tan A - \csc A \sec A (1-2\cos^2 A)= \cot A$

How would I verify the following trigonometry identity? $$\tan A - \csc A \sec A (1-2\cos^2 A)= \cot A$$ My work so far is $$\frac{\sin A}{\cos A}-\frac{1}{\sin A}\frac{1}{\cos A}(1- \cos^2 A- ...
4
votes
1answer
1k views

Remembering exact sine cosine and tangent values?

There exists a common trick to remember exact sine cosine and tangent values. The trick is relatively long, so instead of reposting it, please refer to my answer on this page. Although I have used ...
1
vote
3answers
492 views

Use equalities to derive important trigonometric functions

The trigonometric functions I must know: (A) $\sin(-x)=-\sin x$ (B) $\cos(-x)=\cos x$ (C) $\cos(x+y)=\cos x\cos y-\sin y\sin x$ (D) $\sin(x+y)=\sin x\cos y+\cos x\sin y$ $\sin^2x+\cos^2x=1$ (Use ...
2
votes
3answers
2k views

$(\sin\theta+\cos\theta)^2=1+\sin2\theta$

49) $(\sin\theta+\cos\theta)^2=1+\sin2\theta$ Left Side: \begin{align*} (\sin\theta+\cos\theta)^2=\sin^2\theta+2c\cos\theta\sin\theta+cos^2\theta=1+2\cos\theta\sin\theta \end{align*} This can ...
1
vote
2answers
87 views

Trigonometric Identities To Prove

$\tan\theta+\cot\theta=\dfrac{2}{\sin2\theta}$ Left Side: $$\begin{align*} ...
1
vote
1answer
70 views

Trigonometric Identities

$\dfrac{\sin^2\theta}{1+\cos\theta}=1-\cos\theta$ Right Side: $1-\cos\theta$ either stays the same, or can be $1-\dfrac{1}{\sec\theta}$ Left Side: $$\begin{align*} &= ...
0
votes
1answer
89 views

Verifying some trigonometric identities

Prove the following: 46. $\dfrac{\csc\theta}{\cot\theta}-\dfrac{\cot\theta}{\csc\theta}=\tan\theta\sin\theta$ I got as far as Right Side: $\tan\theta\sin\theta$ to ...
3
votes
2answers
796 views

Solving $\cos^2 \theta + \cos \theta = 2$

Solve the following for $\theta$: $\cos^2 \theta + \cos \theta = 2$ [Hint: There is only one solution.] I started this out by changing $\cos^2\theta$ to ...
0
votes
1answer
231 views

How to decide if a given function is even, odd, periodic, or one-to-one? [closed]

I have a number of exercises (some examples below) that ask to determine some properties of functions: Symmetry with respect to y-axis or origin. (I know how to make a graph, but for some graphs, ...
7
votes
5answers
470 views

A little integration paradox

The following integral can be obtained using the online Wolfram integrator $$ \int \frac{dx}{1+\cos^2 x} = \frac{\tan^{-1}(\frac{\tan x}{\sqrt{2}})}{\sqrt{2}}$$ Now assume we are performing this ...
2
votes
3answers
111 views

How to verify this trigonometric identity?

I am having trouble doing this identity. $$\frac{\cos{A}\cot{A}-\sin{A}\tan{A}}{\csc{A}-\sec{A}} \equiv 1+\cos A\sin A$$ I am stuck I simplified it to. $$\frac{\cos^{2}{A}\div\sin{A} ...
2
votes
2answers
189 views

Verify trigonometric equation $\frac{(\sec{A}-\csc{A})}{(\sec A+\csc A)}=\frac{(\tan A-1)}{(\tan A+1)}$

How Would I verify the following identity. $$\frac{(\sec{A}-\csc{A})}{(\sec A+\csc A)}=\frac{(\tan A-1)}{(\tan A+1)}$$ I simplified it to $$\frac{(\sin{A}-\cos{A})}{(\sin{A} ...
4
votes
2answers
612 views

How do we know Taylor's Series works with complex numbers?

Euler famously used the Taylor's Series of $\exp$: $$\exp (x)=\sum_{n=0}^\infty \frac{x^n}{n!}$$ and made the substitution $x=i\theta$ to find $$\exp(i\theta) = \cos (\theta)+i\sin (\theta)$$ How ...
0
votes
2answers
47 views

Let $S_n=\sum_{k=1}^{n}\frac{\sin\frac{k\pi}{25}}{k}$,how many positive $S_n$?

Let $S_n=\sum_{k=1}^{n}\frac{\sin\frac{k\pi}{25}}{k}$,how many positive $S_n$ are in $S_1,S_2,...S_{100}$
7
votes
4answers
433 views

Nth derivative of $\tan^m x$

$m$ is positive integer, $n$ is non-negative integer. $$f_n(x)=\frac {d^n}{dx^n} (\tan ^m(x))$$ $P_n(x)=f_n(\arctan(x))$ I would like to find the polynomials that are defined as above ...
4
votes
3answers
2k views

Solving the complex equation $\sin(z) = \cos(z)$

To find the complex numbers z satisfying $\sin(z) = \cos(z)$, can I say: $$\sin(z) = \frac{(e^{iz}-e^{-iz})}{2i}=\frac{(e^{iz}+e^{-iz})}{2}$$ and solve for z? So we then reduce this to $$-e^{-iz} = ...
2
votes
2answers
111 views

Homework question regarding inverse function with a cosine

I'm given $f(x)= 3-4\cos(x-2)$ I've gotten to $(-x+3)/4 = \cos(2)\cos(y)+\sin(s)\sin(y)$ But I can't get the $y$ out to create an inverse ...
1
vote
2answers
145 views

Getting a real number from a complex number

I'm attempting to program a formula to say how full an horizontal cylinder is with liquid. Here is the formula I am using with variables from measurements I took: When I use Wolframalpha to solve ...
1
vote
1answer
728 views

Find the equation of each tangent to the curve $r=a\cos3\theta$ which is parallel to the initial line.

Find the equation of each tangent to the curve $r=a\cos3\theta$ which is parallel to the initial line(horizontal axis). here is my steps: $y=r\sin\theta=a\cos(3\theta)\sin\theta$ ...
0
votes
1answer
110 views

Tracing a point on a rotating circle

I have a large circle and a small circle as shown in the image. The distance between the centers of those circles. The Larger circle is rotating about it's center while traveling in direction D such ...
4
votes
2answers
913 views

Find the area enclosed by the loop $r=2(1-\sin\theta)\sqrt{\cos\theta}$

The diagram shows a sketch of the loop whose polar equation is $$r=2(1-\sin\theta),\qquad -\frac{\pi}{2}\leq\theta\leq\frac{\pi}{2}$$ a)Show that the area enclosed by the loop is 16/3. ...
3
votes
1answer
3k views

Find the area enclosed by the curve $r=2+3\cos \theta$.

the question is Find the area enclosed by the curve: $r=2+3\cos \theta$ Here's my steps: since when $r=0$, $\cos \theta=0$ or $\cos\theta =\arccos(-2/3)$. so the area of enclosed by the curve ...
0
votes
1answer
114 views

A trigonometric inequality

How to show that if $0\le\theta\le2\pi$ $|\sum\limits_{n=1}^{p}{\sin{n\theta}}|\le\csc{\frac{\theta}{2}}$ for all integer p?
0
votes
1answer
60 views

Polynomials with roots having the same module and linear dependent arguments

Is it possible for a polynomial with integer coefficients to have some of its roots: $$m_1e^{i\theta_1 \pi}, m_2e^{i\theta_2 \pi}, \ldots, m_ke^{i\theta_k \pi}$$ such that there exist nonzero integers ...
4
votes
2answers
5k views

Finding the height given the angle of elevation and depression.

Please, I need help for this problem. I'm a little confused about it :( From a point A 10ft. above the water the angle of elevation of the top of a lighthouse is 46 degrees and the angle of ...
0
votes
1answer
40 views

Difference Identity problem

I have a homework problem that I don't know what to do with. We were just introduced to sum and difference identities. We've always been provided values in degrees both in class and in homework ...
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vote
2answers
822 views

Write $\csc(x)$ in terms of $\sec(x)$

I was working on trig homework and came across a question that I didn't understand how to even begin to approach. It asked us to use trigonometric identities to write $\csc(x)$ in terms of $\sec(x)$. ...
2
votes
3answers
429 views

Given that $\tan^{-1}(x)+\tan^{-1}(y)+\tan^{-1}(xy)=11/12π$, prove that when $x=1, dy/dx=-1-\sqrt{3}/2$

Given that $x$ and $y$ satisfy the equation: $$\arctan(x)+\arctan(y)+\arctan(xy)=11/12π$$ Prove that, when $x=1, dy/dx=-1-\sqrt{3}/2$. I tried to differentiate both sides: ...
10
votes
1answer
379 views

For a trigonometric polynomial $P$, can $\lim \limits_{n \to \infty} P(n^2) = 0$ without $P(n^2) = 0$?

Disclaimer: The original version of this question focused on $2^n$ in lieu of $n^2$. It is in the hope that the question is easier with $n^2$ that I changed it. I have an always-nonnegative (on the ...
12
votes
2answers
505 views

Evaluating $\displaystyle \int_{0}^{\frac{\pi}{2}}\frac{\sqrt{\sin x}}{\sqrt{\sin x}+\sqrt{\cos x}}dx$

I have to evaluate: $$\displaystyle \int_{0}^{\frac{\pi}{2}}\frac{\sqrt{\sin x}}{\sqrt{\sin x}+\sqrt{\cos x}}dx $$ I can't get the right answer! So please help me out!
5
votes
1answer
405 views

cyclic polygons & trigonometry

At one vertex of a pentagon inscribed in a circle of unit diameter (unit diameter, not unit radius) let the angles between adjacent diagonals be $\alpha,\beta,\gamma$, at the next, ...
5
votes
2answers
666 views

Why is $d\theta/dx$ necessarily $\cos \theta$ in this physics problem? Or am I wrong?

I'm asking this on the math stack exchange because it seems that the key part of this physics problem I'm asking for help on is more related to the geometry of it than the physics of it. I'm ...
2
votes
3answers
154 views

Cute trigonometric triviality

For which values of the coefficient $c$ does the quantity $$ \cos\alpha\cos\beta- c\sin\alpha\sin\beta $$ depend on $\alpha$ and $\beta$ only through their sum? (I'll post a quick answer below. This ...
1
vote
3answers
199 views

find inverse function in given domain

Suppose that we have function $y=\sin(x)$ we need to find its inverse function, assuming that $D(f)=[-\pi/4.\pi/4]$ I know that inverse of $\sin(x)$ is $\arcsin(x)$, it would be answer of a given ...
5
votes
1answer
737 views

Some approximations for $\arccos(1/(1+x))$

I was trying to calculate the maximum ground distance you can see on mountains, with your elvation given. After some simple geometry, I was able to come up with the following formula: Let $h$ be ...
-1
votes
1answer
241 views

Trigonometry word problem involving a window?

A walls is 10 feet tall and is 15 feet from a house. A window in the house is 30.5 feet above the ground.A fire escape slide attaches to the bottom of the window and to the top of the tall of the ...
1
vote
2answers
498 views

Trigonometry word problem involving wheel?

A wheel 5 feet in diameter rolls up with an incline of 18 degrees 20 minutes. What is the height of the center of the wheel above the base of the incline when the wheel has rolled 5 ft up the incline? ...
0
votes
1answer
203 views

Trigonometry word problem involving radius?

Can anyone show me how to solve this problem. The radius of a circle is 21.4 meter. Find the length of the chord subtended by a central angle of 110 degrees 40 minutes and the distance between two ...