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

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1
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1answer
76 views

Calculating $a^{\sin x}=x^{\ln a}$ [closed]

$$a^{\sin x}=x^{\ln a}$$ Can you help me solve this equation for $x$ in terms of $a$?
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4answers
155 views

Solve: $-\frac{1}{\sqrt{2}} < \sin \theta + \cos \theta < \frac{1}{\sqrt{2}}$

The question is: Solve $$-\frac{1}{\sqrt{2}} \lt \sin\theta + \cos\theta < \frac{1}{\sqrt{2}}$$ for values of $\theta$ between $0^\circ$ and $180^\circ$. I realized that: $$\begin{align} ...
0
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1answer
29 views

Asymptote of sine function!

I am reading about asymptotes in my personal reading. I am thinking not all open curves will have asymptotes as I am not able to comprehend an asymptote for Sine Curve. Is it right/wrong?
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4answers
129 views

Proving $\sin 20^\circ \, \sin 40^\circ \, \sin 60^\circ \, \sin 80^\circ =\frac{3}{16}$

The problem is to prove that $$\sin 20^\circ \, \sin 40^\circ \, \sin 60^\circ \, \sin 80^\circ =\frac{3}{16}$$ All my attempts were to get them in $\sin (2A)$ form after eliminating $\sin 60^\circ$ ...
0
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2answers
52 views

Finding nth roots of complex number [duplicate]

If I have a complex number of the form $z = a+bi$, how would I find the complex roots? I know that each root will be equidistant from each other and will form a circle, but I'm not sure how to solve ...
1
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1answer
56 views

How to know which side of the right angled triangle is the base?

If we are given a right angled triangle without any angle or length of any side. How we will find that which side is the base, which side is the perpendicular.
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1answer
36 views

A Sine Wave with a Sine Wave Inside

How do you graph a sine wave that is "squiggly" in the pattern of a sine wave as such that there are points of inflection in the sine wave as if it was a sine wave itself? If anyone can provide a ...
3
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5answers
88 views

Finding argument of complex number and conversion into polar form

How do I find the argument of a complex number, for example $z = 3 + 4i$? I know the polar form of $z$ is $r(\cos\theta + i\sin\theta)$ where $r$ is the modulus of $z$ ($\sqrt{3^2+4^2}$) which would ...
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2answers
35 views

Triangle and Ratio : Find the length of a side.

Let $\theta = \angle CAD, \phi = \angle CDB, \varphi=\angle DBC, \alpha = \angle BCD$ and $\beta=\angle ACD$. Then we have the following system of equations $\theta + \varphi = 90^{\circ},$ ...
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3answers
54 views

Solve $\frac{\tan^3\theta}{1+\tan^2\theta}+\frac{\cot^3\theta}{1+\cot^2\theta} = \sec\theta\csc\theta - 2\sin\theta\cos\theta$ [closed]

$$\frac{\tan^3\theta}{1+\tan^2\theta}+\frac{\cot^3\theta}{1+\cot^2\theta} = \sec\theta\csc\theta - 2\sin\theta\cos\theta$$
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2answers
43 views

Cosine and Sine of Sums

What's a good way to simplify $\sin( \sum\nolimits_{i=1}^{\infty} x_i)$ as the product and sum of $\sin(x_i)$ and $\cos(x_i)$ alone? And the same for $\cos( \sum\nolimits_{i=1}^{\infty} x_i)$?
2
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0answers
21 views

can't figure out multilateration with xyz positions of each post and difference in time

I'm having some real issues figuring out multilateration. I'll start by saying I'm not a math whiz, but I am usually able to figure most things out, but this one has been throwing me through a loop ...
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5answers
130 views

Is there anyway to show $\left| {\frac{{\cos x - \cos y}}{{x - y}}} \right| \le 1$ other than taking derivatives?

The purpose is to show $\left| {\frac{{\cos x - \cos y}}{{x - y}}} \right| \le 1$ for any $x,y\in\Bbb{R}$. Taking partial derivatives with respect to $x,y$ respectively $\frac{{\partial \frac{{\cos ...
3
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2answers
53 views

Ratio of $\frac{\sin x_1 }{\sin x_2}$ where $f(x_1)=f(x_2)$ for a trigonometric sum $f(x)$

If $f(x) = \cos(x+a_1)+\frac12\cos(x+a_2)+\frac14\cos(x+a_3)+\cdots+\frac1{2^{n-1}}\cos(x+a_n)$, where $a_1, a_2, ... a_n$ are some constants and $f(x_1)-f(x_2)=0$, where $x_2 \neq m\pi$, find ...
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2answers
22 views

Prove that $5\cos A+3\cos\left(A+\frac{\pi}{3}\right)+3$ lies between $-4$ and $10$ [duplicate]

After much of solving, I arrive at the expression $$\frac{1}{2}\left[13 \cos A-3\sqrt{3}\sin A +6\right].$$ I can't conclude anything from this. Please help. Hints are welcome.
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3answers
51 views

Prove $\triangle{ABC}$ is isosceles if $\cos A = \frac{\sin B}{2\sin C}$

In $\triangle{ABC}$, $$\cos A = \frac{\sin B}{2\sin C}$$ How to prove that $\triangle{ABC}$ is isosceles?
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1answer
98 views

Why are there 31 solutions to this problem?

How many positive numbers $x$ satisfy the equation $\cos{97x} = x$? What procedure should I use to find the number of possible solutions?
1
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2answers
60 views

When solving $\tan(3x) - \cot(4x)$, how to formulate the answer?

when I solve the following equation: $\tan(3x) = \cot(4x)$ I get the following solution: $x = \frac{\pi}{14} + \frac{\pi n}{7}, n \in \mathbb{Z}$ But as x must be $\neq \frac{\pi}{6} + \frac{\pi ...
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0answers
37 views

how do you find the distance between 2 points with known distances between other points

link to diagram for explanation: http://i.imgur.com/8cmuWib.png I am trying to determine the distance between i and j. These nodes are all placed in a coordinate system. things I know: ...
6
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1answer
53 views

How many triangles exist whose angles are rational and side lengths are roots to quadratic equations?

By "rational angles" I mean a rational degree measure (equivalently, angle a rational multiple of $\pi$). Obviously similar triangles should be counted once. Off the top of my head we have: 30-60-90, ...
1
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1answer
81 views

interesting remainder problem

Let $A=\left( 2\sin{\dfrac{7\pi}{18}}+1 \right)^{2556}$. How can one find the remainder from the division of $\lfloor A \rfloor$ by $17$ ? I have no idea. Thank you.
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1answer
48 views

Constructible real numbers

I'm trying to understand constructible numbers. I know that a real number $r$ is constructible if it can be calculated from 0 and 1 by a finite number of additions, subtractions, multiplications, ...
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2answers
61 views

how to solve nonlinear system of equations

I have $A, B, C, D, E, F.$ I want to calculate a and b from the following system of equations: I know I should solve this system using $3$ equations and $3$ unknowns, but it is not linear. can any ...
2
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2answers
42 views

Prove that $\sin x - x\cos x = 0$ has only one solution in $ [-\frac{\pi}{2}, \frac{\pi}{2}]$

I have to prove that $\sin x - x\cos x = 0$ has only one solution in $\left[-\frac{\pi}{2}, \frac{\pi}{2}\right]$. While it seems obvious that one solution might be $x=0$, I don't know how to do a ...
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3answers
65 views

How to rewrite $\pi - \arccos(x)$ as $2\arctan(y)$?

I get the following results after solving the equation $\sqrt[4]{1 - \frac{4}{3}\cos(2x) - \sin^4(x)} = -\,\cos(x)$, : $$ x_{1} = \pi - \arccos(\frac{\sqrt{6}}{3}) + 2\pi n: n \in \mathbb{Z}\\ x_{2} = ...
0
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1answer
25 views

Resolving $c^2+c^2$

For a program I was making I needed to calculate the diagonal movement in a way that it would have the same speed as normal movement along the axis of the 2d world. My approach was that $speed^2 = c^2 ...
5
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2answers
76 views

What are the ways to solve trig equations of the form $\sin(f(x)) = \cos(g(x))$?

if I have the following trig equation: $$\sin(10x) = \cos(2x)$$ I take the following steps to solve it: I rewrite $\cos(2x)$ as $\sin\left(\frac{\pi}{2} + 2x\right)$ or as $\sin\left(\frac{\pi}{2} ...
0
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1answer
30 views

Standard notation for brackets around trigonometric arguments

I've been going through this site quite a lot lately and I've noticed one thing, the notation $\sin(x), \, \cos(x), \, \tan(x),\ldots \,$ is used frequently. Now, I do tend to bracket the arguments ...
6
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5answers
513 views

When a periodic function is squared (or cubed, and so on…) does it always lose its periodicity?

For instance $$\sin^{2}\left(-\frac{\pi}{6}\right) = \sin^{2}\left(\frac{\pi}{6}\right)$$ i.e., $\sin^2 (x)$ is an even function and loses the $2\pi$-periodicity of $\sin x $. Is this true in ...
0
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1answer
14 views

A question on the radial unit vector in polar coordinates

So, in polar coordinates $$r(t) = \cos(x) \hat{i}+\sin(x) \hat{j},$$ where $x$ is the angle from the beginning of the measurement. I understand how this gives the correct unit vector when we measure ...
2
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0answers
30 views

Calculating truncated cone taper angle from a projected image

I am imagining a truncated cone with taper angle $\phi$, shown in the left of the figure. I view the truncated cone inclined at some angle $\alpha$ from its axis. If $\alpha$ is large enough, I will ...
0
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1answer
25 views

The supremum of the function $f(x)=\frac{1-\cos Nx}{1-\cos x}$

I have the following function: $f(x)=\dfrac{1-\cos Nx}{1-\cos x}$ Where N is integer. I know the function has Sup when x goes to $2n\pi$ $n\in\mathbb{N}$. But is it possible to show this? Thank ...
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6answers
111 views

Finding the value of $\tan 20^\circ$.

Just a small thought popped up in my mind; and now I'm stuck on it. Any idea on how to find the value of $\tan 20^\circ$? I tried doing it by using the multiple angle formulas, but I didn't get an ...
4
votes
1answer
101 views

Find the value of $p$ such that $\cos(P \sin x)=\sin(P\cos x)$ [closed]

Find the smallest positive value of $P$ for which the equation $$\cos(P \sin x)=\sin(P\cos x)$$ is true, where $x\in[0^{\circ},360^{\circ}]$.
0
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1answer
16 views

Direct mapping for Equation of line from point and angle with Y

I want to compute the coefficients of line $Ax+By+C=0$ given a point $x_0,y_0$ and angle $r$ the line makes with positive $y$ direction. Any direct mapping would be more heloful
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2answers
46 views

Perimeter of an octagon inscribed in a circle with given radius

What is the perimeter of an octagon if it's inscribed in a circle with a radius of $5$ inches? The formula I'm using is $\text{P}=2nr\cdot \sin \frac{π}{n}$ where $n$ is the number of sides and $r$ is ...
2
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2answers
35 views

Minimal Trigonometry

Triangle ABC has angle BAC given and the lengths |AB| and |AC| satisfy $$|AB|\cdot |AC| = 1.$$ Let AD be the angle bisector of BAC. Express the length |AD| as a function of the length |AB|. I keep ...
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5answers
107 views

Can someone please check my work?: $\cos^2(x)=1-\sin(x)$

$$\begin{align}\cos^2(x)&=1-\sin(x)\\ 1-\sin^2(x)&=1-\sin(x)\\ (1-\sin x)(1+\sin x)&= 1-\sin(x) \end{align}$$ divide both sides by $1 - \sin(x)$ End up with $1 + \sin(x)$ The answer is ...
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0answers
31 views

At what angle does the stone need to be hit?

In curling, it is often necessary to hit and displace an opponent’s stone to win the end. Olivia would like to hit her opponent’s stone with her own stone. If she releases her stone at the hog line, ...
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2answers
22 views

Function that graphs repeating upper halves of circles

I'm trying to write a periodic function that repeats the upper half of a unit circle, so it would look similar to $|\cos(x)|$ but with the upper half of a circle instead. If anyone could help me out ...
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5answers
233 views

How to remember trig identities?

Suppose I have a trig function $T: \Bbb{R} \rightarrow \Bbb{R}$. I want to be able to derive four basic properties: $$T(x) \cdot T(y)$$ $$T(x) + T(y)$$ $$T(x+y)$$ $$T(cx)$$ where $c$ is some ...
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2answers
25 views

trigonometry mountain elevation problem [closed]

At the foot of a mountain the elevation of its summit is $45^{\circ}$. After ascending $1000\mathrm{m}$ towards mountain up a slope of $30^{\circ}$ inclination, the elevation is found to ...
3
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2answers
80 views

If $\sin(x)=\sin(\pi/4 + x)$, then why isn't $x=x+\pi/4$?

I've been solving a question, If $\cos(x) + \sin(x)=\sqrt{2} \cos(\pi/2 - x)$ then find the value of $x$. We know that $\cos(x) + \sin(x)= \sqrt{2} \sin(\pi/4 + x)$. So, $$\sin(\pi/4 + x) = ...
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0answers
164 views

Solve the equation (very hard)

How to find all irrational solutions of the equation ...
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2answers
53 views

What does circular measurement stand for?

This is a problem in S. L. Loney. I am confused about what this phrase, "The circular measure", stands for in this problem. If the circular measure of two angles of a triangle are respectively 1/2 ...
1
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1answer
68 views

Proving a relation between inradius ,circumradius and exradii in a triangle

Prove that in a triangle $$r^2+r_1^2+r_2^2+r_3^2=16R^2-(a^2+b^2+c^2)$$ where the symbols have their usual meanings. I am looking for a smaller or elegant proof using trigonometry. A geometric proof ...
1
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1answer
53 views

Geometry formulas, how to show identities.

Given $d$ is integer: How do I show: $$\frac{1}{(e^{\frac{2i\pi p}{d}}-1)}=\frac{-i}{2\tan(\frac{\pi p}{d})}-\frac{1}{2}$$ How do I rewrite and show, for $k$ is an integer: $$ ...
0
votes
1answer
32 views

Given only angles and area of triangle, find side length. [closed]

The area of a triangle is $60$ square inches. Find the length of the side included between $A = 25°$ and $C = 110°$. (Round your answer to one decimal place.)
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votes
3answers
81 views

Prove that $\sin^2 \theta + \sin^2 \beta= \sin(\theta + \beta)$ when $\theta+\beta = 90^\circ$ [closed]

If $\theta, \beta$ are two acute angles prove that : $$\sin^2 \theta + \sin^2 \beta= \sin(\theta + \beta) $$ when $\theta, \beta$ are complementary angles, i.e. $\theta + \beta = 90°$. My try... ...
0
votes
1answer
37 views

Peripendicular Line at distance d from point in a given direction

I have a line given by $Ax + By + C= 0$, and a point $x_0,y_0$. From that point $x_0,y_0$ in the direction of the line up to distance $d$, I want to find the equation of the line that is perpendicular ...