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

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1answer
43 views

Calculating the amout of rotation on a line and the length compared to a gear

I am currently working on this steam logo animation, but I got no clue how to calculate the amout of rotation and length of the first beam when attached to a gear. Picture: puu.sh/3okhA.jpg In css, ...
0
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5answers
233 views

solve this problem using Pythagorean identities

$$\frac{\sin(\theta)\cdot\tan(\theta)}{1-\cos(\theta)}= \sec(\theta)+1$$ I need to prove that both sides of the equation are equal to each other, we are supposed to be using Pythagorean identities to ...
0
votes
4answers
896 views

Trigonometric identity: $\frac {\tan\theta}{1-\cot\theta}+\frac {\cot\theta}{1-\tan\theta} =1+\sec\theta\cdot\csc\theta$

I have to prove the following result : $$\frac {\tan\theta}{1-\cot\theta}+\frac {\cot\theta}{1-\tan\theta} =1+\sec\theta\cdot\csc\theta$$ I tried converting $\tan\theta$ & $\cot\theta$ into ...
6
votes
3answers
141 views

prove that $\cos x,\cos y,\cos z$ don't make strictly decreasing arithmetic progression

let $x,y,z\in R$,and such that $$\sin y-\sin x=\sin z-\sin y\ge 0 $$ show that: $$\cos x,\cos y,\cos z$$ don't make strictly decreasing arithmetic progression my idea: we have $$2\sin y=\sin x ...
1
vote
1answer
439 views

How do we define $\sin(\theta)$ or $\cos(\theta)$

On the interval $[0,2\pi]$, how do we define either sine or cosine? Obviously if we have one, the other is straight forward to generate as a phase shift of the other one. To expand a little, we know ...
3
votes
3answers
62 views

Why is this integral zero for every $n \in \mathbb{Z}$? $\int_0^\pi x(\pi -x) \sin 2nx$.

The integral $$\int_0^\pi x(\pi -x) \sin 2nx$$ evaluates to zero, but the function can't be said to be even or odd. What argument, other than pure calculation (as I did) would give the value $0$ ...
3
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4answers
171 views

Prove $\frac{\cos^2 A}{1 - \sin A} = 1 + \sin A$ by the Pythagorean theorem.

How do I use the Pythagorean Theorem to prove that $$\frac{\cos^2 A}{1 - \sin A} = 1 + \sin A?$$
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4answers
207 views

Trigonometric near-identity

The parametrized curve $$ \left( \sec\theta+\csc\theta,\ 2\sqrt{2}\csc(2\theta) \right), \qquad \frac{10}{100} \le\theta\le\frac{142}{100} $$ looks to the naked eye like a straight line. The ...
7
votes
1answer
281 views

Graphs of rational functions of sine and cosine

What do graphs of rational functions of sine and cosine generally look like? (A variety of different shapes, I know. A classification or catalog of them might answer the question, or maybe a ...
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6answers
3k views

Need help in proving that $\frac{\sin\theta - \cos\theta + 1}{\sin\theta + \cos\theta - 1} = \frac 1{\sec\theta - \tan\theta}$

We need to prove that $$\dfrac{\sin\theta - \cos\theta + 1}{\sin\theta + \cos\theta - 1} = \frac 1{\sec\theta - \tan\theta}$$ I have tried and it gets confusing.
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0answers
16 views

Double Angle Identities [duplicate]

Given that $\sin(2x)=\frac 5 {13}$ and $0^\circ < x < 45^\circ$, I need to find $\cos(x)$ and $\sin(x)$. If I work it out, I get $$\begin{gather} \frac 5 {13} = 2 \sin(x) \cos(x) \\ \frac 5 ...
4
votes
2answers
231 views

Maximum of a trigonometric function without derivatives

I know that I can find the maximum of this function by using derivatives but is there an other way of finding the maximum that does not involve derivatives? Maybe use a well-known inequality or ...
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votes
3answers
141 views

How to use double angle identities to find $\sin x$ and $\cos x$ from $\sin 2x $?

If $\sin 2x =\frac{5}{13}$ and $0^\circ < x < 45^\circ$, find $\sin x$ and $\cos x$. The answers should be $\frac{\sqrt{26}}{26}$ and $\frac{5\sqrt{26}}{26}$ Ideas The idea is to use double ...
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3answers
715 views

How to derive compositions of trigonometric and inverse trigonometric functions?

To prove: $$\begin{align} \sin({\arccos{x}})&=\sqrt{1-x^2}\\ \cos{\arcsin{x}}&=\sqrt{1-x^2}\\ \sin{\arctan{x}}&=\frac{x}{\sqrt{1+x^2}}\\ \cos{\arctan{x}}&=\frac{1}{\sqrt{1+x^2}}\\ ...
5
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1answer
156 views

Finite Series - reciprocals of sines

Find the sum of the finite series $$\sum _{k=1}^{k=89} \frac{1}{\sin(k^{\circ})\sin((k+1)^{\circ})}$$ This problem was asked in a test in my school. The answer seems to be ...
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1answer
42 views

A triangular “spot function”

z = (cos πx + cos πy) represents the classical "spot function", made by square cells, used in every laser printer's halftone screening. Does anyone knows the corresponding function to produce ...
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0answers
197 views

Position of a point on a moving unit circle?

Suppose a unit circle is moving in a horizontal line at a speed given by $f(t)$ , and a point on the unit circle is moving counterclockwise at a speed given by $g(t)$ , and the initial position of the ...
5
votes
3answers
797 views

Finding $\sin^6 x+\cos^6 x$, what am I doing wrong here?

I have $\sin 2x=\frac 23$ , and I'm supposed to express $\sin^6 x+\cos^6 x$ as $\frac ab$ where $a, b$ are co-prime positive integers. This is what I did: First, notice that $(\sin x +\cos ...
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1answer
48 views

Derivative of inverse function, why do I get this contradiction?

Consider two functions, $f(x)=\sin x$ $\;$ and $g(x)=\arcsin x$. Then, $f'(x)=\cos x$ $\;$ and $g'(x)=\frac{1}{\sqrt{1-x^2}}.$ We know that $g[f(x)]=x$, so $f'(x)=\frac {1}{g'[f(x)]}$ $\;$. $\:$ ...
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2answers
213 views

Range of $f(x) = \sin(\cos x)$

Problem : Finding the maximum and minimum value of the function : $f(x) = \sin(\cos x)$ My approach : We know that if $f'(x) > 0 $ function attain maximum value by putting $f'(x) = 0$ and ...
2
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1answer
469 views

Simple yet tricky trigonometry

This might seem silly to ask, but how can I solve a trigonometry problem for the unknown $h$ in the form of: $x + 45 = h/\tan 30$ and $x = h/\tan 50$
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1answer
55 views

Find all different integer exponents

Find all different integers that satisfy the following equality: $m(\sin^{n}x + \cos^{n} x- 1) = n(\sin^{m}x + \cos^{m}x - 1), (\forall) x\in\mathbb{R}.$ Case1: $m$ is odd, $n$ is even, then put ...
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3answers
188 views

Confusing Trigonometry Problem

Lets say at an intersection the words "STOP HERE" are painted on the road in red letters 2.5m high. It is important that drivers using this lane can read the letters. How can I find the angle ...
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2answers
2k views

Is it possible to calculate sine by hand?

Without a calculator, how can I calculate the sine of an angle, for example 32(without drawing a triangle)?
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1answer
72 views

Prove that there are no integers $\csc {\frac{j\pi}{n}}-\csc{\frac{k\pi}{n}}=2$

Prove that there are no integers $j,k,n$ with odd $n$ satisfying $$\csc {\dfrac{j\pi}{n}}-\csc{\dfrac{k\pi}{n}}=2$$ This problem from $AMM,1999,10630$, but this solution is very ugly,and it's not ...
2
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2answers
206 views

Trigonometry Problem Solving

How can we estimate the height (h) of a castle surrounded by a moat, using the info below?
3
votes
1answer
193 views

$\arctan$ identities: converting $c \cdot \arctan(\frac{n}{d}) \iff \arctan(\frac{n'}{d'})$

If one has an expression of the form $c \cdot \arctan(\frac{n}{d})$, it can be converted to an equivalent expression of the form $\arctan(\frac{n'}{d'})$ fairly easily, using $c \cdot ...
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3answers
8k views

Calculate Exact Value of $\sin\theta, \cos\theta$ and $\tan\theta$

Having trouble getting a start on this problem, any help would be appreciated! Given point $P = (-3,5)$ is on the terminal arm of angle $\theta$ in standard position. Calculate the exact value of ...
12
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4answers
279 views

Are there two $\pi$s?

The mathematical constant $\pi$ occurs in the formula for the area of a circle, $A=\pi r^2$, and in the formula for the circumference of a circle, $C= 2\pi r$. How does one prove that these constants ...
2
votes
2answers
307 views

Finding the Exact Value

How would one go about finding the exact value of $\theta$ in the following:$\sqrt{3}\tan \theta -1 =0$? I am unsure of how to begin this question. Any help would be appreciated!
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2answers
106 views

What is the derivative of $\frac{x}{2-\tan x}$?

I am too stupid to figure this out so I won't even try anymore $$y = \frac{x}{2 - \tan x}$$ I am sure this will take someone about four seconds to solve, but I spent about ten minutes looking at it ...
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2answers
41 views

How do I show that an angle is a certain value in a triangle with two sides given?

In the following example I am told that angle x is 60° and that I have to prove it is (without a calculator). What is the simplest way of showing that it is true?
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2answers
2k views

Determine depth of a partially filled hemisphere

Recently came across a question in a Year 9 math book of which there was no "working out" supplied and offers now description on how they obtained the answer. The question goes like this: A bowl ...
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1answer
40 views

trigonometric inequality - how to prove it?

Let $ 0 < x < \frac {\pi}{2}$ How to prove it? $$2 \sin x \le x- \frac {\pi}{3} + \sqrt {3} $$
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2answers
43 views

Exponential trigonometrical equation

Find $x$ from $[-\frac{\pi}{2},\frac{\pi}{2}]$ in: $$2^{\sin 3x}-8^{\sin x}=\sin^3{x}$$ I know that $x=0$ verifies the equation, but is it the only solution?
9
votes
2answers
371 views

How do I solve this inequality? $\sin x < 2x^3$

The equation is $\sin x < 2x^3$ The steps I've taken so far are: $\sin x < 2x^3 $ $\sin x - 2x^3 < 0 $ To solve this I should find when the slope is $ 0 $ so I can find the max and ...
0
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1answer
111 views

Why is the length R cosine theta?

Why is the length described as R cosine theta (the top where the Sphere is sliced off)? I've been staring at the geometry for quite a bit & can't figure. Thanks
4
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5answers
177 views

Graphing $\sin(|x|)$?

I'm confused on how the graph is in both quadrant II and III. If $|x|$ is evaluated first wouldn't all the answers be positive, so that when the range of $|x|$ is plugged into $\sin$ wouldn't the ...
5
votes
3answers
167 views

Solve $\tanα+2\tan2α+4\tan4α+8\tan8α+16\tanα=\cotα$ for $\alpha$

My knowledge of trigonometry are still insufficient to resolve this problem. Any help would be greatly appreciated. Solving for $\alpha$: $$\tanα+2\tan2α+4\tan4α+8\tan8α+16\tanα=\cotα$$
6
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1answer
111 views

How prove this $\frac{\sin{(A-B)}\sin{(A-C)}}{\sin{2A}}+\frac{\sin{(B-C)}\sin{(B-A)}}{\sin{2B}}+\frac{\sin{(C-A)}\sin{(C-B)}}{\sin{2C}}\ge 0$

let $0<A,B,C<\dfrac{\pi}{2}$,and $A+B+C=\pi$,prove that $$\dfrac{\sin{(A-B)}\sin{(A-C)}}{\sin{2A}}+\dfrac{\sin{(B-C)}\sin{(B-A)}}{\sin{2B}}+\dfrac{\sin{(C-A)}\sin{(C-B)}}{\sin{2C}}\ge 0$$ my ...
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2answers
2k views

How to calc arc sine without a calculator?

How can I find the arc sine of a sine without using a calculator? Thank you.
6
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3answers
854 views

How to evaluate the trigonometric integral $\int \frac{1}{\cos x+\tan x }dx$

$$\int \dfrac{1}{\cos x+\tan x }dx$$ This can be converted to $$\int \dfrac{\cos x}{\sin x+\cos^2x}dx$$ But from here I get stuck. Using t substitution will get you into a mess. Are there ...
0
votes
1answer
347 views

Evaluation of the integral $\int \cos\omega t\ln\cos\omega t\,dt$

I am trying to evaluate an integral of the form $$ \int \cos\left(\omega t\right) \ln \cos\left(\omega t\right) dt$$ and am unsure how to proceed. I rewrote it as: $$ \textrm{Re} \left\{\int dt ...
0
votes
1answer
73 views

Evaluate the maximum of: $A = \sin A\cdot\sin ^2 B\cdot \sin ^3 C$

Given a triangle ABC. Evaluate the maximum of: $A = \sin A\cdot\sin ^2 B\cdot \sin ^3 C$
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votes
2answers
113 views

Orthogonality of eigenvectors of laplacian

Let $x_i=(\sin i\pi/n,\cdots,\sin (n-1)i\pi/n)$ for $i=1,\cdots,n-1$. I want to show that $x_i \cdot x_j=\delta_{ij} n/2$. Why is it true? I tried $\sin a \sin b=-[\cos(a+b)-\cos(a-b)]/2$ but don't ...
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1answer
31 views

Inverse Trigonometry doubt.

Suppose $\sin y=\sin 2x$, then what will be the solution for $y$? Will it be $y=2x$ or $y=n\pi-2x$ for some $n \in \mathbb{N}$?
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4answers
917 views

Find maximum value of $f(x)=2\cos 2x + 4 \sin x$ where $0 < x <\pi$

Find the maximum value of $f(x)$ where \begin{equation} f(x)=2\cos 2x + 4 \sin x \ \ \text{for} \ \ 0<x<\pi \end{equation}
4
votes
3answers
194 views

$\int \frac{1}{\cos(x)}\,\mathrm dx$

could you help me on this integral ? $$\int \frac{1}{\cos(x)}\,\mathrm dx$$ Here's what I've started : $$\int \frac{1}{\cos(x)}\,\mathrm dx = \int \frac{\cos(x)}{\cos(x)^2}\,\mathrm dx = \int ...
0
votes
1answer
139 views

evaluate the following limit on trigonometry

given that \begin{equation} \lim_{y \rightarrow 0} \frac{\sin y}{y}=1 \end{equation} evaluate the following \begin{equation} \lim_{x \rightarrow 0} \frac{2-2\cos^2 x-2 \cos x \sin ^2 x}{x^4} ...
2
votes
1answer
89 views

minimum value of a trigonometric equation is given. the problem is when the minimum value attains

Suppose the minimum value of $\cos^{2}(\theta_{1}-\theta_{2})+\cos^{2}(\theta_{2}-\theta_{3})+\cos^{2}(\theta_{3}-\theta_{1})$ is $\frac{3}{4}$. Also the following equations are given ...