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

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13 views

Obtaining $\sum_{n=1}^{\infty} a^n \cos{(n\theta)} = \frac{a \cos{\theta}-a^2}{1-2a\cos{\theta}+a^2}$

This is a homework problem. From Fourier Series and Boundary Value problems, Brown/Churchill 8th ed. I should begin with $2\cos{A}\cos{B}=\cos{(A+B)}+\cos{(A-B)}$, substitute with $A=n\theta$ and ...
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1answer
44 views

Limits and Trigonometry

Consider an function $f$ , defined as : $$f^k (\theta) =\sum_{r=1}^n \left( \frac{\tan \left( \frac {\theta}{2^r} \right) }{2^r} \right)^k +\frac 1 3 \sum _{r=1}^n \left( \frac { \tan \left( ...
1
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1answer
32 views

For what values of $x\in \Bbb N$ is $\tan^{-1}\left(\frac{360}{x}\right)$ rational?

For what values of $x\in \Bbb N$ is $$\tan^{-1}\left(\frac{360}{x}\right)$$ rational? Just wondering if there is any method to accomplish this.
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1answer
61 views

for which values of $\theta$ does this equation $x^{\cos\theta} +y^{\sin\theta }=1$ have solutions in integers .?

for which values of $\theta$ does this equation $$x^{\cos\theta} +y^{\sin\theta}=1$$ have solutions in integers ? Note : $x, y$ integers, $\theta$ is real number. Thank you for your help
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3answers
77 views

Solve $\cos3x=\cos4x$

I want to solve the equation $\cos3x=\cos4x$. The given solutions are $x= 0$, $2\pi/7$, $4\pi/7$ and $6\pi/7$. My first approach was to write the whole thing in terms of $\cos x$ this gave, ...
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0answers
16 views

$ABCD$ has area $9$. $M$ is in the middle of $AB$ and the edge $BF$ of length $2$ forms an angle of $60º$, Calculate $[CM,CB,BF]$.

$ABCD$ has area $9$. $M$ is in the middle of $AB$ and the edge $BF$ of length $2$ forms an angle of $60º$. Calculate $[CM,CB,BF]$, knowing that $\mathbb{V}^3$ is oriented by a positive basis. ...
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2answers
30 views

Trigonometry Identity proving

If $\sin(x-y) =\cos y$ prove that $\tan y = \frac{1+ \sin y}{\cos y} $. Is there an error with the question? I don't seem to be able to get the answer. Should it be $\tan x$ instead of $\tan y = ...
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0answers
4 views

Determining pitch and roll angles from the coordinates of a vector

I want to know, given the measurement of an accelerometer at rest (so not really an acceleration but a force per unit of mass) the inclination of this accelerometer, along the X and Y axis. So, In ...
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1answer
16 views

Area of a triangle with one given measurement [on hold]

The hypotenuse of an isosceles right triangle is 4 centimeters long. How long are its sides? Round your answers to two decimal places.
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3answers
65 views

Could someone please explain double-angle identities?

I don't understand how to do maths, mostly because I don't understand why formulae work they way they do, or the reasoning behind equations, etc. I tried to explain the $\sin(2\theta)$ double-angle ...
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2answers
39 views

Triangle with Ratio of Sides Equal to Ratio of Angles

In an equilateral triangle, the side lengths are in ratio 1:1:1, as are the angle measures. Are there also non-equilateral triangles in which the ratio of the side lengths is the same as the ratio of ...
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1answer
39 views

Why is the chain rule applied to derivatives of trigonometric functions?

I'm having trouble to understand why is the Chain rule applied to trigonometric functions, like: $$\frac{d}{dx}\cos 2x=[2x]'*[\cos 2x]'=-2 \sin 2x$$ Why isn't it like in other variable derivatives? ...
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votes
5answers
77 views

Given $\csc\theta=-\frac53$ and $\pi<\theta<\frac32\pi$, evaluate sine ,cosine, and tangent of $2\theta$

If $\csc\theta=\frac{-5}{3}$, what is the exact value of $\tan(2\theta)$, $\sin(2\theta)$, and $\cos(2\theta)$ on the interval of $\left(\pi, \frac{3\pi}{2}\right)$? I think I'm getting the fraction ...
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0answers
21 views

Forces of parallelograms [on hold]

Two forces are applied to an object. The measure of the angle between the $30.2$ pound applied force and the $50.1$ pound resultant is $25$ degrees. a. Find the magnitude of the second applied force ...
3
votes
3answers
66 views

Simple trigonometry equation

The previous class we were doing trigonometry exercises. Before the class finished, our teacher wrote exercises on the table. I am stuck with the following one: $$ \cos(2x) + 1 + 3\sin x = 0 $$ I ...
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3answers
45 views

Are $\cos$ and $\sin$ linearly dependent in $[- π , π]$? [on hold]

Are $\cos$ and $\sin$ linearly dependent on $[- π , π]$ If true, demonstrate; if False show a counterexample.
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2answers
42 views

how to find trigonometric angle of any value? [on hold]

how to find value for this- cot(1.3333) in degrees, without using a calculator? if it is possible please explain the process involved and how to find values of other similar questions.
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4answers
75 views

Prove that $\sin 2\alpha=2 \sin \alpha \cos\alpha$

In this triangle $AD=AC=1$, $BC=a$, $BAC=2\alpha$ I thought $\sin 2\alpha=a$, but I don't know how to continue.
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1answer
13 views

The domain of logarithmic functions with sinx and cosx

I need to solve this equation: $\log_{\cos x} \sin x + \log_{\sin x} \cos x=2$ But in order to solve it, I first need to find the domain. What I did was this: $\cos x\neq1 \wedge \sin x\neq1 $ ...
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3answers
60 views

Basic trigonometry intuition

I have already posted a question regarding the same function here However, now I simply can not grasp why the function has to have two solution sets:$$\cos y=\cos \Bigl(\frac{\pi ...
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4answers
50 views

Evaluate $\sin\left(-\frac{\pi}{6} + \frac{1}{2}\arccos\left(\frac{1}{3}\right)\right)$.

My task is to evaluate $$\sin\left(-\frac{\pi}{6} + \frac{1}{2}\arccos\left(\frac{1}{3}\right)\right).$$ I think I've gotten most of the way there but I keep running into trouble... any suggestions?
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2answers
24 views

Will this equation with $\sin$ and $\arcsin$ cancel?

It can be said that $\arcsin(\sin(x))= x$ are inverses if $x \lt 2\pi$. Can it also be generalized so that $\arcsin(\sin(d\cdot x))= d\cdot x$ if $x \lt 2\pi\cdot d$ for a constant $d$?
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4answers
35 views

How to analytically evaluate $\cos(\pi/4+\text{atan}(2))$

This equals $\cos(\pi/4)\cos(\text{atan}(2))-\sin(\pi/4)\sin(\text{atan}(2))$. I'm just not sure how to evaluate $\cos(\text{atan}(2))$
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0answers
32 views

Solve $\tan(w/2)$ , where $w = \arcsin(2x/(1+x^2)) + \arccos(1-y^2/(1+y^2))$ [on hold]

Solve $\tan(w/2)$ , where $w = \arcsin(2x/(1+x^2)) + \arccos(1-y^2/(1+y^2))$ Also, $|x|<1$ ; $y>0$ ; $xy>1$ The answer is given as $x+y/(1-xy)$
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1answer
32 views

Trigonometry Question $4/\sin 44 = 5/x$ [on hold]

What method would I use to get the answer to $\frac{4}{\sin(44)}=\frac{5}{x}$ and would it be 60.264337990587? This was the answer I have.
2
votes
5answers
42 views

How to evaluate $\cot(2\arctan(2))$?

How do you evaluate the above? I know that $\cot(2\tan^{-1}(2)) = \cot(2\cot^{-1}\left(\frac{1}{2}\right))$, but I'm lost as to how to simplify this further.
2
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3answers
40 views

Show that $\frac {\sin x} {\cos 3x} + \frac {\sin 3x} {\cos 9x} + \frac {\sin 9x} {\cos 27x} = \frac 12 (\tan 27x - \tan x)$

The question asks to prove that - $$\frac {\sin x} {\cos 3x} + \frac {\sin 3x} {\cos 9x} + \frac {\sin 9x} {\cos 27x} = \frac 12 (\tan 27x - \tan x) $$ I tried combining the first two or the last ...
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0answers
27 views

Simplifying cyclometric function

How does one simplify this function? $$ f(x) = \arccos(\frac{\pi}{2} - \sin(x)) $$ A plot in GeoGebra showed a graph that looked like semicircle, so can one expect something in this form: ...
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1answer
30 views

An equality with inverse trigonometric functions

I've stumbled on the equality $$ \tan ^{-1}\left(\frac{3}{4}\right) \left(\pi -\tan ^{-1}\left(4 \sqrt{3}\right)\right)=4 \tan ^{-1}\left(\frac{2}{\sqrt{3}}\right) \cot ^{-1}(3). $$ Out of ...
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votes
2answers
79 views

Geometric intuition for derivatives of basic trig functions

I was inspired by this question to try and come up with geometric proofs for the derivatives of basic trig functions--basically, those that have simple representations on the unit circle ($\sin, \cos, ...
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6answers
62 views

De Moivre's Theorem (Trigonometry)

How to prove that $\cos^4 \theta+\sin ^4\theta=\frac{1}{4}(\cos4\theta+3)$ by using De Moivre's Theorem? I know that $(\cos\theta+i\sin\theta)^n=\cos n\theta+i\sin n\theta$, but how to apply this ...
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5answers
64 views

Question about a trigonometry proof?

I just want to ask how can you prove that 2α is twice the value of α in the following figure that depicts a proof of an arctangent identity (and likewise, for β as well).
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1answer
29 views

Subtraction of trigonometric functions

I was working on a problem booklet and came across the following equation. $$\sqrt2\sin(2x)-\cos(2x)=\sqrt3\sin(2x-a)$$ $a \in \mathbb{R}$ is a specific value that I'm supposed to find, but I don't ...
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3answers
66 views

De Moivre's Theorem (Trigo)

Prove the trigo identity by using method based on De Moivre's Theorem. $\sin^6\theta=\frac{1}{32}(10-15\cos2\theta+6\cos4\theta-\cos6\theta)$ My attempt, Using $z-\frac{1}{z}=2i\sin \theta$ ...
3
votes
4answers
103 views

Why the anti derivative of $\sec(x) \cdot \tan(x)$ is $\sec(x)$?

I have discovered that $$\sec(x) = \frac{1}{\cos(x)}$$ but I do not understand why the indefinite integral of $\sec(x) \cdot \tan(x)$ is $\sec(x)$. I am watching the following videos: ...
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1answer
35 views

angle sine and cosine identities problem 3

Write in terms no greater than one. $$\sin^3x$$ I originally thought the answer was $\sin x\sin x\sin x$, I was wrong. After using these sine and cosine identities, I came up with ...
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2answers
28 views

General solutions for trigonometry equations

I'm taught that how to find the general solution for example $\cos 5\theta=\frac{\sqrt{3}}{2}$. But the exercise given by the book is much more complex than the example. For example, $\sin^2 ...
4
votes
1answer
294 views

Prove this is an isosceles triangle

In a triangle ABC, $\sin B\cdot\sin C=\cos^2(\frac{A}{2})$ Prove that this is an isosceles triangle. Can anyone guide me to prove this? Thanks
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1answer
46 views

Help me prove $\cos A - \sin A = \sin (A \sqrt{2})$, given $\cos A + \sin A= \cos (A \sqrt{2})$. [on hold]

Prove that:$$\cos A-\sin A=\sin A \sqrt{2} \quad \rm{given} \quad \cos A+\sin A= \cos A \sqrt{2}.$$
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4answers
35 views

Find $\theta$ in $\frac{\sin(45º+\theta)}{850}$=$\frac{\sin 30º}{433}$

Find $\theta$ in the equation \begin{equation*} \frac{\sin (45º+\theta)}{850}=\frac{\sin 30º}{433}. \end{equation*} I know how to use the sum and difference but i still can't get the value of theta. ...
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1answer
18 views

Period of a solution in a trigonometric equation

This is more of a general question, which keeps confusing me when solving trigonometric equations. When is the period $k\pi$, and when is it $2k\pi$? For example, if I need to solve $\tan x=1$, is ...
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1answer
41 views

Integration by parts prove integral of cos^n x dx [on hold]

I'm having a problem with one of my questions. How can I prove that $\begin{align}\int\cos^n x dx&=\sin x\cdot\cos^{n-1}x+(n-1)\int\sin^2x\cos^{n-2}x dx\end{align}$ ?
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0answers
36 views

Trigo Study plan

In what order of topics is probably the most effective in learning trigonometry for starters... where should I first start? and steps in between to De Moivre's Theorem (which is the last topic)... ...
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1answer
27 views

Explanation of two argument variant for arctan

Can someone please explain why $$\tan^{-1}\left(\frac{y}{x}\right)$$ has the additional conditions based on what the value of x and y are? I'm most specifically interested in the second equation: ...
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4answers
58 views

How do I solve the trigonometric equation $1 - \sin^2x - \cos(2x) = \frac{1}{2}$?

Solve for $x$ when $1-\sin^2x - \cos 2x = \dfrac{1}{2}$. I can' t change it into a form I can work with. It is rather complicated.
4
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5answers
74 views

find all possible solutions

The set of all $x$ in the interval $[0,\pi]$ for which $2\sin^2x-3\sin x+1 \geq 0$, is _________________. I have tried by factoring it first and then comparing it with the inequality. My final ...
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votes
2answers
42 views

Trigonometry problem.

If $ \sin\theta = n\sin(\theta + 2\alpha)$, then $\tan(\theta + \alpha) $ is equal to? I tried evaluating $n$, however I got no conclusive answer. I tried expanding $\sin(\theta + 2\alpha)$, but to ...
0
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1answer
40 views

Trigonometric identity [on hold]

I have troubles solving the following problem: Assume that $\alpha, \beta$ and $\gamma$ are the three angles in triangle. Show that: $$\cot \biggl( \frac{\alpha}{2}\biggl) + \cot \biggl( ...
2
votes
2answers
47 views

Establishing a trigonometric identity for $n\in\mathbb{N}$

The original problem was showing that this infinite sum converges to $\tan\theta$: $$\sum_{n=1}^\infty \frac{\tan\dfrac{\theta}{2^n}}{\cos\dfrac{\theta}{2^{n-1}}}$$ One hint was given: the series ...
0
votes
4answers
44 views

How to convert $\cos4\theta$ into $\cos3\theta$

How do i show that: $\cos 4θ = − \cos 3θ$ for each of the values θ = $\frac{\pi}7, \frac{3{\pi}}7, \frac{5{\pi}}7, \pi.$ How is $\cos4\theta$ related to $\cos3\theta$? Can someone please explain..