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

Value of $cos\theta$ when $\theta$ is very small

What happens (actually why) to the value of $cos\theta$ when $\theta$ is small enough that its higher powers that is $\theta^2$ (and more) can be neglected?
3
votes
3answers
47 views

Factorize Trigonometric Equation

I have a problem with the following trigonometric equation: $$3\sin(x)^2 - 2\sin(x)\cos(x) - \cos(x)^2 = 0$$ It's from the book Engineering Mathematics 7th edition by Stroud. The book is giving the ...
2
votes
3answers
49 views

How can I solve the simultaneous equations that arise in solving $\cos(z)=2$.

If I have $\cos(z)=2$ I can say $\cos(a+ib)=2$ using double angle ideas $\cos(a)\cos(ib)+\sin(a)\sin(ib)=2$ using Euler's formula $\cos(a)\cosh(b)+i\sin(a)\sinh(b)=2$ equating real and imaginary ...
2
votes
1answer
47 views

Trigonometric identities with multiplication

Why aren't there Trigonometric identities with multiplication inside the function? For example for $\sin(xy)=?$.
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1answer
878 views

How to find how far something has traveled. trig help

Sandra is riding the Ferris wheel, and her height can be modeled by the equation $H(t) = 25 \cos (\pi/14) t + 31$, where $H$ represents the height of the person above the ground in feet at $t$ ...
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3answers
88 views

Does $\tan x\cdot \cos x$ equal $\sin x$?

Is it true that $\tan x\cdot \cos x = \sin x$? If I put $x=30$ in my calculator then I don't get the same answer as $\sin 30$, why is this? Don't the two cosines cancel out? I'm probably missing ...
1
vote
1answer
25 views

Trigonometric Equation, quadratic using two functions

I am struggling to know how to solve this equation as it involves more than one type of trigonometric function, I know how to do it with one repeated function. If a solution could be explained, that ...
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2answers
46 views

How to prove this trigonometry

I need to prove that $$\cos^2(\beta -\gamma)+ \cos^2( \gamma - \alpha) +\cos^2(\alpha -\beta) = 1+2 \cos(\beta- \gamma) \cos( \gamma - \alpha)\cos(\alpha -\beta) $$ To do this I have used the ...
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1answer
28 views

Proof of Compound Angle from Ptolemy's Theorem

I have a query regarding a proof I'm reading on the additive Sine compound angle formula, which uses Ptolemy's theorem. http://www.cut-the-knot.org/proofs/sine_cosine.shtml I'm looking at the ...
1
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1answer
28 views

$K$ is a region in $\mathbb{R}^2$ where the area is $5$

Say that $K$ is a region in $\mathbb{R}^2$ where the area is $5$. Let B = \begin{pmatrix} 3 & 8 \\ 4 & 6 \end{pmatrix} Find the area of the region B$K$. Any starting hints? Is it possible ...
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1answer
136 views

Has the age at which we teach Mathematics changed over the last two centuries?

My experience of learning Advanced Trigonometry and Calculus is that it was done to 17 and 18 year olds (School Curriculum in Australia). I assumed that it was similar in the UK, US and Europe. In ...
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1answer
24 views

Proving $\sin ((n-1/2)\phi) + \sin(\phi/2)=\sin({n+1 \over 2}\phi)$

I am trying to show that $\sin ((n-1/2)\phi) + \sin(\phi/2)=\sin({n+1 \over 2}\phi)$ I tried to apply $\sin (x-y) = \sin x \cos y - \cos x \sin y$ to it and I got $\sin ((n-1/2)\phi) + ...
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1answer
21 views

Find the largest segment

I have seven lines with different measures. The length of each line it's a positive integer and the shortest length is equal to 1 cm. It is known that's impossible to choose three of them that makes a ...
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0answers
68 views

Link between a cubic polynomial and a trig identity

Alright, so I am told to prove that: $$\tan (3A) = \frac{3\tan(A)-\tan^3(A)}{1-3\tan^2(A)}$$ This can be pretty easily done by applying the $\tan$ addition formula, taking the angles $2A$ and $A$, ...
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votes
2answers
33 views

Finding the measure of a base angle of an isosceles triangle given its side lengths [on hold]

I know this is looked down upon, but it is a simple question I got jammed on and can't figure out why I kept getting the wrong answer. I just want to see solution, simple question for most of you ...
4
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2answers
33 views

Roots of unity, where $\omega^3 = 1, \omega \neq 1$.

Say that $\omega^3 = 1$ and $\omega \neq 1$. Find the value of $(1 - \omega + \omega^2)(1 + \omega - \omega^2)$. I'm not very good at the roots of unity. May I have a couple of hints to get started? ...
3
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1answer
70 views

Trouble solving $\int\frac{x}{\sqrt{x^2-6x}}$

I need to solve the following integral $$\int\frac{x}{\sqrt{x^2-6x}}$$ I started by completing the square, $$x^2-6x=(x-3)^2-9$$ Then I defined the substitution variables.. ...
7
votes
1answer
157 views

Evaluating $\int_0^\pi \frac{x}{(\sin x)^{\sin (\cos x)}}dx$

Evaluate $$\int_0^\pi \frac{x}{(\sin x)^{\sin (\cos x)}}dx$$ I tried using by parts and complex numbers along with series expansion but I was unable to find the answer. Please Help!
3
votes
3answers
37 views

Find the least degree Polynomial whose one of the roots is $ \cos(12^{\circ})$

Find the least degree Polynomial with Integer Coefficients whose one of the roots is $ \cos(12^{\circ})$ My Try: we know that $$\cos(5x)=\cos^5x-10\cos^3x\sin^2x+5\cos x\sin^4x$$ Putting ...
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5answers
46 views

Find an algebraic expression for sin(arccos(x))

I have a question that asks me to find an algebraic expression for sin(arccos(x)). From the lone example in the book I seen they're doing some multistep thing with the identities, but I'm just not ...
2
votes
3answers
135 views

Prove that $2\sum_{k=1}^n\cos kθ= \frac{\sin\left(n+\frac12\right)θ}{\sin\fracθ2}-1$ [closed]

Prove that $$2\sum_{k=1}^n\cos kθ= \frac{\sin\left(n+\frac12\right)θ}{\sin\fracθ2}-1$$ By using $$e^{iθ}+e^{2iθ}+\cdots+e^{niθ}=\frac{e^{iθ}(1-e^{inθ})}{1-e^{iθ}}$$
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1answer
90 views

Help with this trigonometry problem?

Is there an easier way of doing this problem: A square tower stands upon a horizontal plane. From a point in this place from which three of its upper corners are visible their angular elevations ...
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3answers
48 views

Show that: $\sinh^{-1}(x) = \ln(x + \sqrt{x^2 +1 } )$

could someone Please give me some hint of how to do this question thanks
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votes
6answers
123 views

Evaluate $\lim_{x→0}\left(\frac{1+\tan x}{1+\sin x}\right)^{1/x^2} $

I have the following limit to evaluate: $$ \displaystyle \lim_{x→0}\left(\frac{1+\tan x}{1+\sin x}\right)^{1/x^2} $$ What's the trick here?
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4answers
90 views

Trigonometric Limit: $\lim_{x\to 0}\left(\frac{1}{x^2}-\frac{1}{\tan^2x}\right)$

I cannot figure out how to solve this trigonometric limit: $$\lim_{x\to 0} \left(\frac{1}{x^2}-\frac{1}{\tan^2x} \right)$$ I tried to obtain $\frac{x^2}{\tan^2x}$, $\frac{\cos^2x}{\sin^2x}$ and ...
0
votes
5answers
48 views

How to prove $\displaystyle\lim_{x \to 0} \dfrac{\sin^{-1} x}{x} = 1$?

How to prove this? Is there any geometrical proof? I have proved , btw, $\displaystyle\lim_{x \to 0} \dfrac{\sin x}{x} = 1$ by Sandwich Theorem and little geometry.
2
votes
6answers
73 views

Solve for $x:1 + \tan^2(x) = 8\sin^2(x)$

I have a tricky problem , I tried several methods and I can't seem to get a definite answer. $1 + \tan^2(x) = 8\sin^2(x), x \in [\frac{\pi}{6} , \frac{\pi}{2}]$ I got to ...
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1answer
37 views

trigonometry - identities and formula, proving

Prove: $$\sin(x) - \sin(x) \cos^2(x) = \sin^3(x).$$ I swear it is easy, but I don't know what I'm forgetting to look at?
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3answers
944 views

General term of Taylor Series of $\sin x$ centered at $\pi/4$

What is the general term for a Taylor series of $\sin(x)$ centered at $\pi/4$? It should be $(-1)^{[??]} \times \sqrt{2}/2 \times \frac{(x-\pi/4)^n}{n!}$ What power is $(-1)$ supposed to be raised ...
0
votes
1answer
36 views

Solving triangle

If side $a$ is known and the angles are given as functions of two variables (let's call them $x$ and $y$), what is the easiest way to find $y$ as a function of $x$. To make things easier, let one of ...
2
votes
3answers
2k views

How to find all the solutions to cos$(z)=0$

How would you go about finding all the solutions to $\cos(z)=0$, where $z\in \mathbb{C}$? I have $$\cos(z)=0 \implies (e^{iz})^2=-1 \implies ...
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1answer
24 views

Inverse cosine of a complex number, take $\cos z=\sqrt{2}$ for $z$

If I am given $\cos z=\sqrt{2}$ for $z$ and asked to solve it using the following: $$ \cos^{-1} z =-i \log\sqrt{z+i(1-z^2)} $$ I've only gotten as far as taking $\cos z=\sqrt{2}$ and changing it to ...
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4answers
85 views

Find the exact value of $\cos(11\pi/12)$.

This question I look at as being similar to $\sin(7\pi/12)$. You can break it up using the special triangles into $3\pi/12 + 4\pi/12$. However with this one, I can't find one of the angles in which ...
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3answers
62 views

How to prove $\sqrt\frac{1 - \sin x}{1 + \sin x} = \frac{1}{\cos x} - \tan x$

Prove $\sqrt\frac{1 - \sin x}{1 + \sin x} = \frac{1}{\cos x} - \tan x$ I tried but I couldn't figure it out, give me a hint please.
23
votes
4answers
702 views

Is $\sum_{k=1}^{m-1}\frac{1}{\sin^2\frac{k\pi}{m}}=\frac{m^2-1}{3}$ true for $m\in\mathbb N$?

Question : Is the following true for any $m\in\mathbb N$? $$\begin{align}\sum_{k=1}^{m-1}\frac{1}{\sin^2\frac{k\pi}{m}}=\frac{m^2-1}{3}\qquad(\star)\end{align}$$ Motivation : I reached ...
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votes
1answer
67 views

How prove that $\frac{1}{\sin^2\frac{\pi}{2n}}+\frac{1}{\sin^2\frac{2\pi}{2n}}+\cdots+\frac{1}{\sin^2\frac{(n-1)\pi}{2n}} =\frac{2}{3}(n-1)(n+1)$ [duplicate]

How prove that sum $$\frac{1}{\sin^2\frac{\pi}{2n}}+\frac{1}{\sin^2\frac{2\pi}{2n}}+\cdots+\frac{1}{\sin^2\frac{(n-1)\pi}{2n}} =\frac{2}{3}(n-1)(n+1)$$
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2answers
60 views

show that $(1 +\sin{A}) + \cos^2{A} = 2(1 + \sin{A})$? [on hold]

It is an identity. The question is asking me to show that the left hand side is equal to the right hand side. $$(1 +\sin{A}) + \cos^2{A} = 2(1 + \sin{A})$$
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1answer
160 views

Express $C_n = \cosh(0) + \cosh(1) + \cosh (2) + \dots + \cosh(n)$

Could someone give me some hint of how to do this question please. I've been stuck for more than $3$ hours on this question.
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2answers
52 views

finding angle value inside this triangle

I need a method to calculate the angle X in the image below, I know its value (30 degree) but how ?!! thank you.
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0answers
24 views

Dirichlet integral using real Analysis

The teacher made this approach to solve the Dirichlet integral , $$ J_n= \int_0^\frac{\pi}{2} \frac{\sin(2nx)}{\sin x}\:\mathrm{d}x,\quad I_n = \int_0^\frac{\pi}{2} \frac{\sin(2n+1)x}{\sin ...
1
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1answer
27 views

Trig and Geometry problem

I have this problem to solve. There is a triangle ABC containing a line segment bisecting Angle C with length s. The side opposite angle A is length a, across angle B is length b and the measure of ...
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3answers
58 views

Complex Number to a power

I asked this question yesterday, but the answers did not actually answer what I wanted to know since I asked the question in the wrong way. I have $e^{i\frac{2014\pi}{12}}$. I know Euler's formula, ...
2
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1answer
424 views

Solving for radius of a combined shape of a cone and a cylinder where the cone is base is concentric with the cylinder?

I have a solid that is a combined shape of a cylinder and a concentric cone (a round sharpened pencil would be a good example) Know values are: Total Volume = 46,000 Height to Base Ratio = 2/1 ...
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1answer
25 views

Find an equation for a sinusoid with minimum and maximum

Here's my problem: Find an equation for a sinusoid that has a minimum at (30°,-1) and an adjacent maximum at (75°,7). Please help! I've tried everything I can think of, but I'm really drawing a ...
5
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1answer
142 views

Why are logarithms of trigonometric functions useful?

I have noticed that in many trigonometric tables the logarithm of the trigonometric values are given. Why this is given and not the actual values of the trigonometric functions? For example, instead ...
14
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0answers
534 views

On Ramanujan's Question 359

In JIMS 4, p.78, Question 359 was asked by Ramanujan. If, $$\sin(x+y) = 2\sin\big(\tfrac{1}{2}(x-y)\big)\tag1$$ $$\sin(y+z) = 2\sin\big(\tfrac{1}{2}(y-z)\big)\tag2$$ prove that, ...
2
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0answers
21 views

Can trigonometric functions for double precision be implemented in terms of those for single precision?

In some program environments like GLSL there is support for single and double precision numbers for arithmetic and square roots computation, but only single precision trigonometric functions are ...
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0answers
20 views

Cartesian extremities of a 3d segment

I have a segment in 3d space and I want to calculate its extremities. I know the cartesian coordinates (x,y,z) of the segment's middle point, the segment's length L and the segment's orientation using ...
4
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0answers
47 views

How prove $\root 4\of{\frac{1}{2}\sin x\cos z}+\root 4\of{\frac{1}{2}\cos x\sin z}=\root{12}\of{\sin 2y} $?

Let $\sin(x+y) = 2\sin\left(\dfrac{x-y}{2}\right)$ and $\sin(y+z) = 2\sin\left(\dfrac{y-z}{2}\right)$. How prove $\root 4\of{\frac{1}{2}\sin x\cos z}+\root 4\of{\frac{1}{2}\cos x\sin ...
2
votes
2answers
88 views

Lowering powers of $\cos^2x \sin^4x$

First, I will be straight up, this is a homework question. I need to write $\cos^2 x \sin^4 x$ in terms of cosine to the first power. I know that $\sin^4x$ = $$ \frac{3-4\cos 2x+\cos 4x}{8}$$ from ...