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

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2answers
38 views

Differentiate $y = \arccos( sin( x ) )$

I have found a different solution, so not urgent, I just can't make the solution below work completely: Using a right angled triangle, let $\sin(x) = a/c$ --> opposite over hypotenuse. $y = ...
2
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1answer
29 views

How do I find 2x2 orthonormal diagonalizing matrices using only trigonometry?

I have a matrix $A=\begin{bmatrix} a & b \\ c & d \end{bmatrix}$ (where all values are known), and I eventually want to diagonalize it into: $$ A=UDV^T $$ for orthonormal U and V. If I ...
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1answer
33 views

Calculate the derivative of sin5x using limits

Well, that's it. How do you calculate $\frac{d}{dx} {\sin5x}$ using the limit formula for derivatives? $$\lim_{h \to 0} \frac {f(x+h)-f(x)}h$$ I managed to get a lot of sines and cosines using ...
0
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0answers
23 views

Is there a 4 pointed star that is regular?

I am studying about the area of a 4 pointed star, I wonder if there is really a 4 pointed star that is regular? what could be the characteristics of a regular 4 pointed star?
2
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1answer
40 views

Integral of $\frac1{\cos^n x}$

Hi guys I have already proven for an assignment that: $$\int\cos(x)^n dx=\frac{1}{n}\cos(x)^{n−1}\sin(x) + \frac{n-1}{n}\int\cos(x)^{n−2}dx$$ Now we have been asked to calculate ...
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1answer
25 views

Turntable Photography problem, Concerning set rotation and intervals.

For a project I have to take pictures of an object on a rotating turntable. Setup is as follows: Camera with separate flashes are in front of a turntable taking pictures of a object on the ...
1
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2answers
37 views

General formula for $\sin\left(k\arcsin (x)\right)$

I'm wondering if there's a simple way to rewrite this in terms of $k$ and $x$, especially as a polynomial. It seems to me to crop up every so often, especially for $k=2$, when I integrate with trig ...
2
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3answers
38 views

Integrating $\int_{-\pi}^{\pi} \frac{ d\theta}{w - sin \theta}$

I know that the integral $$\int_{-\pi}^{\pi} \frac{ d\theta}{w - sin \theta} = \frac{2\pi}{\sqrt{w^2-1}}$$ where w, is an arbitrary constant and at some point you must do the substitution $$u = tan( ...
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1answer
19 views

Restricting the domain of the trigonometric functions in identities

I have this formula in my textbook: $ \sin ^2 \alpha = \frac{\tan ^2 \alpha}{1+\tan^2 \alpha} $ where $\alpha \neq \frac{\pi}{2}+ k\pi $ I think that the restriction for the angle is wrong and it ...
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1answer
17 views

Root of function involving trig and exponential

Would anyone know an analytical solution to finding the root of $$ f(x) = \sin(x^2) - e^x $$ in $[-1,1]$? I'm writing a simple root finding program and thought I'd try this as a test case, but ...
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0answers
4 views

Finding the 2 point coordinates for a known edge.

Say I have an edge A'B' which is a vector (5,3 9). How can I find the individial points A' and B' from A'B'. I translated the points A and B by a vector then combined them to make the edge AB. Then ...
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3answers
55 views

How do I show algebraically that the period of the tangent function is $\pi$?

How do I show that the positive real number $p$ for which $\tan (x+p)=\tan (x)$ is equal to $\pi$? In essence how do I prove the period of the tangent function is $\pi$? Please bear in mind I am a ...
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3answers
51 views

If SinA= SinB what does A equal

I know that if Tan A =Tan B then A = B + nπ This is because Tan has a periodicity of π What is the equivalent formula for Sin A = Sin B and Cos A = Cos B Please explain also why these formulas ...
3
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1answer
43 views

Calc 2: Integration by Parts w/ trig identities

$$\int e^{3\theta}\sec^4(e^{3\theta})\tan^{11}(e^{3\theta})d\theta$$ I just want to make sure that I'm doing this correctly so that I can understand the material. I would also appreciate any tips or ...
2
votes
3answers
82 views

Integral $\int_0^\pi \frac{x\,\operatorname dx}{a^2\cos^2x+b^2\sin^2x}$

Integrate: $$ \int_0^\pi \frac{x\,\operatorname dx}{a^2\cos^2x+b^2\sin^2x} $$
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0answers
14 views

Trig Question Lemma Application

A trig lemma states that if $a^2 + b^2 = 1$, there exists such an angle that a = cos theta, b = sin theta. This lemma is the converse of a = cos x, b = sin x. I am then given an application ...
2
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1answer
69 views

How to prove theorem using Euler's formula?

I'm having a great deal of trouble with this proof. "Prove $\cos θ + \cos 3θ + \cos 5θ + \cdots + \cos [(2n-1)θ] = \dfrac{\sin 2nθ}{2 \sin θ}$. Prove $\sin θ + \sin 3θ + \sin 5θ + \cdots + \sin ...
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4answers
62 views

How to simplify $(\sin\theta-\cos\theta)^2+(\sin\theta+\cos\theta)^2$?

Simplify: $(\sin \theta − \cos \theta)^2 + (\sin \theta + \cos \theta)^2$ Answer choices: 1 2 $ \sin^2 \theta$ $ \cos^2 \theta$ I am lost on how to do this. Help would be much appreciated.
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1answer
38 views

(complex variables) Find roots of a complex variable such that $1+\omega^m+\omega^{2m}+ \cdots + \omega^{(n-1)m} = 0$

I need some help understanding the intuition behind the following question: Consider the root of $z^n=1$ given by $\omega = \cos\frac{2\pi}{n} + i\sin\frac{2\pi}{n}$. For which integers $m$ is ...
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4answers
26 views

Proving a trigonometric identity (Homework)

$$\frac{(\sec A - \tan A)(\sec A + \tan A)} {\csc A-\cot A} \equiv \cot A + \csc A $$ So I started by using DOTS (Difference of two squares) on the numerator on the left hand side. This gave me: ...
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2answers
21 views

(complex variables) Express $\cos3\phi$, $\cos4\phi$, and $\cos5\phi$ in terms of $\cos\phi$ and $\sin\phi$.

I'm not sure what the intuition is supposed to be behind this question. This is my attempt at $\cos3\phi$. Does this look agreeable? We can use the identity $e^{iz}=\cos z + i \sin z$. Let $z = 3 ...
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2answers
40 views

how to prove that $\sin(90+v) = \cos(v)$

I need to prove that $\sin(90+v) = \cos v$ and that $\cos(90+v) = -\sin v$ So I did the following steps to prove these statements $\sin(90+v) = \sin(90-(-v)) = \cos(-v) = \cos(v)$ $\cos(90+v) = ...
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2answers
31 views

Why is that for any trigonometric function $f, f(2\pi + \theta )=f(\theta )$ for any value of $\theta$ [proof reading]

Here was the question asked to me :: Why is that for any trigonometric function $f, f(2\pi + \theta )=f(\theta )$ for any value of $\theta$ I spontaneously said that it was because of their very ...
0
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1answer
20 views

Finding the velocity of an object from its horizontal and vertical components?

Let's say a ball is thrown and it experiences typical projectile motion (moves in a parabolic arc etc.) and the only information we know are the equations for the horizontal and vertical components of ...
0
votes
1answer
14 views

Show P, Q and R are non collinear

If P $\equiv$ $(-sin(\beta - \alpha), -cos\beta)$, Q $\equiv$ $(cos(\beta - \alpha), sin\beta)$ and R $\equiv$ $(cos(\beta - \alpha + \theta), sin(\beta - \theta))$, where $$0 \lt \alpha, \beta, ...
0
votes
2answers
47 views

Trig:What's the graph of $\,\,\sin x .\sin x$?

I've been googling the graph of $\,\,\sin x .\sin x$ to see a visual of a trig problem i'm working on: $\sin^2x=\frac 14$ intervals $0$ to $2 \pi$. you have to use this equation to solve: ...
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5answers
61 views

Limits without L'Hopitals Rule

Evaluate the limit without using L'hopital's rule a)$$\lim_{x \to 0} \frac {(1+2x)^{1/3}-1}{x} $$ I got the answer as limit = 2/3... but I used L'hopitals rule for that... How can I do it another ...
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2answers
44 views

Why is the unit circle the preferred circle to derive values of trig functions?

I am reviewing basic trigonometry and came across a nice example that uses an equilateral triangle (sides of length r), cut in half vertically, to demonstrate that cos(60) = r/x = r/(r/2) = 1/2. From ...
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1answer
31 views

How is $r(\theta) = \sin \frac\theta2$ symmetric about the x-axis?

I understand how it is symmetric about the $y$-axis. because $r(-\theta) = \sin \left(-\frac\theta2\right)=-\sin \left(\frac\theta2\right)=-r(\theta)$ But how is it symmetric about $x$-axis?
0
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2answers
21 views

find the cosine of the angle ABC

I have problem with solving this task. I know that the answer might be A. But only with calculator by calculating the angles. can someone explain me or give me a hint to solve it. cos^-1 (5/13) = ...
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1answer
33 views

Limits and Trigonometry (Squeeze Theorem) [on hold]

So the question states to use the Squeeze Theorem to evaluate the following limits. (a) $\lim\limits_{x \to 1}\quad (x-1)\sin \left(\frac{\pi}{x-1}\right)$ (b) $\lim\limits_{x\to 0^-}\quad x^3 ...
0
votes
1answer
21 views

Explain trigonometry rewrite

While looking at a solution to a longer task I found this part that confuses me. How is this rewrite done? As it is presented in a one step way it should be trivial but I can't see it. $$ \left| \sin ...
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4answers
119 views

How can I prove this question concerning trigonometry? [on hold]

Prove that, for some constant $B$, $$4\cos(x) - 3\sin(x) = 5\cos(x+B).$$ Then, estimate the value of $B$.
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0answers
12 views

Simultaneously solving trigonometric equations

Let $N\in\mathbb N$. Given $\theta_1,\ldots, \theta_N\in [0,2\pi)$ I would like to prove that there exist $\rho\in\mathbb R_+$ and $\varphi\in[0,2\pi)$ such that $$ f_\ell(\rho,\varphi):=\theta_\ell ...
1
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4answers
98 views

$\text{Prove that}$ $\frac{\sin(\frac{n+1}2)*\cos(\frac n2)}{\sin\frac 12} \ge\frac n2$

Prove that$$\frac{\sin\left(\frac{n+1}2\right)\times\cos\left(\frac n2\right)}{\sin\left(\frac 12\right)} \ge\frac n2$$ So far I've switched up the problem and gotten it down to all sin functions. I ...
0
votes
2answers
15 views

Partial derivative of trig function

I need some assistance on the following calculus problem: Let $$w = 2\cot(x)+y^2z^2$$ $$x = uv$$ $$y = \sin(uv)$$ $$z = e^u$$ Find $\frac{\partial w}{\partial u}$ for $u = \frac{1}{4}$ and $v = ...
1
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2answers
42 views

How do you solve this trig/geometry question?

In a quadrilateral $ABCD$, if $\sin\left(\frac{A+B}2\right)\cos\left(\frac{A-B}2\right) + \sin\left(\frac{C+D}2\right)\cos\left(\frac{C-D}2\right) = 2$ then $\sin\left(\frac A 2\right) ...
-1
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1answer
47 views

Prove this trigonometric identity? [on hold]

Prove that $(\tan^2 \theta -\sin^2 \theta) = (\tan^2 \theta) \cdot (\sin^2 \theta)$
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2answers
34 views

How to write $a^{ix}$ in terms of $\sin(x)$ and $\cos(x)$?

We know that $e^{ix} = \cos(x) + i\sin(x)$ and the plot of $2^{ix}$ seems to have sinusoidal behavior. http://goo.gl/Xfg2wp Can we claim that we can write $a^{ix}$ in terms of $\sin(x)$ and ...
3
votes
3answers
40 views

Proving that $\dfrac{\tan(x+y)-\tan x}{1+\tan(x+y)\tan x}=\tan y$

Edit: got it, silly mistakes :) I need to prove that $\dfrac{\tan(x+y)-\tan x}{1+\tan(x+y)\tan x}=\tan y$ $$=\frac{\tan x+\tan y-\tan x+\tan^2x\tan y}{1-\tan x\tan y+\tan^2x+\tan x\tan y}$$ ...
0
votes
1answer
64 views

Fermat's Last Equation

Sorry this is an amateur question but I was wondering since Andrew Wiles solved Fermat's Last Theorem what effect does this have any impact on Geometry. Does this prove in a sense Higher Order right ...
1
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1answer
18 views

Calculating quadrant facing from a rotational matrix and two 3d vectors

I am working on a space-ship simulator, and having trouble with facing arcs between two space objects. Each object has a rotation matrix defined as follows: ...
0
votes
1answer
24 views

equation for the radius of a circle that is tangent to two lines and passing through a specific point on one of the lines?

I'm interested in finding the equation for the radius (and optionally the center point) for a circle that is tangent to two lines and passing through a specific point on one of the lines. So far, I've ...
1
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1answer
46 views

Trying to find an $\arctan(x/y)$ identity.

I have this equation : $$\theta = \arctan\left(\tfrac xd\right) + \arctan\left(\tfrac yd\right).$$ $\theta$ is an angle and I am trying to express $d$ as a function of $\theta$. So is there a way ...
1
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3answers
29 views

Implicit differentiation with trig function

I have the following expression which I need to implicitly differentiate: $$ xy^2 + x^2 + y + \sin(x^2y) = 0 $$ I'm a little confused as I'm not entirely sure what to do with the trig function. ...
0
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1answer
36 views

Calculating the points of a annular sector type shape.

The problem involves a circle inside a square sharing a common center point. The circle is always smaller than the square so that their edges never intersect. Then an annular sector (see cyan shape in ...
0
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1answer
19 views

Finding the vertical shift of a sinusoidal function

I'm currently studying sinusoids, I've been given a graph with a few key points and have been told to find a cosine function which fits it. When it comes to finding the vertical shift of the graph the ...
1
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2answers
44 views

$m\cos^2{\theta} + n\sin^2{\theta} < l \implies \sqrt{m}\cos^2{\theta} + \sqrt{n}\sin^2{\theta} < \sqrt{l} $

Prove that $m\cos^2{\theta} + n\sin^2{\theta} < l \implies \sqrt{m}\cos^2{\theta} + \sqrt{n}\sin^2{\theta} < \sqrt{l} $ for every $m, n, l >0$.
2
votes
2answers
22 views

Definite integral of trig function

I'm looking for some assistance on the following problem: Let $$ T(x) = \int_{4r^3}^{4} tsin(t^3)dt $$ Find $$T'(r)$$ I'm struggling to find the antiderivative of the sine function, particularly as ...
3
votes
6answers
100 views

Find the first derivative $y=\sqrt\frac{1+\cosθ}{1-\cosθ}$

$$y=\sqrt\frac{1+\cosθ}{1-\cosθ}$$ my professor said that the answer is $$y'=\frac{1}{\cosθ-1}$$ she said use half angle formula but I just end up with ...