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

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17 views

Perimeter of an ellipse intuition help

I am aware that you can take the circumference of an ellipse using an elliptic integral and haven't looked much into it, but that seems to be a bit extreme and i was taking a personal look at conic ...
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0answers
5 views

Least period of a nonzero sum of two sinusoidals.

Let $a,b,\omega_1,\omega_2,\phi_1,\phi_2\in\mathbb R$ where $a,b,\omega_1,\omega_2>0$. Let $f,g\colon\mathbb R\to\mathbb R$ by $f(t)=a\cos(\omega_1t-\phi_1)$ and $g(t)=b\cos(\omega_2t-\phi_2)$. ...
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1answer
25 views

equation solver online

can you tell me please if is there an online or software tool that will solve equations like -8*sin3x + 5*cos3x = 4.3 for 0< x <360? that I will just type equations like the above and it will ...
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2answers
48 views

Solving Integrals w/Trig

I need to solve the following integral: $$\int \sin^2(x)\cos^2(x) dx$$ This problem belongs to math notes that can be found here. Here are the steps listed to solve the equation. I can solve to a ...
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2answers
24 views

Trigonometry: How to determine the Period

I'm still kinda confused with solving the period on the diagram above. Amplitude= $3$ Max = $3$ Min = $-3$ Period = ? $y=a\cos(bx+c)$ Value of $a$ = $3$ Value of $b$ = ? Value of $c$ = ?
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1answer
61 views

Solve this tough fifth degree equation.

$$x^5+x^4-12x^3-21x^2+x+5=0$$ I think it can be solved by trigonometric ways but how?
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1answer
17 views

Compound Angles with an unknown [on hold]

$4.5 \sin\theta + 1.5 \sin(\theta + \alpha) = V_m \sin(\theta + \phi)$ Can this equation be solved without the value of $\alpha$ ? I'm looking for the values of $V_m$, $\theta$ and $\phi$.
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1answer
44 views

Solution to trigonometric derivative

Version 2 For \begin{align} &x(t)\text{:=}\cos (t)+\cos (2 t)+1&\\ &y(t)\text{:=}\sin (t)+\sin (2 t)&\\ \end{align} how would I go about proving that the solutions to \begin{align} ...
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2answers
13 views

Use of small approximations to get the following formula

I cannot see how the following formula has been found $\displaystyle \cos(\theta + \frac{v\epsilon^2 \Omega}{L}+O(\epsilon^4))-\cos(\theta) = -\epsilon^2\frac{\Omega v}{L}\sin(\theta)+O(\epsilon^4)$ ...
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4answers
63 views

Trigonometric Expression Simplification

Could someone explain how to simplify $(\cos(x)-\csc(x))/(\sin(x)-\sec(x))$? Any help would be appreciated.
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2answers
67 views

Why is $x^5 \sin x$ an odd function?

Why is $x^5 \sin x$ an odd function? Is the result just wrong? Because $f(-x)= (-x)(-x)(-x) \sin(-x) = (-x)(-x)(-x)(-x)(-x) (-\sin x) = (-x^5)(-\sin x) = x^5 \sin x$
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1answer
49 views

Given the matrix $A^k$, how to get $A^{k+1}$?

Given: $$A^k = \left(\begin{array}{rr} \cos kx & \sin kx \\ -\sin kx & \cos kx\end{array}\right)$$ $$A^{k+1} \overbrace{=}^? \left(\begin{array}{rr} \cos kx & \sin kx \\ -\sin kx & ...
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3answers
33 views

Maximum of subtended angle $\theta$

Following Problem, from Jim Fowler's Mooculus class: A painting is mounted on a wall. The bottom of the painting is 5 feet above eye level, and the top of the painting is 14 feet above eye level. If ...
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2answers
29 views

Simplfy trigonometric functions by only considering integer inputs?

I have the below function which only takes integer input, $$ 2 \sqrt{3} \sin \left(\frac{\pi t}{3}\right)+\sqrt{3} \sin \left(\frac{2 \pi t}{3}\right)-\sqrt{3} \sin \left(\frac{4 \pi ...
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3answers
23 views

finding and angle and coordinate point

"For a given angle $θ$ and a circle of radius $r$ and center $(h,k)$, recall that we can determine the Cartesian coordinates $(x,y)$ of the point on the circle determined by $θ$ and $r$, where ...
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1answer
19 views

Trigonometry graphs sinusoidal waves

i need help on this questions. I couldn't figure how to determine for both question A and B. But i have the answers for them, i just don't understand how the amplitude is 3 and so on.
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1answer
30 views

Exact value of polynomial at trigonometric argument

Given that $$\cos 8\theta= 128\cos^8 \theta −256\cos^6 \theta +160 \cos^4 \theta −32\cos^2 \theta +1$$ Find the exact value of: $$4x^4 −8x^3 +5x^2 −x$$ where $x=\cos^2 ...
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1answer
20 views

Finding the square roots of a complex number.

Express $z=4\sqrt2(1+i)$ in modulus/argument form. Hence find the two square roots of $z$ and mark their representations on an Argand Diagram. So far I've worked out the mod/arg form of the ...
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0answers
17 views

Lagging or Leading trigonometric functions.

Consider the function $f(x) = 2 * \sin(0.5 * x)$. Now suppose I want to create a function which is similar to the mentioned but to "lag" the mentioned function by $45$ degree angle, then which of the ...
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1answer
50 views

Is there a quicker way to write $\cos (n\theta)$ in terms of $\cos \theta$?

Im writing $\cos 8\theta$ in terms of $\cos \theta$ using De Moivre's Theorem $$\cos 8\theta= \Re {(\cos\theta+ i \sin \theta)^8}$$ Let $s=\sin \theta$ and $c=\cos \theta$ $$=c^8 ...
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2answers
34 views

Struggling to find the second derivative of this function's first derivative

So I've found the first derivative of this function but now I have to find the second derivative. I've tried everything but I cannot seem to get it. Here's the original function: $x = a sec(θ)$, $y = ...
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0answers
12 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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2answers
15 views

Product to sum formulas

Write the product as a sum. cos 4x cos 2x this is what i tried 2{cos2xcosx} = 2[1/2 cos(2x+1x)+ cos(2-1)] = 1[cos(3x)+cos(1x)] = cos 3x + cos x
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0answers
18 views

Triple integral containing definite integral and exponentials with trigonometric functions

I am attempting to solve the following integral analytically: $$ \int_{z=5i}^{z=1} \int_{t=\csc^{-12}(z)}^{t=2} \int_{\theta=\sin^{t}(z)}^{\theta=t^2} {[\mathrm{e}^{t\cos(\mathrm{e}^{i \theta})} + ...
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0answers
23 views

for what value of x is arcsin(sin(x)) = x true

for what value of x is arcsin(sin(x)) = x or sin(arcsin(x))= arcsin(sin(x)) true I know that the value is between -1 and 1 Could someone explain me why?
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2answers
73 views

Why “$\lim\limits_{x\rightarrow \infty} \frac{x+\sin x}{x}$ does not exist” is not an acceptable answer?

Find the limits: $\lim\limits_{x\rightarrow \infty} \frac{x+\sin x}{x}$ Since the numerator and denominator tends to infinity as $x$ tends to infinity, then applying Lhopital's rule: ...
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1answer
23 views

Is a trigonometric function applied to a rational multiple of $\pi$ always algebraic?

Specifically, just to talk about cosine, is it true that $\cos(\frac{a\pi}{b})$ is algebraic for integers $a$ and $b$? Looking at this post and the link to trigonometric constants in the comments, it ...
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3answers
65 views

How to prove that tg 55º<$\pi/2$

How to prove that tg 55º<$\pi/2$? I checked it on a calculator, but how to prove it though? Is it some trigonometric substitution?
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1answer
30 views

Which identity is being used to get $\sin(wa)\cos(wt)=\sin(w(a+t))+\sin(w(a-t))$?

Which identity is being used to get $\sin(wa)\cos(wt)=\frac{\sin(w(a+t))+\sin(w(a-t))}{2}$? Couldn't find it among the trigonometric identities.
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0answers
27 views

Looking for proof of formula in WolframMathWorld article [duplicate]

I came across the formula (24) in the WolframMathWorld article on Web page http://mathworld.wolfram.com/TrigonometryAngles.html where no source of the proof could be identified by me. The formula is ...
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3answers
37 views

Calculating a limit with infinitely many terms

I've encountered this limit : $$\lim_{n\to\infty} \frac1n \left(\sin\left(\frac{\pi}{n}\right) + \sin\left(\frac{2\pi}{n}\right)+\cdots+\sin{\frac{(n-1)\pi}{n}}\right)$$ Wolfram gives the value: ...
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1answer
41 views

Arctan(f(x)) is almost the same as Erf(f(x)) for many f(x). Is the just coincidence or is there a reason?

For example: Arctan(x) is almost Erf(x) (subtle differences in absolute value and curve) Arctan(x^50) is almost Erf(x^50) (difference in absolute value) and many others, so we can conclude: ...
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1answer
8 views

What is and what represents a convergents function in polynomial form?

$$\mathbf{convergents}(cos(1), 20)$$ What exactly is a convergents function and what, that series of fractions is representing ? There is an use for this in numerical linear algebra ? Feel free to ...
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0answers
5 views

Calculating distance of camera in 3D environment

I have a stage 840x840 px in size. My viewport is 840x840 px and so is my cube. I want the face of my cube to fit exactly the space of the viewport and so the flash stage. How can I calculate the ...
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2answers
27 views

Calculate angle betwen two lines

I have been trying to find the best solution to this problem, but my math is pretty bad. What I want to do is calculate the "Angle" in radians, I have all the 3 co-ordinates and all the 3 lengths ...
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0answers
26 views

Basis of Trigonmetric Polynomials Help

Write the following trigonometric polynomials in terms of the basis functions: $\cos^2(x)$ $\cos^2(x) \sin^3(x)$ Is there a certain way to solve these types of problems because I'm very unsure on ...
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1answer
54 views

Is it possible to expand $\sin(2x+1)\cdot\sin(2x+1)$?

Is it possible to treat it as a binomial?
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3answers
18 views

Finding an angle $\theta$ in a complex number

If we know that $z = \frac{1}{\sqrt2}(\cos\theta+i\cdot\sin\theta)$ and also that $z = \frac{(\sqrt3-1)+i(\sqrt3+1)}{4}$ How can I find $\cos\theta$ and $\sin\theta$? Using a calculator it gives me ...
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2answers
25 views

Trig Identity Confusion

Solving a problem I came across $\cos^2(t) - \sin^2(t) + 1$. The back of the book has the next step answer as $\cos(2t) -1$. Using the double angle identity how is it possible to receive the $-1$?
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2answers
16 views

Am I going about this wrong? Complex expression to polar form

I have the expression below, which I'd like to write in polar form. $$z = \frac{i}{{1+\frac{i(\sqrt3-1)}{1+i}}}$$ Own process My process was very tedious; and I also wouldn't solve the final part ...
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1answer
38 views

De Moivre's Theorem for proving

I have been asked by my lecturer to answer this question but I'm having problems doing so. The question is: Prove that $$\cos (5\theta) = 16\cos^5\theta - 20\cos^3\theta + 5 ...
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29 views

Condition for trigonometric inequality

I want to prove the following statement: Suppose $\frac{1}{4}(\cos(\theta_1)+\cos(\theta_2))^2+\lambda^2(a\sin(\theta_1)+b\sin(\theta_2))^2\leq 1$ holds for all $\theta_1,\theta_2\in[-\pi,\pi]$, then ...
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2answers
45 views

Why is the sine and cosine always between $-1$ and $1$?

Why is the sine and cosine always between $-1$ and $1$? If I would have circle with a radius other than $1$, then it wouldn't be between $-1$ and $1$ anymore, would it? This also ties in with another ...
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5answers
42 views

Proof of $\arcsin x \le 2\arctan x$?

I am looking for a proof for the following 'fact': $$ \arcsin x \le 2\arctan x \quad \forall x\in[0,1). $$ I put fact between single quotes, as the only proof I found is a plot by wolframalpha. I know ...
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1answer
15 views

Using sine law determine missing angel x

Okay so I need to use SINE to determine the missing angle. (X). I know one angle is 85 degrees. And the sides are 25mm, 43.8 mm, and 36 mm, can someone please tell me what I have to put into my ...
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2answers
52 views

How is $\tan^{-1}(a/b) = \tan^{-1}(a) - \tan^{-1}(b)$?

I'm having a problem proving: $\tan^{-1}(a/b) = \tan^{-1}(a) - \tan^{-1}(b)$ Thanks!
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3answers
62 views

The value of $\int_0^{2\pi}\cos^{2n}(x)$ and its limit as $n\to\infty$

Calculate $I_{n}=\int\limits_{0}^{2\pi} \cos^{2n}(x)\,{\rm d}x$ and show that $\lim_{n\rightarrow \infty} I_{n}=0$ Should I separate $\cos^{2n}$ or I should try express it in Fourier series?
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23 views

Calculating originally arc approximated by cubic bezier curve

I have an cubic bezier curve, which is representing an arc by an approximation. The approximation was calculated with the kappa constant: $$ \\k = \frac43*(\sqrt{2}-1) $$ This means, that the ...
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3answers
16 views

Cyclic quadrilaterals - finding the size of an angle

I know this might seem like a really simple question, but I really don't understand where I am going wrong. I am familiar with cyclic quadrilaterals as well as their properties, but this question ...
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1answer
23 views

Finding Y coordinate of third triangle point when X coordinate and two other points are already known

Suppose you know the coordinates for points A and B of a triangle. We can refer to those coordinates as (Ay,Ax) and (By,Bx). Also, suppose you know the X coordinate for point C (Cx) but do not know ...