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

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0
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2answers
30 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 = ...
0
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0answers
11 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 ...
0
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2answers
14 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
0
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0answers
10 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})} + ...
0
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0answers
20 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
68 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: ...
2
votes
1answer
22 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 ...
3
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3answers
53 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?
1
vote
1answer
28 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
24 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
36 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: ...
0
votes
1answer
37 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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votes
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 ...
0
votes
2answers
25 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 ...
0
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0answers
24 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 ...
2
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2answers
53 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 ...
1
vote
2answers
24 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
15 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 ...
0
votes
1answer
36 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 ...
1
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0answers
26 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 ...
0
votes
2answers
44 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 ...
2
votes
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 ...
0
votes
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 ...
1
vote
2answers
51 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!
1
vote
3answers
60 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?
0
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0answers
21 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 ...
0
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3answers
15 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 ...
0
votes
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 ...
1
vote
1answer
9 views

Find all of the exact solutions of the equation and then list those solutions which are in the interval [0, 2pi)

This is for trigonometric equations and inequalities: Find all of the exact solutions of the equation and then list those solutions which are in the interval [0, 2pi) Cos(9x)=9
1
vote
1answer
18 views

Finding Trig Functions

I am given $$\cos(\beta) = \frac{\sqrt{15}}{8}$$ and I am asked to find the indicated trig function of $$\sin(90^{\circ}- \beta)$$ I know $$\sin(\beta)= \frac{7}{8}$$ but I don't know how to go about ...
0
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0answers
17 views

The cosine rule, Bearings [on hold]

two forest rangers leave base and travel to different parts of the forest. Simone averages 70km/hr along a straight track in a direction 25 degrees, While bruce averages 80km/hr in a direction 135 ...
0
votes
2answers
27 views

Limit as x approaches 0 from the left: $\lim_{x \to 0^{-}} \sin^{-1}\left({\frac{1}{2+e^\frac{1}{x}}}\right)$

Help me find the limit as x approaches 0 from the left: $$\lim_{x \to 0^{-}} \sin^{-1}\left({\frac{1}{2+e^\frac{1}{x}}}\right)$$ Thanks,
3
votes
2answers
52 views

Prove $\lim_{x\rightarrow 0}\cos (x)=1$ with the epsilon-delta definition of limits

Prove $$ \lim_{x\rightarrow 0}\cos (x)=1 $$ with the epsilon-delta definition of limits
2
votes
1answer
45 views

Moving a point around a circle

we're currently working on a game which involves a character that rotates around a point. We are using a rotation matrix to rotate a given a point (x,y) around another point by first translating to ...
2
votes
0answers
13 views

How to compute uniformly distributed points on an ellipse

The ellipse can be parametrized in polar coordinates by $$r(\theta)=\frac{1}{a+\cos\theta}$$ up to a scaling factor, and $a>1$. Suppose we measure $S$, the distance along the ellipse from the ...
-1
votes
1answer
21 views

Trigonometric problem with $\cos{\alpha}$ and $\sin{\alpha}$

The problem is $$3 \cos{\alpha} = 3 - 2\sin{\alpha}.$$ Find the value of angle alpha I have some trouble solving this. I don't know where to start.
-3
votes
1answer
31 views

If $y = 2\sin(x)-\sin^2(x)$ and $x = 2\cos(x)-\sin(x)\cos(x)$ what is $\frac {dy}{dx}$? [on hold]

If $y = 2\sin(x)-\sin^2(x)$$\ \ \ x = 2\cos(x)-\sin(x)\cos(x)$ What would $\frac {dy}{dx}$ equal to? so $\frac {dy}{dx}=2\cos(x)-\frac {2\cos(x)\sin(x)}{-2sin(x)}$ ... ? what would $y'$ of ...
0
votes
0answers
23 views

Determine sine wave frequency from two arbitrary points

If I have only two arbitrary points on a sine wave, what would be the simplest method for determining the frequency of the sine wave? The frequency is unknown. The bandwidth is restricted, the time ...
11
votes
2answers
122 views

How does one evaluate $\int \frac{\sin(x)}{\sin(5x)} \ dx$

The below problem is taken from Joseph Edwards book Integral Calculus for beginners. How does one show: $$5 \int \frac{\sin(x)}{\sin(5x)} \ dx= \sin\left(\frac{2\pi}{5}\right) \cdot ...
-1
votes
1answer
27 views

Is $\cos(\arctan(-x) - \arcsin(y))$ the same as $\cos(\arctan(-x)) - \cos(\arcsin(y))$?

Is $\cos(\arctan(-x) - \arcsin(y))$ the same as $\cos(\arctan(-x)) - \cos(\arcsin(y))$? Because if I put this on my calculator I got different solutions.
4
votes
1answer
38 views

How to prove this inequality relating to trigonometric function?

In a triangle, A, B, C are three corners of the triangle, try to prove that : $$\root 3 \of {1 - \sin A\sin B} + \root 3 \of {1 - \sin B\sin C} + \root 3 \of {1 - \sin C\sin A} \geqslant {3 \over ...
3
votes
4answers
48 views

$ \sin x \geq \frac{2x}{\pi}, x \in [0, \frac{\pi}{2}]$

I have tried the following: $$ f(x) = \sin x-\frac{2x}{\pi} \\ f'(x)= \cos x-\frac{2}{\pi} \\ f''(x) = -\sin x \leq 0 $$ But this doesn't seem to be heading in the right direction as it would appear ...
1
vote
0answers
19 views

Is a sine wave plus the sum of its odd harmonics symmetrical around the x axis at half the period of the fundamental?

I have a function such that $$x(t)=A_1 \sin(2 \pi f t+\phi_1)+A_2 \sin(2 \pi (3f) t+\phi_2)+...+ A_n \sin(2 \pi ((2n+1)f) t+\phi_n)$$ Is such a function symmetric around the point that is half ...
3
votes
3answers
46 views

Using $x=\tan \theta$ to solve $\int x\sqrt{1+x^2}\,\mathrm dx$

I'm having a lot, I repeat, a lot of trouble with Calculus II, particularly trigonometric substitution. At the moment, I'm extremely confused as to how to integrate $\int x\sqrt{1+x^2}\,\mathrm dx$ ...
4
votes
1answer
31 views

Find the missing angle of similar triangle

Find the missing angle $\theta$ in the triangle below given that $R>r$, $l\geq R$, $0< \theta < \frac{\pi}{2}$. Attempted Solution I attempted to use similar triangles to find the angle ...
6
votes
5answers
59 views

The limit as $x$ goes to infinity of $x-x\cos(4/x).$

I would like to determine $$\lim_{x\to \infty} \left(x-x\cos \frac4x\right)$$ How do I even start this? I can't plug in infinity to $4/x,$ can I? That will make it zero? Is the answer infinity then?
1
vote
2answers
40 views

How to find the exact value of $\tan(\sec^{-1} 4)$?

I would like to know if there is a general method to solve equation looking like this: $$\tan(\sec^{-1} 4)$$ without using a calculator (you have to find the exact value)? How to proceed?
0
votes
1answer
22 views

Prove this trig identity?

I'm having trouble proving the following identity: $$A\cos(\omega t) + B\sin(\omega t) = \sqrt{A^2 + B^2} \cos(\omega t - \arctan(\tfrac{B}{A}))$$ Does anyone know how this can be done? Thanks!