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

learn more… | top users | synonyms (2)

2
votes
1answer
65 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 -28c^6(1-c^...
0
votes
2answers
54 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
votes
1answer
50 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
votes
2answers
32 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
2
votes
2answers
141 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: $\lim\limits_{...
7
votes
2answers
151 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
votes
3answers
127 views

How to prove that $\tan 55^\circ<\pi/2$

How to prove that $\tan 55^\circ<\pi/2$? I checked it on a calculator, but how to prove it though? Is it some trigonometric substitution?
1
vote
1answer
37 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.
1
vote
0answers
30 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 ...
4
votes
3answers
57 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: $2\...
0
votes
1answer
188 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: Arctan(...
0
votes
1answer
10 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 ...
2
votes
1answer
171 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
42 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 of ...
2
votes
1answer
62 views

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

Is it possible to treat it as a binomial?
1
vote
3answers
49 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
36 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 $-1$?
0
votes
2answers
30 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 ...
1
vote
1answer
51 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 \cos\theta\text{...
6
votes
0answers
120 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
703 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 ...
1
vote
5answers
96 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 ...
1
vote
2answers
62 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
72 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
votes
1answer
225 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
votes
3answers
44 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
445 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
59 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
22 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
votes
2answers
48 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
407 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
333 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 ...
3
votes
0answers
52 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 $\...
12
votes
2answers
283 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 \log\left\{\...
4
votes
1answer
47 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
278 views

How to show $ \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 $...
3
votes
3answers
65 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
48 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
125 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
665 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
28 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!
3
votes
0answers
64 views

How to evaluate the following integrals

$$\int\limits_0^{\frac{\pi }{2}} {{x^2}{{\ln }^2}\left( {\sin x} \right)\ln \left( {\cos x} \right)dx} ,\int\limits_0^{\frac{\pi }{2}} {x\ln \left( {\sin x} \right){{\ln }^2}\left( {\cos x} \right)dx} ...
2
votes
2answers
263 views

Need some help solving high-school level trignometry question.

here it is. I've tried solving it multiple ways but it gets too complicated. Is there any way to solve this?
0
votes
1answer
35 views

Trigonometry rewrite

How do you go from $$ \frac{ \sin z }{e^z -1} $$ to $$ \frac{\cos z}{e^z}$$ My first approach was to use eulers formula. But that made the calculations quite hard. I'm guessing this can be done ...
1
vote
1answer
186 views

Absolute extrema of $\sin x+\cos y+\sin (xy)$ on a square

I want to find the absolute extrema of the function $f(x,y)=\sin x+\cos y+\sin (xy)$ on $\{ (x,y) \mid 0\le x\le 2\pi,~0\le y \le 2\pi \}$. I tried by finding the gradient of the function $f$, but it ...
1
vote
2answers
55 views

Some help with sin and cos

I'm having trouble to understand the following equalities in these two equations, i.e. how to apply the addition formulas. Firstly: $$ \frac {1- \frac {sin^2(\frac x2)} {cos^2(\frac x2)}} {1+ \frac {...
0
votes
1answer
37 views

Trigonometric ratios

I'm stuck with a problem. Given is a triangle $\Delta ABC$ with $\angle A = 35°, BC=3$ and $AC=5$. I need to find the two possible values for $\angle C$. I only managed to found one angle. I did the ...
1
vote
1answer
28 views

$\tan \{\frac{1}{2} \sin^{–1} (2x/ 1 + x^2) + \frac{1}{2} \cos^{–1} (1 – y^2/1 + y^2) \}$ is equal to.

$$ \tan \left\{ \frac{1}{2} \arcsin \frac{2x}{1 + x^2} + \frac{1}{2} \arccos \frac{1 – y^2}{1 + y^2} \right\} $$ is equal to. Note: i think $\sin a=2x/1+x^2$, $\cos b=(1 – y^2/1 + y^2)$
-1
votes
1answer
25 views

How many milliliters of liquid to fill [duplicate]

A right circular cone has a depth of 103 mm and a top diameter of 82.4 mm. The cone contains water to a depth of 30.0 mm. How many more millilitres of liquid need to be added in order to fill the ...
1
vote
0answers
280 views

Find the length of the longer diagonal on a trapezium with only 2 sides stated.

Im at a loss here, i know i have to divide the trapezium, but im still not sure which calculation is relevant to it then. Thanks in advance.