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

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-1
votes
1answer
66 views

Find the exact value of $2\left(\cos\frac{4\pi}{19}+\cos\frac{6\pi}{19}+\cos\frac{10\pi}{19}\right)$

$$S=2\left(\cos\frac{4\pi}{19}+\cos\frac{6\pi}{19}+\cos\frac{10\pi}{19}\right)$$ I've tried the sum of $2 \cos$ but can't find any solution. Please help!
-1
votes
2answers
30 views

How to find intersections of sine and cosine functions with $X$ axis

I've been struggling with this question for a few days, because I've been able to find the said intersections, but based on suppositions, rather than on mathematical process. For example, if I have ...
1
vote
0answers
31 views

Prove trigonometric inequality with sin

Let $ n\in \mathbb{N}^{*},x\in \mathbb{R} $. Prove that $ sin^{2}(x)\cdot sin^{2}(2x)\cdot ...\cdot sin^{2}(2^{n}x)\leq \left ( \frac{3}{4} \right )^{n},\forall x\in \mathbb{R} $. The only result I ...
0
votes
3answers
35 views

sum of all Distinct solution of the equation $ \sqrt{3}\sec x+\csc x+2(\tan x-\cot x) = 0\;,$

The sum of all Distinct solution of the equation $\displaystyle \sqrt{3}\sec x+\csc x+2(\tan x-\cot x) = 0\;,$ Where $x\in (-\pi,\pi)$ and $\displaystyle x\neq 0,\neq \frac{\pi}{2}.$ ...
2
votes
4answers
126 views

Extended $\lim_{x \rightarrow 0}{\frac{\sin(x)}{x}} = 1$ limit law?

So I've learned that $\lim_{x \rightarrow 0}{\frac{\sin(x)}{x}} = 1$ is true and the following picture really helped me get an intuitive feel for why that is I have been told that this limit is ...
0
votes
2answers
29 views

Proof of a trigonometric relation

In the solution of the ambiguous case for plane triangles, in which the sides $ a, b$ and angle $ A $ are given, how to prove that $ (c+c')/2 = b\cos A $ Where $c$ and $c'$ are the corresponding ...
0
votes
0answers
34 views

Is there an analytic solution to find zeroes of a polynomial plus sin()?

Is there an analytic solution to find the zeroes of an equation of the form: $$0 = at^2+bt+c+\sin(mt^2+nt+o)$$
2
votes
3answers
50 views

How to work with trig functions when dealing with limits tending to a point, without using L'Hôpital's rule

Past Paper Question: For each of the following functions f , determine whether $\lim_{x\to a}f(x)$ exists, and compute the limit if it exists. In each case, justify your answers. a) $f(x)= ...
4
votes
1answer
33 views

Sketching the graph of $\cos(4\pi t-\pi)$

I am struggling to sketch the graph of $\cos(4\pi t-\pi), -3<t<3$. Here is what I have tried: $w=4\pi$ $f=2$ $T=0.5$ So the graph will start from $0$, and form two cycles between $0$ and $1$, ...
2
votes
2answers
41 views

arccot limit: $\sum_{r=1}^{\infty}\cot ^{-1}(r^2+\frac{3}{4})$

I have to find the limit of this sum: $$\sum_{r=1}^{\infty}\cot ^{-1}(r^2+\frac{3}{4})$$ I tried using sandwich theorem , observing: $$\cot ^{-1}(r^3)\leq\cot ^{-1}(r^2+\frac{3}{4})\leq\cot ...
1
vote
1answer
41 views

Is there an easy way of showing $8\cos^3(12^o)-6\cos(12^o)=\phi$?

Is there an easy way of showing (1) (1) $$8\cos^3(12^o)-6\cos(12^o)=\phi$$ with out substituting into the equation? ...
-2
votes
2answers
63 views

Help with $\cos^{\frac 32}(x)$

I'm looking at putting the following in an excel formula but need help. $\cos^{\frac 32}(x)$ So how do you calculate $\cos$ to the power of $(3/2)$? Thanks!
0
votes
0answers
20 views

Basis of Trigonometry and sine of an angle and of a number.

Since equiangular triangles are similar, the ratios of its sides are proportional. So, consider two right triangles $ ABC $ and$ A'B'C'$ with the same angles. Then $\sin \alpha = \frac{AB}{BC} = ...
-1
votes
2answers
52 views

If $\tan\left(22 \frac{1}{2}^\circ\right)=\sqrt2 -1$, prove $\tan\left(11 \frac{1}{4}^\circ\right)=\sqrt{4+2\sqrt2} -\sqrt2 -1$ [closed]

If $\tan\left(22 \frac{1}{2}^\circ\right)=\sqrt2 -1$, then prove that $$\tan\left(11 \frac{1}{4}^\circ\right)=\sqrt{4+2\sqrt2} -\sqrt2 -1$$ please help me I could not even get to the first ...
0
votes
2answers
62 views

New Golden Ratio Construct with Geogebra using Square and Triangle with Same Base Width. Geometric proof of golden section?

The below construct of the golden ratio, based on the ratio of segment c to segemnt b, is so very close to PHI. Geogebra gives the value of 1.61957 instead of 1.61803. Might anyone have any insight ...
0
votes
1answer
32 views

Getting the coordinates of the center of a circle bisecting two other circles.

We have circles $C_1$ and $C_2$ with centers $(-d,0)$ and $(d,0)$, radii $a_1<d$ and $a_2<d$ respectively. If circle $D$ with radius $r$ (and with centre not necessarily on the x-axis) bisects ...
0
votes
1answer
42 views

Geometric interpretation of the ratio of the sides of a triangle.

In right triangle trigonometry, the sine of an angle $A$ is defined as the ratio of two lengths, the opposite leg $a$ and the hypotenuse $c$, that's to say, $\sin A= \frac{a}{c}$? My question is: ...
0
votes
2answers
31 views

Trigonometry word problem (involving wires)

A guy wire $78$ feet long runs from the top of a pole $56$ feet high to the ground and pulls on the pole with a force of $290$ pounds. What is the horizontal Pull on the top of the poll? I am not ...
2
votes
2answers
60 views

write down the expression for $\sin (15°)$ using the double angle formula.

show that $\sin 15^\circ=\frac {\sqrt3 -1}{2\sqrt2}$ using $\cos2A=1-2\sin^2A$ However I got $\sin 15^\circ= \sqrt{\frac {2-\sqrt 3}{4}}$ instead.
0
votes
1answer
27 views

Scalar product is 0 in any triangle

How can we prove that the following scalar product relation holds in any triangle? $$\left [-\overrightarrow{AB}\tan B (\tan A +2\tan C)+\overrightarrow{AC}\tan C (\tan A+2\tan B)\right ]\cdot \left ...
1
vote
1answer
62 views

Area of Pentagon question

Suppose that a regular pentagon circumscribes a circle of radius $r$. Show that the area of the pentagon is $5r^2\tan(36°)$. I know that the area of a triangle is $\frac{1}{2}bh$, where $b$ is the ...
0
votes
1answer
40 views

Find the trigonometric sum $S = \cos(2\pi/13) + \cos(6\pi/13) + \cos(8\pi/13)$

Find the sum $S = \cos(2\pi/13) + \cos(6\pi/13) + \cos(8\pi/13)$ I have been thinking to multiply both sides by $2\sin(2\pi/13)$ or something else, but I haven't found a solution. Please a hint!
1
vote
2answers
28 views

Help with proof that that $f(x)=\cos x\,\,\,[0,\pi]\rightarrow[-1,1] $ is one to one and onto

Is there a way to prove that $f\,:\,[0,\pi]\rightarrow[-1,1]\,\,\,\,f(x)=\cos x$ is one to one and onto ? I know that for $[0,\pi]$ cosine is strictly increasing function therefore it has to be one ...
1
vote
2answers
42 views

Solving a system of equations which contain sin and cosine terms.

Hello my question is the following: Solve the given system of equations: $$E=\frac{l_{p}}{\pi}\sqrt{\sin^{2}\left(\frac{\pi y_{1}}{l_{p}}\right)+\sin^{2}\left(\frac{\pi ...
1
vote
3answers
56 views

Show geometrically or algebraically $(\sqrt2-1)a+(\sqrt3-1)b<c$

Pythagoras theorem $$a^2+b^2=c^2$$ Show geometrically (Addressing to @blue a Trigonographer) (1)$$(\sqrt2-1)a+(\sqrt3-1)b<c$$ Or algebraically (general users) (@BLue the Trigonographer) ...
-4
votes
0answers
36 views

Trigonometric functions. prove that : prove that $\cos x + \cos y + \cos z + \cos (x+y+z) = 4\cos(x+y/2)\cos(y+z/2)\cos(x+z/2)$ [closed]

Prove that : $$\cos x + \cos y + \cos z + \cos(x+y+z) = 4\cos(\frac{x+y}{2})\cos(\frac{y+z}{2})\cos(\frac{x+z}{2})$$ Do not bother. the problem is quite easy and have been solved. Thank you all ...
4
votes
3answers
68 views

Right Triangle and Circle Theorem

Let $ABC$ be a triagnle such that $\angle BAC$ is a right angle. Suppose $D$ is a point lying on $BC$ such that $BD=1$, $DC =3$ and $\angle ADB=60^{\circ}$, find the length of $AC$. I was told that ...
3
votes
1answer
37 views

The Inverse Trigonometric Functions

I know that if $y=\sin x$ then $\arcsin y=x$; that is, $\arcsin$ is used for the inverse and $\arcsin$ is not a function if we don't restrict the domain of $y=\sin x$ but I don"t get that what is the ...
0
votes
2answers
41 views

Graphing the translated trig function $y = 2\cos(3(x - 1)) + 4$

So for the trig function $$ y=2\cos(3(x-1))+4 $$ I'm not sure how to draw the graph with a shift of $1$ to the right, would it be preferable to sketch it in radians? Also am I supposed to expand the ...
1
vote
3answers
61 views

Angle between squares at which they just touch along the circumference of a circle

Say I have two squares whose centers fall along the circumference of a circle. The circle has radius $x$. The squares have the same height and width $y$. The height of one square is parallel to the ...
1
vote
2answers
58 views

How do you solve for $\alpha$ in this trigonometric equation?

I have $$\tan (2\alpha) = \frac {4n^2}{4n^4-1}$$ And I want to solve for $\alpha$. So far I have tried applying the inverse tangent to both sides and dividing by two, but the book says that the answer ...
5
votes
2answers
67 views

Why am I getting two different answers (and the textbook a third) on this 3D trig problem?

Simone is facing north and facing the entrance to a tunnel through a mountain. She notices that a $1515$ m high mountain is at a bearing of $270^\circ$ from where she is standing and its peak has ...
2
votes
3answers
57 views

Solving $5\sin x=1+2\cos^2x$, without sketching a graph

$$5\sin x=1+2\cos^2x$$ "Solve, for $0 \le x <360$" I know using rearrangement $x= 30°$. I also know using a graph sketch and knowledge of symmetry that the other solutions within the range can be ...
0
votes
0answers
16 views

Did I interpret this wiki article on spherical interpretation correctly?

In Lua pseudocode, I believe the wikipedia article here is saying that the formula is used in the following way: ...
-2
votes
1answer
22 views

If $\cos\pi\theta$ is algebraic and $\theta$ is irrational, what is the set of possible $\theta$?

I know that $a= \cos \pi \theta$ is an algebraic number ($\theta$ is rational). I want to prove that if $\cos\pi\theta$ is rational, then the possible only possible values of $\theta$ are $0,±1/2,±1$ ...
1
vote
0answers
23 views

Transform the system of trigonometric equations

How to extract $\ell$ and $L$ from the following system of equations: $$\alpha=\arctan {R_E \cos \ell \sin L \over R_0 + R_E(1 - \cos \ell \cos L) }$$ $$\beta=\arctan {R_E \sin \ell \cos \alpha \over ...
2
votes
1answer
34 views

parallelogram diagonals in a relationship with basic geometry

This was a question in my textbook for homework a while ago but not even the teacher can find the solution using only basic geometry (further rules below). Basic only since it's in the section where ...
4
votes
3answers
101 views

Why is $5\tan(54^\circ) = \sqrt{25 + 10\sqrt{5}}$ and $\tan\left(\frac{\pi}{5}\right) = \sqrt{5 - 2\sqrt{5}}$?

On the Wikipedia Page about Pentagons, I noticed a statement in their work saying that $\sqrt{25+10\sqrt{5}}=5\tan(54^{\circ})$ and $\sqrt{5-2\sqrt{5}}=\tan(\frac {\pi}{5})$ My question is: How would ...
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votes
0answers
13 views

Implementation of elementary functions [closed]

I am on a project to implement floating point sine and cosine functions using VHDL. Even though i tried. but i need to have something that implement sine cosine and hyperbolic functions simultaneously ...
2
votes
0answers
83 views

Integrate $\cos^2(\pi x)\cos^2(\frac{n\pi}{x})$

I need to integrate $\int\cos^2(\pi x)\cos^2(\frac{n\pi}{x})dx$. There's no limit for $x$ but if it helps you can assume $\frac{n}{10} \le x \le n$. Walfram calculator can gives the following ...
1
vote
2answers
89 views

Prove $\sin^2(10 ^\circ)-\sin^2(20^\circ)-\sin^2(40^\circ)=-\frac{1}{2}$ identity

10 degrees $$\sin^2(10^\circ)-\sin^2(20^\circ)-\sin^2(40^\circ)=-\frac{1}{2}$$ $$\cos^2(10^\circ)-\cos^2(20^\circ)-\cos^2(40^\circ)=-\frac{1}{2}$$ Why are they both have same answer? The only ...
0
votes
1answer
35 views

Generic formula for third point of triangle knowing the other two points and all the side lengths

My question is similar to this one, but the solution provided makes use of some 'properties' that will not be true for all triangles. For example, if point A is not in the origin, or point B is not in ...
2
votes
5answers
78 views

How to prove that $\int_0^{\infty}\cos(\alpha t)e^{-\lambda t}\,\mathrm{d}t=\frac{\lambda}{\alpha^2+\lambda^2}$ using real methods?

Could you possibly help me prove $$\int_0^{\infty}\cos(\alpha t)e^{-\lambda t}\,\mathrm{d}t=\frac{\lambda}{\alpha^2+\lambda^2}$$ I'm an early calculus student, so I would appreciate a thorough answer ...
13
votes
2answers
136 views

Egg vs. chicken: trig functions, exponential, real and complex

This is something I was shaky about when I took calculus, real analysis, and then complex analysis. Specifically, is the following chain of definitions circular in any way? Define the set ...
1
vote
0answers
75 views

Evaluating $\int_0^\infty \sin x^2\, dx$ with real methods and without gamma functions?

I know that $$\int_0^\infty \sin x^2\, dx = \sqrt{\frac{\pi}{8}}$$ but all of the methods I've found seem to be too complicated for an early calculus student. Is there any method of calculating this ...
0
votes
2answers
59 views

How does the way we define cos or tan have anything to do with degrees of the angle?

So sine of angle $A$ is just a ratio. It is the ratio of the length of the opposite or perpendicular of angle $A$ and the hypotenuse. Cosine of angle $A$ is also just a ratio. It is the ratio of the ...
0
votes
2answers
63 views

How to describe $\sec^{-1}{x}$ in terms of $\tan^{-1}{x}$?

How to describe $\sec^{-1}{x}$ in terms of $\tan^{-1}{x}$? I tried the following: $y=\sec^{-1}{x}\Longleftrightarrow x=\sec{y}$ $x^2=\tan^2{y}+1$ $\tan{y}=\pm\sqrt{x^2-1}$ ...
2
votes
3answers
42 views

How to show that a limit exisits, and evaluate the limit of trig functions as $x$ tends to point

For each of the following functions $f$, determine whether $\lim_{x\to a}f(x)$ exists, and compute the limit if it exists. In each case, justify your answers. $$f(x)= x^2\cos\frac{1}{x} (\sin x)^4, ...
2
votes
0answers
55 views

The representation of trigonometric functions of irrational multiples of $ \pi $ in closed form

It is know that, if $\alpha\pi$ is an irrational multiple of $\pi$, where $\alpha$ is an algebraic number that can be expressed by radicals, then the trigonometric functions of $\alpha\pi$ cannot be ...
0
votes
2answers
63 views

How to prove the following curious approximation to $\sin (A/2)$

In the article "Concerning haversines in plane trigonometry" by G. W. Evans, the following approximate formula is mentioned without proof: $ \sin \frac{A}{2} \approx \sqrt{\frac{c-b}{2c}}$ Here $c $ ...