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
30 views

Formula for the general term of the Taylor series of $\tan(x)$ at $x = 0$

Today we were taught different expansions; one of them was the series expansion of $\tan(x)$, $$\tan(x)=x+\frac{x^3}{3}+\frac{2x^5}{15} + \cdots .$$ So, with curiosity, I asked my sir about next term. ...
1
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0answers
31 views

A curious approximation to $\cos (\alpha/3)$

The following curious approximation $\cos\left ( \frac{\alpha}{3} \right ) \approx \frac{1}{2}\sqrt{\frac{2\cos\alpha}{\sqrt{\cos\alpha+3}}+3}$ is accurate for an angle $\alpha$ between $0^\circ$ ...
1
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1answer
43 views

Writing answer for $2\cos x+\sin2x=0$

$$2\cos x + \sin 2x = 0\implies \cos x = 0 \; \&\; \sin x = -1$$ So, the solution my book provides is $x = π/2 + 2nπ\; \&\; x = 3π/2 + 2nπ$ Why is $3π/2$ (for $\cos$) not included in the ...
1
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0answers
26 views

Basic trigonometry truck problem

I've got a decent grasp on trig but this one problem annoys me so much I need to get it answered. Here it is: "A truck with a long back tray can scrape the road when it travels up steep driveways. If ...
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0answers
31 views

Solution to System of Complicated Differential Equations

I'm looking for a solution to this set of complicated differential equations: $$\begin{align} \dfrac{dθ}{ds} & = \dfrac{\cos θ}{r} − z\\ \dfrac{dz}{ds}& = − \cos θ \\ \dfrac{dr}{ds} &= ...
3
votes
3answers
51 views

Switch from $a\cdot \sin(t) + b \cdot \cos(t)$ to $c \cdot \cos(t+f)$

How could I switch from $a\cdot \sin(t) + b \cdot \cos(t)$ to $c \cdot \cos(t+f)$? Thank you for your time.
4
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1answer
39 views

If A,B,C,D are angles of a quadrilateral and $\sin (A/2)\sin (B/2)\sin (C/2)\sin (D/2)=1/4 $ prove that $A=B=C=D=\pi/2 $

If A,B,C,D are angles of a quadrilateral and $\sin (A/2)\sin (B/2)\sin (C/2)\sin (D/2)=1/4 $ prove that $A=B=C=D=\pi/2 $. How should I proceed? Any suggestions might be helpful.Thanks.
1
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2answers
212 views

Integrating trig function

I'm stuck at this problem: $$ \int{\sqrt{(\sin^2 x)^2 + (2\sin x \cos x)^2}dx} = \int{\sqrt{\sin^2 x \sin^2 x + 4\sin^2 x \cos^2 x} dx}$$ I tried a few trig identities: $\sin^2 x = \frac{1-\cos ...
2
votes
3answers
80 views

Solving for $x$ in $\tan(3x) \tan (2x)= 1$

If $$\tan(3x) \tan(2x)= 1$$ Then $x$ is equal to Attempt: I used the '$\tan$' identity but it showed no results. The identity: $$\frac{\tan(2x)+\tan(3x)}{1-\tan(2x)\tan(3x)}$$
2
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1answer
56 views

Find the smallest number p for which the equation $\cos (p\sin (x))=\sin(p\cos (x))$

Find the smallest number p for which the equation $\cos (p\sin (x))=\sin(p\cos (x))$ has a solution. x belongs to $[0,2\pi] $ Any hints for this please.Don't know how to proceed.
-2
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1answer
13 views

Solving basic Trigonometry equations on a Ti-84 Plus [on hold]

I am trying to solve some equations for example: tanθ = 2/4 or tan15 = x/30 but I do not know how I would use a Calculator to solve these as I do not know how to format them or which symbols to use ...
0
votes
2answers
80 views

Determining “% slopes” of lines

My previous posts were unsuccessful. I need help from a math person so I can write a c#.net application. I know c#.net but my math skills aren't as strong, which is why I'm here. I need the ...
9
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2answers
149 views

Prove that $\sin x \cdot \sin (2x) \cdot \sin(3x) < \tfrac{9}{16}$ for all $x$

Prove that $$ \sin (x) \cdot \sin (2x) \cdot \sin(3x) < \dfrac{9}{16} \quad \forall \ x \in \mathbb{R}$$ I thought about using derivatives, but it would be too lengthy. Any help will be ...
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1answer
48 views

Find all the numbers of $x,y$ that satisfy $\tan ^4(x)+\tan^4(y)+2\cot^2(x)\cot^2 (y)=3+\sin^2 (x+y) $

Find all the pairs of $x,y$ that satisfy $$\tan ^4(x)+\tan^4(y)+2\cot^2(x)\cot^2(y)=3+\sin^2 (x+y) $$ I tried applying AM-GM inequality on the first two terms but then I am stuck.Please help. $x,y$ ...
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2answers
68 views

Why we not check conditions while solving questions?

Note:Down ward problem is just an example to express my question(I know the both solution of problem are insufficient but the first solution is in my 10+2 book and second one is mine which is ...
2
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0answers
45 views

Prove that $\tan \left(\frac{3\pi}{11}\right)+4\sin \left(\frac{2\pi}{11}\right)=\sqrt {11}$ [duplicate]

Prove that $$\tan \left(\frac{3\pi}{11}\right)+4\sin \left(\frac{2\pi}{11}\right)=\sqrt {11}$$ Hints please !
0
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1answer
35 views

How does $ \frac{1}{x}\left(\frac{\pi}{2} - \arctan\frac{1}{x}\right)$ simplify to $\frac{1}{x} \arctan x $?

Here is a solution I read when trying to solve a problem, and I can't figure out how it jumped in this step here: $$ \frac{1}{x}\left(\frac{\pi}{2} - \arctan\frac{1}{x}\right) = \frac{1}{x} ...
3
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0answers
32 views

Are these Infinite Series Representations of Special Functions?

I am not sure how to google the answer for this question. Anyway, in trying to compute the velocity of a charged particle in an electromagnetic field, I came across these two infinite series ...
0
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1answer
34 views

Solving for $\cos a$

Given that $\sin(\frac{a}{2})=\frac{1}{4}$ and $405^{\circ} < \frac{a}{2} < 450^{\circ}$ find $\cos a$ I tried the following $\sin \frac{a}{2} = \sqrt{\frac{1-\cos a}{2}} \Longrightarrow ...
1
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1answer
67 views

A Golden Ratio Symphony! Why so many golden ratios in a relatively simple golden ratio construction with square and circle?

Note: this construction is a vastly expanded version of my earlier construction here: Have you seen this golden ratio construction before? Three squares (or just two) and circle. Geogebra gives PHI or ...
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5answers
40 views

Finding a Coordinate on a circle using radius, angle, and origin

I am trying to calculate a point on a circle using an angle and a different point. With this picture, I know the origin O, the radius r, the angle A, and the point B. Now I want to find the point ...
2
votes
4answers
76 views

Solution of $4 \cos x(\cos 2x+\cos 3x)+1=0$

Find the solution of the equation: $$4 \cos x(\cos 2x+\cos 3x)+1=0$$ Applying trigonometric identity leads to $$\cos (x) \cos \bigg(\frac{x}{2} \bigg) \cos \bigg(\frac{5x}{2} \bigg)=-\frac{1}{8}$$ ...
0
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2answers
58 views

Show that $\cos (\sin \theta)>\sin (\cos \theta)$

For all $\theta$ in $[0,\pi/2]$ I need to show that $\cos (\sin \theta)>\sin (\cos \theta)$. In my book it is done like $cos (\theta)<\pi/2- sin (\theta) $.Then they took sine on both sides ? ...
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2answers
33 views

Prove that $x-a \sin(x)=b$ has one real solution, where $0\lt a \lt 1 $

$a,b \in \mathbb{R}$. Prove that $x-a\sin(x)=b$ has one real solution, where $0\lt a \lt 1 $. I need some sort of starting hint as to how to prove this. I can define $g(x)= x-a\sin(x)-b$ but more ...
1
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0answers
12 views

Calculate random distribution of triangles within a rectangle. [on hold]

I am a programmer trying to find a solution to a problem regarding images and JavaScript. I want to create a function that when given an image will calculate a random set of triangles of different ...
1
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1answer
22 views

Prove $1-cos(2\pi k/n)=2sin(kπ/n)sin((n−k)\pi/n)$

It looks like a sum to product question, however, I couldn't find a way to make it work. Prove $$1-cos\frac{2\pi k}{n}=2sin\frac{k\pi }{n}sin\frac{(n−k)\pi }{n}$$
1
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1answer
42 views

Solve $2x\cos x+(1-x^2)\sin x=0$

I can't solve: $$2x\cos x+(1-x^2)\sin x=0$$ The solution must be $(k-1)\pi<x_k<(k-1)\pi+\frac{\pi}{2}$ for $k=2,3,4,\ldots$ Any hint? Many thanks!
1
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0answers
46 views

A curious trigonometric equality

Let's consider the following expression: $(1)\cos(15\sqrt{2}^\circ) = \frac{1}{2}\sqrt[\sqrt{2}]{\frac{\sqrt3}{2}+\frac{i}{2}} +  \frac{1}{2}\sqrt[\sqrt{2}]{\frac{\sqrt3}{2}-\frac{i}{2}}$ The left ...
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1answer
38 views

Finding angle with only cosine expressions

I am trying to find the values of angles "a" and "b" based on the following expression, and I am simply trying to figure if it is possible based on just that. "x" is a scalar value: ...
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1answer
59 views

How can this equation produce an integer as a result? $\frac{1}{1+\cos(40^0)}+\frac{1}{1+\cos(80^0)}+\frac{1}{1+\cos(160^0)}=18$

How can this equation produce an integer as a result? $$\frac{1}{1+\cos(40^0)}+\frac{1}{1+\cos(80^0)}+\frac{1}{1+\cos(160^0)}=18$$ $\cos(x)$ are irrational values apart from a few.
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1answer
55 views

New Golden Ratio Conjecture with Triangle and Square: It is very close, but is it really the golden ratio?

Geogebra gives me 1.616 for the ratio of the blue segment p to the red segment q instead of the golden ratio 1.618 for the construction shown below, so it could be close to PHI, but no cigar. This ...
2
votes
1answer
73 views

Prove that Triangle ABC is an equilateral triangle iff $\tan{A}+\tan{B}+\tan{C} = 3^\frac32$.

This question is picked from AM GM HM inequalities, so this is to be proved form that concept only, I think it isn't possible because there is no inequality, but if it is please tell me how.
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0answers
53 views

What are the practical applications of this trigonometric identity?

On various occasions people have asked here how to prove that $$ \text{if } \alpha+\beta+\gamma = \pi \text{ then } \frac{\sin(2\alpha) + \sin(2\beta) + \sin(2\gamma)} 2 = ...
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0answers
23 views

Polar coordinate - Write an equivalent rectangular equation…

Write an equivalent rectangular equation $y=-x$ It should be simple, but in my attempt, I always come up short with coordinates $\sqrt{2}$ for both $x$ and $y$. My $r$-variable is $2\sqrt{2}$. Is my ...
1
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4answers
73 views

Chain rule to differentiate $\sin ^2\frac{x}{2}$

I have this equation $$\sin ^2(\frac{x}{2})$$ Using the chain rule $ M'(N(x)).N'(x)$: $$\begin{align*} &M= (\sin \frac{x}{2})^2 \\ &N= \frac{x}{2}\end{align*}$$ That makes $$2\sin ...
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0answers
18 views

Find all the equilibrium solutions of $x' = \cos(x^2)$ and determine stability.

One can determine equilibrium solutions of autonomous EDO's setting the derivative equal to zero. So, in this case, all the equilibrium solutions take the form $x^* = \pm \sqrt{\frac{\pi}{2} + k\pi}$, ...
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0answers
33 views

Trigonometric integral $\int_{[-\pi,\pi]^2}{\frac{1-e^{-in\cdot\theta_1}}{1-\cos(\theta_1)\cos(\theta_2)}\,d\theta_1\,d\theta_2}$

I am trying to compute the following integral (see here). Since it seems to be the wrong approach, I am trying to calculate another one which I hope it will give me what I am looking for. My point is ...
3
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3answers
63 views

What is the fallacy of this trigonometrical proof that $1=-1?$

I have this equation which I solved- $$\sin^4x-\cos^4x=1$$ $$\implies -\cos^4x=1-\sin^4x$$ $$\implies-\cos^4x=(1+\sin^2x)(1-\sin^2x)$$ $$\implies-\cos^4x=(1+\sin^2x)\cos^2x$$ ...
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3answers
36 views

Pentagon circumscribes a circle. Prove that its area is $5r^2\tan(36^\circ)$

Suppose that a regular pentagon circumscribes a circle of radius r. We are supposed to show proof, using the trigonometric area of the triangle (1/2)bhsin(36°) that the area of the pentagon is ...
2
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1answer
16 views

Ration of sum and Product of Trigonometric expression.

If $A,B,C \in \mathbb{R}$ and $\displaystyle\cos(A-B)+\cos(B-C)+\cos (C-A)=-\frac{3}{2}\;,$ Then $\displaystyle \frac{\sum \cos^3(\theta+A)}{\prod\cos(\theta+A)}\;, $Where $\theta \in ...
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2answers
22 views

Maximum Height is giving me negative

Hey guys for this parametric equations its giving me negative Question is: A dart is thrown from a point 5 feet above the ground with an inital velocity of 58 ft/sec and angle of elvation of 41∘. ...
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1answer
24 views

Parametric Problem: Throwing a Dart <Test Review>

Yup it's me ... Parametrics, who would have thought xD! Anyways, again ... I am doing review and I really need this grade to get an A in math class; that's why I am asking questions here. And you guys ...
1
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0answers
19 views

Solve the equation $7\sin\theta+3\cos\theta=4$ for all solutions in the interval $0°\leq\theta\leq360°$, giving $\theta$ to the nearest $0.1°$

Using the Addition Formulae \begin{align} 7\sin\theta+3\cos\theta & = 4 \\ \Rightarrow \sin(\theta+23.2°) & = \frac{4}{\sqrt{58}} \\ \end{align} \begin{align} \sin(\theta+23.2°) & = ...
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2answers
39 views

Determining exact value of $\cos (A+B)$ in a specific quadrant

The question reads: Angles $A$ and $B$ are obtuse angles in quadrant 2 (II). If $\csc A = 3$ and $\tan B$ = -1/3, determine the exact value of $\cos (A+B)$. How would I take on this question? ...
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1answer
33 views

How to deal with identities in which one expression, but not the other, evaluates to “undefined” in particular instances

I haven't touched trigonometry for awhile and whilst flicking through an old set of notes I came across the following expression: $$\csc(x) \cdot \sec(x) \cdot \sin^2(x)$$ I'm aware that this can ...
1
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3answers
64 views

Solving a trig equation that is quadratic?

I have to solve for $x$ given $$\tan^2 x = 2 + \tan x\;\;\;\;\;\;0≤x≤2\pi$$ I brought it all to one side and set it all equal to zero like: $$\tan^2 x - \tan x - 2 = 0$$ What am i supposed to do ...
1
vote
5answers
80 views

How would you verify the following trig identity $\frac{\sin(x)}{\sin(x)+\cos(x)}=\frac{\sec(x)}{\sec(x)+\cos(x)}$

How to verify the following trig identity. $$\frac{\sin(x)}{\sin(x)+\cos(x)}=\frac{\sec(x)}{\sec(x)+\cos(x)}$$ I started with the right side and multiplied the numerator and denominator by ...
0
votes
0answers
14 views

Finding altitude and azimuth with an accelerometer and magnetometer

I posted this in the astronomy stack exchange forum, but considering that it is a very math intensive question I figured there could also be people on here that could help. For a project with my ...
0
votes
1answer
47 views

Calculating the absolute value of sum of rational numbers [on hold]

If $\sqrt{9-8\cos40}=a+b\sec40$, and $a$ and $b$ are rational numbers, then $\lvert a+b\rvert =\,{}$?
0
votes
0answers
16 views

Using GPS coordinates in trillateration

for a project we need to find a certain position. The info we have : 3 surrounding positions and the distance between those positions and the point we are looking for. We've got a setup like this ...