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

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1answer
42 views

Localizing a point using distance measurements to four points in 3-D

This article explains how to do trilateration step by step. I need to extend this process to 3-D. As far as I know, I need four distance measurements in order to calculate a fifth point's coordinates. ...
6
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1answer
124 views

How prove $\left(\sum\cos{\frac{2k-1}{p}\pi}\right)\cdot\left(\sum\cos{\frac{2k-1}{p}\pi}\right)$

Question:let $p$ be an odd prime number,let $A$ be the set of the (postive and less than $p$) quadratic residues modulo $p$,and $B$ be the set of the (positive and less than $p$ quadraric non-residues ...
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1answer
89 views

Why there are two different values of θ for same quadrant?

Let Sin θ = 1/2 is function. Let us find its solution set. sine is +ve in I and II quadrant with reference angle π/6 θ = π/6 (I quadrant) Now here is my problem. We can use π-θ and (π/2)+θ to find ...
17
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2answers
928 views

A tricky integral

I'm trying to find the exact value of $$\int_{\frac{1}{\sqrt{3}}}^{\sqrt{3}} \frac{\arctan{(x^2)} }{1+x^2} \, dx$$ Ostensibly, I'd want to use this: $$\frac{d}{dx}\arctan{(x)}=\frac{1}{1+x^2}$$ But ...
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1answer
518 views

solving a non-linear (trigonometric) system of equations with two equations and two variables

I'm trying to solve the following system of equations: $$l_1*sin(\alpha)=l_2*cos(\gamma)+l_3*sin(\beta)$$ $$l_2*sin(\gamma)+l_1*cos(\alpha)=l_3*cos(\beta)+l_4$$ with the unknowns $\beta$, $\gamma$ ...
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2answers
47 views

Solving for x Algebraically

So my problem is: $$\arcsin (x) = \arccos (5/13)$$ ^ Solve for $x$. How would I begin this problem? Do I draw a triangle and find the $\sin(x)$ or is there a more algebraic way of doing this? ...
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3answers
56 views

Solve $6\sin x \cos 2x\ge 0$

How can I solve the following inequality: $${6\sin x\cos 2x\ge 0}$$ Can you give me an explicit explanation of how this exercise can be understood. I have no problems with trigonometric equations, but ...
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2answers
66 views

Why is the integral of $\cos(x) - \sin(x)$ from $0$ to $π/4$ equal to $\sqrt2 - 1$?

Why is $$\int_ {0}^{π/4} {\cos(x)} - {\sin(x)} \ \mathrm{d}x=\sqrt2 -1$$ This answer popped up on a problem I was doing and it piqued my interest. Can anyone help me out?
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3answers
45 views

$\frac d{dx}\cos x \space\mathrm{vs}\space\frac d{dx}\cos(-x)$

Like it says on the tin. I thought that the rule for deriving $sin{x}$ and $\cos x$ was simply the chain rule; $\displaystyle\frac d{dx}f(g(x))=f'(g(x))g'(x)$; applying to $\cos(x)$, this appears to ...
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1answer
40 views

Is there an upper bound on the number of distinct integer outputs a trig function can have?

"Usually" when you plug integers into trig functions you don't get integers as output. I'm interested in trig functions that pass through integer lattice points. Let $f$ be a linear combination of ...
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3answers
587 views

Odd $\sin/\cos$ integral

How to evaluate $$\int \frac{\sin^3 x}{\cos^5x}dx\ ?$$ I've tried various substitutions with $\sin x = u$ or $\cos x = u$, I've tried using Euler's formula which result in too heavy calculations and ...
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3answers
152 views

Integral with goniometric functions $\int(1+\cos^2x-\sin^2x)dx$

I am solving this example: Transcription: \begin{align} &\int(1+\cos^2x-\sin^2x)dx=\int(1+1-\sin^2x-\sin^2x)dx=\int(2-2\sin^2x)dx=\\ ...
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2answers
42 views

Writing answers to trigonometric equation

I wonder how to write answers to trigonometric equations in more elegant form. For instance if we have $ \displaystyle \sin x = \frac{\sqrt{2}}{2} \vee \sin x=-\frac{\sqrt{2}}{2}$ then I write four ...
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2answers
255 views

derivative of trigonometry for cos to the power of 3

This appears in my homework and I don't know how to do it, could you help me please? $f(x) = \cos ^3 (4x + 1)$ with $0 < x < 1 $ Find the derivative of $f(x)$. I know the derivative of $\cos ...
8
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4answers
676 views

Evaluate $\int\frac{1}{\sin(x-a)\sin(x-b)}\,dx$

I'm stuck in solving the integral of $\dfrac{1}{\sin(x-a)\sin(x-b)}$. I "developed" the sin at denominator and then I divided it by $\cos^2x$ obtaining ...
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1answer
140 views

If this relation holds, then is the triangle equilateral?

Let $ABC$ be a triangle. If $$\sum_{cyc}\frac{BC}{4AC\cos^2({\frac{\angle BAC}{2})}+BC}=\frac{3}{4}$$ then the triangle is equilateral? We can check if we set $\widehat{BAC}=\pi/3$ and $AB=BC=CA$ that ...
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3answers
84 views

Prove this trigonometric identity [closed]

Prove the following trigonometric identity: $16\sin ^2\left(θ\right)\cos ^3\left(θ\right)=2\cos \left(θ\right)-\cos \left(3θ\right)-\cos \left(5θ\right)\:$
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1answer
70 views

Prove $\cot(x) +\cot(\frac{\pi}{3}+x) + \cot(\frac{2\pi}{3}+x) = \frac{3-9\tan^2x}{3\tan x-\tan^3x}$

As the title says. I tried simplifying the LHS but got: $$\frac{\tan^2x+7\tan x- \sqrt{3}}{\tan^2x-\sqrt{3}}$$ What should I do next?
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1answer
116 views

Solid Trigonometry

V ABCD is a pyramid with the vertex V situated perpendicularly above the centre of the square base ABCD. If $\theta$ is the angle between the edge VA and the base, and $\phi$ is the angle between the ...
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1answer
59 views

New Angle When Opposite Side is Halved

Suppose you have a right triangle with any length sides. The value of one of the angles is $\theta$ and the opposite side is a. If I change the triangle so that the new length of side a is $\frac a2$, ...
3
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1answer
133 views

Inequality problem about sides of a triangle and the semiperimeter

Let $a,b,c$ the sides of a triangle and $s$ be the semi perimeter. Then show that $$ a^2+b^2+c^2 > \frac{36}{35}(a^2+\frac{abc}{s}) $$ I tried it doing in many ways using some ...
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4answers
96 views

Integral of $16/(1-\cos8x)$

Can someone please help me with this question: $$ \int \ \frac{16}{1-\cos8x} \ \ dx \ \ . $$ I tried substitution by letting $u=1-\cos8x$, it got messy after the substitution. I used the ...
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4answers
334 views

Prove this identity: $\frac{2\sin^4x+\cos^2x-2\cos^4x}{3\sin^2x-1} =1$

I am stuck on this identity $$\frac{2\sin^4x+\cos^2x-2\cos^4x}{3\sin^2x-1} =1$$ I began working on the left side trying to get things to cancel out or equal one by the Pythagorean identities. I am ...
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2answers
98 views

Rotate $z = 0$ plane in 3D

I have 100 points on $z=0$ plane. I want to rotate those points, such that they lie on any plane $P(a,b,c,d)$, preserving distances. Hence, I need a rotation matrix. For instance, if my points are ...
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1answer
42 views

Fairly simple trig question

Three points at coordinates $(0,c)$, $(p,q)$, $(0,d)$ respectively. The angle at $(p,q)$ between $(0,c)$ and $(0,d)$ is $θ$. Find $d$. P.s. This isn't homework.
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2answers
213 views

An isosceles triangle has sides of 17 centimeter and 20 centimeter and 20 centimeter. Find the magnitude of the angles trigonometry

An isosceles triangle has sides of 17 centimeter and 20 centimeter and 20 centimeter. Find the magnitude of the angles
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79 views

Solve $ \sin \pi x = (x+1)^2$

Is it really to solve it? I just need it because this solution is interval of integration. One solution is $-1$, but i need other
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1answer
197 views

Using Trigonometry vs. Geometry

I have a friend who is trying to build a wooden retaining wall around a trampoline. He wants the wall to be in the shape of a regular polygon with sides between $24"-30"$. He asked me to figure out ...
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1answer
34 views

Khan's proof of the derivative of the inverse cosine function

In one of his thousands of videos, Khan used to prove the derivative of the inverse cosine function. But I don't understand one of his steps. Basically, he wants to find $\frac{d}{dx}\cos^{-1}x$ so ...
0
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1answer
163 views

Calculating coordinates of corners of polygon

In the image below, I have a Google Earth-based application I am working on and a specific requirement is to calculate the bounds of a drawn polygon given the center($P$) coordinates $(a,b)$, ...
2
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4answers
257 views

Solve $\cos x+8\sin x-7=0$

Solve $\cos x+8\sin x-7=0$ My attempt: \begin{align} &8\sin x=7-\cos x\\ &\implies 8\cdot \left(2\sin \frac{x}{2}\cos \frac{x}{2}\right)=7-\cos x\\ &\implies 16\sin \frac{x}{2}\cos ...
0
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1answer
64 views

How do you solve $1+sin(2\theta)=0.0368$

$$ 1+sin(2\theta)=0.0368 $$ I'm studying for my physics final and I can not remember how to solve the above equation. I thought I could just move the 1 to the other side, and then take the inverse ...
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3answers
492 views

Evaluate $\lim_{x \to 0} \frac{2 - \cos(3x) - \cos(4x)}{x}$

How do you evaluate $$\lim_{x \to 0} \frac{2 - \cos(3x) - \cos(4x)}{x}?$$ I have looked at this problem for a while and cannot think of a way of algebraically determining this limit although I know ...
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1answer
41 views

An Inverse Cosine Problem

Here is my problem: $$ \sin(\cos^{-1} \frac{2}{5} ) $$ I know how to do it for the most part; I just draw a triangle with sides 2,5 and √21 and I then find the sine (opposite/hypotenuse) of the ...
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3answers
150 views

sin (x) + cos (x) = 0. Why this equation has only one solution set?

The equation "sin (x) + cos (x) = 0" has only one solution set "$x=\frac{3\pi }{4}+\pi n$". Why it has not solution set "$x=\frac{7\pi }{4}+\pi n$"? Although it satisfy the equation. Please help ...
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2answers
49 views

Knowing the hypotenuse and the direction of the adjacent, how would I get the length of the adjacent

I have a good understanding of the basic SohCahToa trig functions, but this kinda stumped me, since I don't have two parts of the information that is needed, here is a example image: ...
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3answers
41 views

Trigonometry Question, Please Help

Find $\sin\theta$ if $\tan\theta$=4; 180< $\theta$ <270. I just don't know what to do. I'm not sure if I missed a section or what. I have online Trig and it's confusing.
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1answer
45 views

Prove that $\frac{\pi}{2}-x<\tan^{-1}(x)<\frac{\pi}{2}-x+\frac{x^3}{3}$

Prove that for every $x>0$, it is true: $$\frac{\pi}{2}-x<\tan^{-1}(x)<\frac{\pi}{2}-x+\frac{x^3}{3}$$ We can split it into two statements: $\frac{\pi}{2}-x<\tan^{-1}(x)$ ...
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1answer
39 views

$2\pi^2(x-1)^2+4a\cos(2\pi x)-9a^3=0$ For which $a$ has only one solution…

For which values of real parameter $a$ the following equation has only one solution: $$2\pi^2(x-1)^2+4a\cos(2\pi x)-9a^3=0$$ Frankly I have no idea and I hope you'll give me some understandable hint ...
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2answers
110 views

How can this problem of trigonometry be solved?

If $$\sin A + \cos A + \tan A + \cot A + \sec A + \csc A = 7$$ then prove that $$\sin 2A \quad\text{ is a root of }\quad x^2 - 44x - 36 = 0$$ I have no idea how to solve it. Plz help.
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1answer
91 views

Solving Triangle Word Problems

A ship leaves port for a remote island 34 miles west and 60 miles south of the port. Find the distance and bearing to the remote island. What would the diagram look like to this problem? and what ...
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1answer
53 views

Sin and Cos relationship with Triangle sides

In a triangle ABC, ${sinA < \frac{a}{c}}$ and ${cosA > \frac{b}{c}}$. Which of the statements below are always false regarding triangle ABC? ABC is an acute triangle ABC is an isosceles ...
2
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1answer
61 views

what are the integral sides possible?

Let a triangle have sides $a$,$b$,$c$ and $c$ is the greatest side , triangle ABC is obtuse in nature having integral sides. Find the smallest perimeter possible Given A=2B where A and B are angles of ...
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1answer
296 views

Trigonometry when to divide and times

OK, so I am revising for an upcoming test and am a bit confused. I have previously learn Trigonometry, however that was around a year ago, and i have completely forgotten it now. I went on to ...
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2answers
370 views

Reduction formula tricky problem (Further Maths: F3)

$\int \:e^{ax}\cos ^n\left(x\right)dx$ I just cannot get it to reduce, I keep ending up with too many species in the next integral to use parts again. I have important Further Pure F3 exam in a month ...
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1answer
333 views

How do I calculate the height of a cross section of a circle?

I'm working on an LED lighting project and have discovered that it involves a little math... I'm mounting LEDs to plexiglass facing away from the surface I want lighted. I'm looking at cutting a ...
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1answer
271 views

Is this a good way to prove that $3^x+4^x =5^x $ has $x=2$ as the only real solution?

Divide both sides of the equation $3^x+4^x=5^x$ by $5^x$. $$ \Rightarrow \frac { 3^x }{ 5^x } +\frac { 4^x }{ 5^x } =\frac { 5^x }{ 5^x }$$ $$\tag1 \Rightarrow \left( \frac 3 5 \right)^x + \left( ...
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1answer
34 views

proving a range of an angle if the tangent of $y=sinx$ creating an angle $\alpha$ with the positive direction of the $X$ axis.

Hey i need to prove that if the tangent of $y=sinx$ creating an angle $\alpha$ with the positive direction of the $X$ axis ($\alpha$ is in radians) so $\frac{3}{4}\pi\leq \alpha\leq \pi$ or $0\leq ...
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0answers
34 views

Is it possible to calculate the width of this table

Is it possible to calculate the length of x (width of the table) using the given values and any information that can be inferred from the image. If not, what is the best estimate that can be found.
0
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2answers
73 views

What will be the solution to: $2^{\cos x}=|x| $Find $x$.

Well, the question is this really. Find the solution to the eqn. $2^{\cos x}=|x| $ Can it done by plotting on the $\cos$ curve? If we square and take log on both sides, I think we get something ...