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

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2
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3answers
559 views

Calculate wine volume in a horizontal barrel using a dipstick

I suck at math, but still need a way to mark a dipstick to measure the volume of wine in a barrel. This question has been asked, but the only answer is to cryptic for me to understand! My barrel has ...
2
votes
1answer
65 views

Is there an equation to find the angle of the diagonal in a rectangle?

If we have a rectangle of length 5 and height 5 the angle of the diagonal would be 45°. We know this is true but how can we arrive at this conclusion mathematically?
1
vote
5answers
66 views

Trigonometric identity involving double angles

If $\alpha$ and $\beta$ are acute angles and $\displaystyle{\cos2\alpha=\frac{3\cos\beta-1}{3-\cos2\beta}}$, then prove that $\displaystyle{\tan\alpha=\sqrt{2}\tan\beta}$. I tried this question by ...
2
votes
1answer
27 views

Characterizing functions which satisfy de Moivre's theorem

Let $f$ and $g$ be two non-zero functions $ \mathbb R \to \mathbb R $ which are continuous and differentiable everywhere. Furthermore, say that for all integer $n$: $$ (f(\theta) + i g(\theta))^n = ...
3
votes
2answers
1k views

Proof of trigonometric identity using vector calculus

Question: Using vector calculus, show that $\sin (A+B) = \sin A \cos B + \cos A \sin B$ I have no idea how to even attempt the question. A small hint to help me get started would be greatly ...
1
vote
1answer
259 views

How would I normalize the slope of a line?

Assuming I have different lines with different slopes, I would like to compare the slope of each line as relative to one another. The program I am currently writing needs to compare the slopes of the ...
1
vote
3answers
158 views

Proving a second derivative

Given that $$y = \sin^3 x + \cos^3 x$$ prove that $$\frac{d^2 y}{dx^2} = \frac{3}{2} (\cos x + \sin x)(3 \sin 2x - 2)$$ I began with differentiating the equation as it is and it took me around ...
1
vote
2answers
145 views

How come that $(\cos2x + 1) = (2 \cos^2 x – 1 + 1)?$

I really am not able to find any identity which results in this expression, any help regarding how we obtain the right-hand side of the equation $$\cos2x + 1 = 2 \cos^2 x$$ would be really ...
0
votes
1answer
497 views

Why is (2 sin 5x cos 5x) (2 sin x cos x) = sin 10x sin 2x? [closed]

The question says it all, why? Also, those are not powers, all of them are co-efficient, thanks. PS This is actually not a homework, was a solved example I was going through and I got confused.
1
vote
2answers
59 views

Trigonometry - Answers in Multiple Quadrants

May I please get some help with this question? Find the value of $\sec \theta $ if $\tan \theta = 0.4$ and $\theta$ is not in the 1st quadrant. Now, here is my current working: $\frac{\pi}{2} < ...
2
votes
1answer
56 views

Evaluate the following integral: $\int_0^{\pi} \sqrt{\frac{2}{\pi}}\sin(nx)\sqrt{\frac{2}{\pi}}\sin(mx) dx$

Show that for $n,m = 1,2,3, ...$: $$\int_0^{\pi} \sqrt{\frac{2}{\pi}}\sin(nx)\sqrt{\frac{2}{\pi}}\sin(mx) dx = \delta_{mn},$$ where $\delta_{mn}=\begin{cases} 0 & m \neq n \\ 1 ...
0
votes
3answers
598 views

If $\sec \theta+\tan \theta= \sqrt{3}$ then the positive value of $\sin \theta$

If $\sec \theta+\tan \theta=\sqrt{3}$ then the positive value of $\sin \theta$ Note: $1/\cos\theta+\sin\theta/\cos \theta=\sqrt{3}$ $\sin\theta=\sqrt{3}\cos \theta-1$ squaring on both sides we get ...
3
votes
3answers
341 views

Evaluating $\int \frac{1}{x\sqrt{9x^2-1}}\,dx$

I try to integrate $$\int \frac{1}{x\sqrt{9x^2-1}}\,dx$$ let $u=x^2,\quad \quad du=2x\,dx,\:\quad \:dx=\frac{1}{2x}\,du$ $$ \begin{align} & \int \frac{1}{x\sqrt{9u-1}}\frac{1}{2x}\,du \\[8pt] = ...
1
vote
4answers
44 views

Find $\frac{dy}{dx}$ for $x=2\theta+sin2\theta$ and $y=1-cos2\theta$

The parametric equations of a curve are $$x=2\theta+\sin2\theta,\:y=1-\cos2\theta.$$ Show that $\frac{dy}{dx}=\tan\theta$. I can use the chain rule to get ...
0
votes
2answers
34 views

Find angle $\alpha$ from a complex vector

I'm trying to solve this problem from a Russian book: Find the angle which is needed to rotate the vector $3\sqrt{2} + i2\sqrt{2}$ to obtain the vector $-5+i$. EDIT: $\tan\dfrac{\pi}{6} \neq ...
7
votes
1answer
126 views

Showing that $ \int_{0}^{\pi / 4} \arctan \! \left( \sqrt{\frac{\cos 2x}{2 \cos^{2} x}} \right) \mathrm{d}{x} = \frac{\pi^{2}}{24} $.

I was wondering if an expert in integration could kindly solve the following problem, which was posed in a mathematics competition (I can’t remember which one) and was unsolved by any participant. ...
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2answers
208 views

Calculate $n$ points having equal cartesian distance over a single sine wave

I'd like some help figuring out how to calculate $n$ points of the form $(x,\sin(x))$ for $x\in[0,2\pi)$, such that the Cartesian distance between them (the distance between each pair of points if you ...
0
votes
1answer
98 views

How to determine a general arithmetic sequence formula for two intersecting trig function

I have equations out of two trigonometric functions. For example $\cos(4\alpha$) = -$\sin(5\alpha)$ $\tan(0.5\alpha$) = 2 $\sin(\alpha)$ How can I determine a general arithmetic sequence formula ...
7
votes
5answers
154 views

Are all triangles where “$a^2 = b^2+ c^2$”, right-angled?

For a right angle triangle, you can say that the square of the hypotenuse is equal to the sum of the squares of the other two sides. Does the converse hold, ie. can you also say that, if the square ...
2
votes
3answers
89 views

Can cos(n!) in degrees tend to one if n>6?

does cos(n!) in degrees tend to 1. consider cos(n!)=cos(n*...*6!),6!=720=360*2.So this is like rotating on the plane n*...7*2 times so cos(n!)=1,When n>6 .Does this proof hold even when n tends to ...
-1
votes
3answers
94 views

How to evaluate an integral of the form $\int \frac{dx}{-ax^2 + b}$?

I need to evaluate $\int \frac{dx}{-ax^2 + b}$ while both $a$ and $b$ are positive. I was blocked while I was trying $ x=\tan\theta $ which turned $ dx=\sec^2\theta\, d\theta $ This method didn't ...
6
votes
2answers
149 views

Trigonometric integral: $\int_{25\pi/4}^{53\pi/4} \frac{1}{(1+2^{\sin x})(1+2^{\cos x})}\,dx$

Is it possible to evaluate the following in a closed form? $$\int_{25\pi/4}^{53\pi/4} \frac{1}{(1+2^{\sin x})(1+2^{\cos x})}\,dx$$ I found the above definite integral at I&S but the solution is ...
1
vote
1answer
79 views

How to solve this trigonometry equation?

Show that $3(\sin θ - \cos θ)^4 + 6(\sin θ + \cos θ)^2 + 4(\sin^6 θ + \cos^6 θ)$ is independent $θ$. This question was there in one of the cbse sample papers for class x . Tried many methods but ...
1
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2answers
30 views

Prove that if $y= 2x+\sin(y-2x)$, then $\frac{dy}{dx}$ is equal to $2$ using implicit differentiation

If ${y= 2x+\sin(y-2x)}$, then $\frac{dy}{dx}$ is equal to $2$. I solved it and got stuck with ${y'}= \frac{2-2\cos(2x-y)}{1-\cos(2x-y)}$. I end up getting zero as my numerator. Do I still need to use ...
0
votes
1answer
190 views

If $\tan^2\theta=1-e^2$, then the value of $\sec\theta+\tan^3\theta \csc \theta$ is

If $\tan^2\theta = 1-e^2$,then the value of $\sec\theta$ + $\tan^3\theta \cdot \csc\theta$ is... NOTE: $1/\cos\theta +\cos^3\theta/\sin^3\theta \cdot 1/\sin\theta$ multiply: $\sin^2 ...
1
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1answer
140 views

$\cos \alpha$/$\cos \beta$=a, $\sin \alpha$/$\sin \beta$=b then $\sin^2\beta$ is

Trigonometry help: $\cos \alpha$/$\cos \beta$=a, $\sin \alpha$/$\sin \beta$=b then $\sin^2\beta$ is. Note: $\sin \beta$ =b/$\sin \alpha$
2
votes
2answers
96 views

Finding the value of $\frac{\cos^4\beta}{\cos^2\alpha} + \frac{\sin^4\beta}{\sin^2\alpha}$.

Trigonometry $\dfrac{\cos^4 \alpha}{\cos^2 \beta}+ \dfrac{\sin^4\alpha}{\sin^2\beta} = 1$ then the value of $\dfrac{\cos^4\beta}{\cos^2\alpha}+ \dfrac{\sin^4\beta}{\sin^2\alpha}$ is? NOTE: can ...
3
votes
2answers
120 views

How to simplify the integral of $\int\frac{\cos(8x)}{\cos(4x)+\sin(4x)}dx$?

So I am trying to integrate this problem $\int\frac{\cos(8x)}{\cos(4x)+\sin(4x)}dx$, and my professor went over it in class and went from $\int\frac{\cos(8x)}{\cos(4x)+\sin(4x)}dx \rightarrow ...
3
votes
3answers
203 views

Integral of $\cos(\cos x)$ over $[0,2\pi]$

How to compute the following integral? $$\mathcal{J}_2=\int_{0}^{2\pi}\cos(\cos t)\,dt$$ I'm trying to compute this integral, but I have no idea of how to do it, can someone help me?
0
votes
1answer
69 views

Obtuse Angle using sine function

An airplane flew 490 miles at a bearing of N65°E from airport A to airport B. The plane then flew at a bearing of S38°E to airport C. Find the distance from A to C if the bearing from airport A to ...
0
votes
1answer
40 views

Sine Curve on the Y axis

This is a pretty simple question I have. I am trying to generate a line like this ? http://i58.tinypic.com/16izj0w.png Basically I want to rotate the sine curve 90 degrees and have that curve on the ...
2
votes
1answer
66 views

Triangle with two angles separated by side length

If a triangle has two angles (30◦ and 50◦ respectively) separated by a side of length 8, is it possible to find the lengths of the other two sides using Sine Law or Cosine Law? If not, why not? If ...
2
votes
0answers
203 views

How to find the approximate basic frequency or GCD of a list of numbers?

I could't actually summarize the question in the title, so I'll explain my situation. I want to tell the integer numbers which act as the best approximate basic frequencies of a list of real numbers: ...
0
votes
2answers
25 views

Range of a function - trigonometric

Question: Find the range of the function: $$\sin^4 x + \cos^4 x$$ I really have no idea how to initiate this question. Please help me find a solution!
6
votes
4answers
881 views

Find the side of an equilateral triangle given only the distance of an arbitrary point to its vertices

Triangle $ABC$ is an equilateral triangle and $P$ is an arbitrary point inside it. The distance from $P$ to $A$ is $4$ and the distance from $P$ to $B$ is $6$ and the distance from $P$ to $C$ is $5$. ...
1
vote
1answer
190 views

Using jacobian to solve a nonlinear system of equations?

I have to solve a system of nonlinear equations using jacobian but I'm not sure how to solve for the solutions. I remember one of my friends doing $Ax = B$; where jacobian matrix was $A$, but im not ...
0
votes
4answers
72 views

Physics problem, stuck in algebra.

I end up with the equations; $$u=u_1' \cos(a)+u_2' \cos(b)$$ $$u_1' \sin(a)=u_2' \sin(b)$$ $$u^2=u_1'^2+u_2'^2$$ I have to show that $$a+b=\frac{\pi}{2}$$ $x'$ isn't the derivative of $x$, it's a ...
2
votes
1answer
59 views

Evaluating an inv. tan function

The problem: Evaluate the inv. function by sketching a unit circ., finding the angle, and eval. the correct pair on the circle. Function: $\tan^{-1}(-1)$ I found a solution for this, but my ...
1
vote
2answers
50 views

Simplifying a trigonom. expression

The problem: Simplify the expression. Specify the range of $x$ for which the simplification holds: $\cos(\tan^{-1}x)$. So we know that, $\tan^{-1}x$ is the angle $\theta$ for which $\tan\theta ...
1
vote
3answers
108 views

Rotating an object around any origin

I want to extend my program that generates PDF and I need like to rotate an object (for example -30deg clockwise): 1: original 2: rotated object (origin is bottom left) The first problem is, that ...
3
votes
2answers
153 views

How to prove this $\frac{\sin{(nx)}}{\sin{x}}\ge\frac{\sqrt{3}}{3}(2n-1)^{\frac{3}{4}}$

let $n<\dfrac{\pi}{2\arccos{\dfrac{c}{2}}},c\in (0,2),c=2\cos{x}$, show that $$\dfrac{\sin{(nx)}}{\sin{x}}\ge\dfrac{\sqrt{3}}{3}(2n-1)^{\frac{3}{4}}$$ where $0<x<\dfrac{\pi}{2}$ My idea: ...
0
votes
1answer
18 views

Find catesian coordinate of T-point $P(-\frac{65\pi}{2}) $

Find the Cartesian coordinates of T-point $P(-\frac{65\pi}{2}) $. It is easy when there is no negative sign. I don' t know how to do with negative sign.
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vote
2answers
774 views

Why must closest approach occur when relative velocity is perpendicular to motion?

The first part i) I can solve correctly, but I need some advice and intuition on how to solve the second part ii). Here is the mark-scheme for the question: But for part ii) I do not understand ...
1
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0answers
56 views

Would this thinking about the dot product hold?

Background today I completed the chapter on the dot product of vectors. But in trying to figure out exactly what the dot product is. I came to the conclusion that it can be interpreted as the length ...
2
votes
5answers
165 views

If angular velocity $\omega=\sqrt{\frac{3g\sin\theta}{2a}}$ can I find angular acceleration $\alpha$ by differentiating $\omega$?

It was my understanding that angular acceleration is the derivative of angular velocity. The reason I ask is Thanks.
2
votes
5answers
1k views

Showing the $n$-th derivative of $\cos x$ by induction

I was asked to show that the $n$-th derivative of $\cos x$ is $\cos(\frac{n\pi}{2} + x)$. My progress : By induction, I proved it was true for $n=1$. Then I assumed it was true for $n = k$ so now I ...
-1
votes
1answer
55 views

How to find slope on line that known only point and angle

How to find slope on line that known only point and angle Image will describe more clearly I'm wont to find the orange line slope to find point on it ( b , c , d ) suppose that A and angle are ...
3
votes
7answers
11k views

Why $\sin(n\pi) = 0$ and $\cos(n\pi)=(-1)^n$?

I am working out a Fourier Series problem and I saw that the suggested solution used $\sin(n\pi) = 0$ and $\cos(n\pi)=(-1)^n$ to simply the expressions while finding the Fourier Coefficients $a_0$, ...
-1
votes
2answers
53 views

How do I change $\cos(\frac{\theta}{2})$ into $\cos(\theta)$ in an equation?

Just give me an example. eg. $\cos(\frac{\theta}{2})=\frac{1}{2}.$ I want to make $\cos(\frac{\theta}{2})$ become $\cos(\theta)$. Thanks.
1
vote
2answers
48 views

simplification of the area of a hyperbolic circle (BONOLA, S 53)

I'm trying to understand the S-53 of "Non-Euclidean Geometry" (BONOLA, R.) in which the formula for the area of a circle of radius r: $$2\pi k^2(\cosh\frac rk -1)$$ is somehow reduced by only applying ...