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

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4answers
64 views

Is $\sin 2x$ the same thing as $2\sin x$?

Is $\sin2x$ the same thing as $2\sin x$? I am unsure whether it is valid to bring out the two outside the sine.
1
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1answer
40 views

Differentiation Proving

Can someone please help me solve this question. Provide a hint? If $$\cos\frac x 2\cos\frac x 4\cos\frac x 8\cdots=\frac{\sin x}x$$ then prove that $$\frac{\sec^2(x/2)}4 + \frac{\sec^2(x/4)}{16} ...
1
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0answers
15 views

Nonnegativity on a special domain entails nonnegativity on the whole plane

Let $Q$ be a real bivariate polynomial such that $Q(x,\tan(x))\geq 0$ for any $x\not\in\{\pm\frac{\pi}{2}\}+(2\pi){\mathbb Z}$. Does it necessarily follow that $Q$ is nonnegative on the whole of ...
2
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3answers
67 views

$\int \dfrac{\cos x}{\left(\cos (2x)\right)^{3/2}} dx$

Wolfram gives this nice result: $$\int\frac{\cos x dx}{\cos^{3/2}2x}=\frac{\sin x}{\sqrt{\cos 2x}}+\text{constant}$$ I have tried writing $\cos 2x = \cos^2x - \sin^2x $ and doing Weierstrass ...
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5answers
117 views

Prove $3(\sin x-\cos x)^4 + 6(\sin x+ \cos x)^2 + 4(\sin^6 x + \cos^6 x) -13 = 0$

Q) Prove that $3(\sin \theta-\cos \theta)^4 + 6(\sin \theta+ \cos \theta)^2 + 4(\sin^6 \theta + \cos^6 \theta) -13 = 0$ Source: Trigonometric Functions, Page 5.9, Mathematics XI - R.D. Sharma ...
0
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0answers
18 views

solving 3D trigonometry

A 7m tall flag pole is placed exactly 5m in front of a north facing vertical wall. When the sun is north-west, the shadow of the pole just touches a vertical wall. Calculate the angle of elevation of ...
0
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1answer
20 views

When total time gets minimized?

We want to get from $\displaystyle{C}$ to $\displaystyle{A}$. The path $\displaystyle{C \to B \to D \to A}$ can be done with constant velocty $\displaystyle{w}$. So that the time to get there gets ...
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0answers
55 views

Calculate area of this figure

I have an homework assignment where I have to calculate area of the figure underneath. I used the following formula to calculate the result $\frac {130 \cdot 55 \cdot sin35}{2} = 4188 m$ and then ...
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2answers
55 views

Why are the sines and cosines of something resulting in the wrong anser?

For any developers, I use the following code: ...
0
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3answers
53 views

Find the sum $1+\cos (x)+\cos (2x)+\cos (3x)+…+\cos (n-1)x$ [duplicate]

By considering the geometric series $1+z+z^{2}+...+z^{n-1}$ where $z=\cos(\theta)+i\sin(\theta)$, show that $1+\cos(\theta)+\cos(2\theta)+\cos(3\theta)+...+\cos(n-1)\theta$ = ...
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1answer
24 views

Integrate a trigonometrical expression [closed]

I know that this may sound a silly question but is the following which is the integral: $\int(1+(\cot^2)x)dx$
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votes
3answers
140 views

What is the order of operations in trig functions?

Is $\sin(x)^2$ the same as $\sin^2(x)$ or $\sin(x^2)$? I thought it would mean the former interpretation, $\sin^2(x)$, rather than the latter, but my teacher and I had a long argument on this and in ...
0
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1answer
44 views

3 snooker balls in a box

A regulation snooker ball is 52.5 mm in diameter. What are the minimum internal dimensions of a cube that can exactly contain 3 of them? I'm sure there must be an easy answer, but I'm not a ...
1
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1answer
54 views

Evaluation of $\int_{-\pi}^{\pi} \cos(ax) \sin^n(bx) dx$

As it is a kinda famous integral I thought I would find something on MSE but I didn't so here I am. If there is, link it in the comments and I will delete the question. How do I evaluate ...
0
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2answers
65 views

Simplify the trigonometric expression

Simplify the expression $$\left(1-\frac{\cos61^{\circ}}{\cos1^{\circ}}\right) \left(1-\frac{\cos62^{\circ}}{\cos2^{\circ}}\right)\cdot ...\cdot \left(1-\frac{\cos119^{\circ}}{\cos59^{\circ}}\right)$$
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1answer
35 views

How many points to span a goniometric wave and how to construct the goniometric function

I have two questions concerning the spanning of a simple trigonometric function: What is the minimum number of points to define/span a "simple" trigonometric wave in two dimensions? Is it possible ...
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1answer
31 views

Solve the equation $\left | \tan x \right | = 2 \cos^2x$

Solve the equation $\left | \tan x \right | = 2 \cos^2x$
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1answer
11 views

Find the speed of an object given two vectors

Here's the question: An airplane flies horizontally from east to west at 304 miles per hour relative to the air. If it flies in a steady 50 mile per hour wind that blows horizontally toward the ...
0
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0answers
19 views

Signed angle bewtween two normals

I have two abitrary planes in 3D space which share two vertices. Each plane has a unit normal and boh planes follow the same 'handedness' which describes the 'up' side of the plane. The line between ...
0
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1answer
16 views

Figuring out the radius from chord or arc

Are you able to figure out the radius of a circle by any chord and or arc?
2
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0answers
16 views

Polygons inscribed in circles, with integer sides and integer radius

Is there a simple characterization for an integer partition $(s_1,\dots,s_k)$, such that a polygon with these sides is inscribed in a circle with integer radius? This is what I got so far: All ...
1
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1answer
33 views

Find the angle of a triangle

I'v tried to solve this problem but did not get the right result. Triangel PQR is PQ = 5,0 cm, QR = 6,3 cm and RP = 7,4 cm. Calculate angle P. I tried to solve it by using by using the following ...
2
votes
4answers
109 views

Find the midpoint between two points on the circle

I want to place a new point in the middle of the two points which are on the circle outline (Arc). I have the coordinates $(x,y)$ of the center of the circle, the two red points and the radius of the ...
2
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1answer
29 views

How find the least value of the expression: $M = \cot^2 A + \cot^2 B + \cot^2 C + 2(\cot A - \cot B)(\cot B - \cot C)(\cot C - \cot A)$?

Consider all triangles $ABC$ where $A < B < C \leq \frac{\pi}{2}$. How find the least value of the expression: $M = \cot^2 A + \cot^2 B + \cot^2 C + 2(\cot A - \cot B)(\cot B - \cot C)(\cot C - ...
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1answer
127 views

Prove |cos(x−1)|+|cos(x)|+|cos(x+1)|≥3/2

I'm working on an induction proof, but I keep coming up against a brick wall. While working through the induction proof process I keep ending up with $$|\cos(m)|\ge\frac12$$ ,but clearly this isn't ...
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3answers
38 views

Two ways to find the cosine of 180 degree angle

I found a question how to find the value of cos 180, then we all know that its answer is equal to cos 0, which give us 1 as answer. I myself think that the idea of cos 180 is equal to 1 is : ...
0
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1answer
11 views

indefinite trig substitution integral

I am not able to understand how to get the solution for an integral. Substituting something like $\frac{1}{12}tan(\theta)$ seems to be the right thing to do, but I can't figure it out from there. ...
1
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2answers
33 views

Sum of two trig function's identity

We all know that $\sin(x) + \sin(y) = 2\sin((x+y)/2)\cos((x-y)/2)$ But is there an identity for $\sin(x) + z\sin(y) = ?$ Or do I need to figure it out using Euler's formula $\sin(x) = (e^{ix} - ...
1
vote
1answer
44 views

Will $\arccos(x)$ always give me the angle I am looking for?

Perhaps a dumb question, but I've never had trigonometry classes so I am much behind every one else. If $\cos\theta = x$, will $\arccos(x)$ always give me the "first" angle in positive direction with ...
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0answers
11 views

Finding directions of the internal forces within a freely jointed rigid object? [migrated]

Consider the the following problem: Here is a force diagram showing the situation: For part (i) Taking moments about $B$ for $BC$ gives $84.5Lcos\beta=2LT$ So $T=39$N For part (ii) Resolving ...
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2answers
29 views

Lengthy Product of trigonometric ratios

What is the value of the product $\sin(10) \sin(20) \sin(30) \sin(40) \sin(50) \sin(60) \sin(70) \sin80$, where all the angles are in degrees? Solve using complex numbers. I found this in a book of ...
0
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1answer
31 views

Accumulation points of trigonometric sequences

I am interested if the following sequences have accumulation points: $$x_{n} = \sin(2+\frac{1}{n})$$ and $$x_n = \tan (n)$$ Specifically for the first sequence is $1$ and $-1$ accumulation points? ...
11
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5answers
240 views

Proof: $\cos^p (\theta) \le \cos(p\theta)$

I came across this problem when I was at a book store inside of a book made to prepare Berkeley graduates to pass a mandatory exam. I wanted to buy the book, but, alas, I didn't have the money (forty ...
1
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3answers
63 views

How do I simplify $\arccos(x)−\arcsin(x)$ for $x$ in $(−1,1)$

How do i simplify $\arccos(x)−\arcsin(x)$ for $x$ in $(−1,1)$ i got somewhere that... $\sin(x)= \cos(\frac{\pi}{2}-x)$ so $\arccos(\sin(x))+x=\frac{\pi}{2}$ substituting that $\sin(x)=t ...
1
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2answers
81 views

How do i simplify $\cos(4x)\cos(3x)−4\sin(x)\sin(3x)\cos(x)\cos(2x)$?

How do i simplify $\cos(4x)\cos(3x) − 4\sin(x)\sin(3x)\cos(x)\cos(2x)$ ? I tried plugging in the double angle formulas $\cos(x+3x)$ and $\cos(x+2x)$ and went nowhere please help me. Also maybe ...
1
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3answers
57 views

Solve $\sin 2x = \sin(x-\frac{\pi}6)$

I got a this equation to solve and $\sin(2x) = \sin(x-\frac{\pi}{6})$ They was nice to give me for possible answers it could be. x = $\frac{\pi}{6}$ x = $\frac{\pi}{18}$ x = $\frac{7\pi}{18}$ x = ...
2
votes
1answer
80 views

How do I understand Pythagorean theorem

1) I understand the formula $$\frac{BC}{AB}=\frac{BH}{BC},\ \frac{AC}{AB}=\frac{AH}{AC}$$ But I can't understand the formula is obtained $$BC^2=AB\times BH \ \text{and}\ AC^2=AB\times AH$$ Why if ...
0
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1answer
85 views

Measuring Earth Radius using distance between two tall buildings

Measuring Earth's radius using two tall building heights H and h with a ground separation s and also when the building tops are collinear with the horizon: Concluding result of concurrent thread " ...
2
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3answers
42 views

Evaluating $\int \:x\csc \left(x^2\right)\cot \left(x^2\right)dx$

I try to evaluating $\int \:x\csc \left(x^2\right)\cot \left(x^2\right)dx$ let $u=x^2,\quad \quad du=2xdx,\:\quad \:dx=\frac{1}{2x}du$ then i get $\int \:x\csc \left(u\right)\cot ...
7
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3answers
416 views

Help with determining irrationality of a number?

I am trying to prove: If $\cos(\pi\alpha) = \frac{1}{3}$ then $\alpha \in \mathbb{R} \setminus \mathbb{Q}$ So far, I've tried making it into an exponential, since exponentials are easier to ...
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0answers
56 views

heights and distances [closed]

Background: I live on fifth floor of a building and observe four straight horizontal beams as one line at a certain level above ground on a radio transmission tower exactly projected on the horizon ...
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1answer
9 views

Questions regarding angle of incline of truncated cones

My question is a little more complicated than what the title says. I am trying to take a truncated cone and form an accurate 2D net drawing of it so it can be cut out and folded into the same 3D ...
2
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3answers
53 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
44 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?
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5answers
48 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
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1answer
20 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
224 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 ...
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1answer
21 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
148 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
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2answers
131 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 ...