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

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3
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3answers
79 views

How is $\sqrt {2+\sqrt {2+\sqrt {2+}}} … n $ times = $2\cos( π/2^{n+1})$?

How is $\sqrt {2+\sqrt {2+\sqrt {2+}}} ... n $ times = $2\cos( π/2^{n+1})$? No idea. Please help. I found this identity in a solution of a problem related to limits. Also if any more identities ...
0
votes
1answer
26 views

Finding a 3rd point in a 3D triangle with known plane, two points and lengths of each side

I have a very similar problem to the below question. right triangle in 3D space, vectors, line intersection? Rather than having the unit vector $A$ I have the lengths $i_2$ to $i_3$ and $i_1$ to ...
0
votes
0answers
28 views

Trig. substitution problem, getting from $\sec^3()$ to next step with an ln [duplicate]

I need help as to how they get from the step in red to the step outlined in blue. I'm not seeing how they get the ln in there...
0
votes
4answers
72 views

How does $1 + \tan^2x = 1/\cos^2x$?

I am unable to see why $$1 + \tan^2 x= 1/\cos^2x$$ I have looked into the topic anad I am familiar with the reciprocal ratios of cosec, sec, and cot. but cannot derive how this statement makes sense. ...
3
votes
2answers
50 views

How to explain why the angle between two vectors in $\mathbb{R}^n$ is defined the way it is.

It is given in couple of the textbooks I have seen that they just define the angle between two vectors $\vec{x}, \vec{y} \in \mathbb{R}^n$ to be $\theta$ such that $$ \cos \theta = \frac{\vec{x} ...
1
vote
1answer
37 views

Trig Integration, how did they get this answer?

Here is the full question for context: And here is the full answer given: I don't understand how they go from the first part in the red box to the next, I don't have a clue.
1
vote
1answer
37 views

why value of Trigonometric ratios of angle and its reference angle are same?

I'm learning Trigonometry right now with myself and at current about how to find trigonometry ratios of angles greater than $90^\circ$. I came to know that for finding trigonometric ratio of these ...
2
votes
3answers
52 views

Finding $g'(x)$ of the following.

So I need help solving the following question. $$\int_3^{x^5} \sin(t^2)\,dt$$ I tried to do it and got $5x^4\sin(x^2)$, but the correct answer is $ 5x^4\sin(x^{10})$. Where did the $x^{10}$ come ...
2
votes
1answer
40 views

How do I solve $∫4cos^2(x) dx$? [duplicate]

I have the basic idea of how to work out the integral of a trig function, but am having trouble in applying the concept. Would really appreciate it if someone could help me. Thanks!
0
votes
3answers
47 views

Definite integral $\int_1^{\pi/8}(x-1)\sin 4x\,dx$

I know, it might be silly, but I can't figure out what to do with $\sin 4x$ in $$\int_1^{\frac{\pi}{8}} (x-1)\sin 4x\,dx$$ Source. I guess $u = x - 1$ and $\sin 4x=2\sin 2x\cos 2x$. But what's ...
1
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1answer
45 views

Dimensions of a rectangle containing a rotated rectangle

Given sides a, b, and an arbitrary rotation Θ (0 - 360 degrees) around the centerpoint of the rectangle, how would I calculate sides A and B of a containing rectangle?
0
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2answers
24 views

If $x=r\cos(\theta)$ then in $?=r\cos(\theta+a)$ what is $?$ equal to?

What I mean by $r\cos(\theta+a)$ is that it's the same function $r\cos(\theta)$ but it's translated by $a$ units, if this makes any sense. I just want to know what it means in terms of $x$.
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2answers
33 views

Need better explanation for part of trig substitution problem with intengral

I've been stuck on this problem for a long time, and have finally managed to get all but the part with the.... bounds? of the integral figured out. I can't seem to get my head around what's going on ...
-3
votes
1answer
33 views

How do I calculate the angle between two sides of a polygon? [closed]

So I got a polygon and I have all of the points. What I need, is to find all internal angles of this irregular polygon. How do I do that?
1
vote
2answers
29 views

least value of expression

What is the least value of $\csc^2(x)+36\sec^2(x)$ ? So I differentiated and when simplifying we get $\tan^4(x)=1/36$, but that is giving me max value i think. Can someone just help me . Or is ...
0
votes
1answer
31 views

The inequality used to study a trigonometric function's sign has no solutions

The function, sign of which we are asked to study, is: $$f(x) = 2\sin(x)\cos(x) - 1,$$ or, simplified: $$f(x) = \sin(2x)-1,$$ without using its graph to do so, since the exercise itself asks, in ...
1
vote
4answers
118 views

Proving Trig Identities (Complex Numbers)

Question: Prove that if $z = \cos (\theta) + i\sin(\theta)$, then $$ z^n + {1\over z^n} = 2\cos(n\theta) $$ Hence prove that $\cos^6(\theta)$ $=$ $\frac ...
4
votes
0answers
86 views

Trigonometric Expression for $1 + \cos \alpha + \cos 2\alpha + \cdots + \cos n \alpha$ using complex numbers

This question is not a duplicate because I am asked here to use the fact that $1 + \cos \alpha + \cos 2 \alpha + \cdots + \cos n \alpha = Re (1 + z + z^{2} + \cdots + z^{n})$, where the question this ...
4
votes
2answers
52 views

Proving trig identity using De Moivre's Theorem

Question: Prove $$\cos(3x) = \cos^3(x) - 3\cos(x)\sin^2(x) $$ by using De'Moivres Theorem So far (learning complex numbers at the moment) that De Moivre's theorem states that if $z$ $=$ ...
3
votes
3answers
74 views

Limit of the floor function of $\frac{x}{\sin(x)}$

Alright this looks like a very simple problem at the first go. I need to find $$\lim_{x\rightarrow0^+} { \left\lfloor{\frac{x}{\sin (x)}}\right\rfloor}$$ So since I know the inner function's graph ...
2
votes
1answer
23 views

Value of $z$ in $\sin z=\frac{1}{2}\sqrt{2}(1-i)$?

I'm trying to find the value of $z$ in $\sin z=\frac{1}{2}\sqrt{2}(1-i)$. I have tried with the equation $\sin z=\frac{e^{iz}-e^{-iz}}{2i}=\frac{1}{2}\sqrt{2}(1-i)$ but I get stuck and can't make it ...
0
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2answers
38 views

Find out the general solution of $\tan x + \cot 2x = 2$.

I want to solve for $x$ in $$\tan x + \cot 2x = 2.$$ I tried to write in terms of $\sin x $ and $\cos x$ but couldn't get the answer. Any help is appreciated.
0
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1answer
15 views

Proving Equations (Complex Conjugates)

The question is: $z$ is a complex number given by $z$ $=$ $sin$$(\theta)$ $+$ $i(1-cos(\theta))$, $-\pi < \theta < \pi $ Show that if $w$ $=$ $\frac 1{z-i}$ then $w$ $=$ $z^* + i $ where ...
1
vote
2answers
27 views

Given diagonals, lower base, and height, find the legs and upper base of isosceles trapezoid

Given an the height, base, and diagonals of an isosceles trapezoid, how am I to find the upper base and the legs? I know I can find the area of the triangles made by the diagonals, but how is that ...
3
votes
1answer
83 views

What is the standard deviation of this random variable? (I want to check my calculation)

Consider the random variable (orientation angle) $0\le\theta\le 2\pi$ with the following PDF where $\theta_0$ is the mean orientation angle: $(n\in\mathbb Z , n\ge 0)$ ...
3
votes
7answers
66 views

Evaluating the following limit: $\lim _{x\to \frac{\pi }{4}}\left(\tan\left(2x\right)\tan\left(\frac{\pi }{4}-x\right)\right)$

I don't find the right identities for this $$\lim _{x\to \frac{\pi }{4}}\left(\tan\left(2x\right)\tan\left(\frac{\pi }{4}-x\right)\right)$$ Someone can help me ? Thanks.
2
votes
0answers
38 views

How to tell if a polynomial has exact trigonometric or logarithmic roots?

The polynomial $$ 64x^7 -112x^5 -8x^4 +56x^3 +8x^2 -7x - 1 = 0 $$ has seven roots, x = {1, $-\dfrac{1}{2}, \cos \dfrac{2n\pi}{11}$}, where n={1,2,3,4,5}. Is there any way to tell if an arbitrary ...
0
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1answer
24 views

Cartesian to Spherical coordinate conversion specific case when Φ is zero and θ is indeterminant

Following is the conversion for spherical to cartesian coordinate \begin{align} x &= r \cos\theta \sin\varphi \\ y &= r \sin\theta \sin\varphi \\ z &= r \cos\varphi \end{align} and we are ...
0
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1answer
30 views

Solve equation for t

$$s = 2 \ln|\tan(t) + \sec(t)|$$ I tried to solve it and got a quadratic equation which turned out to equal $arcsin(\dfrac{-2 \pm e^s}{2(1+e^s)})$ This doesn't seem right. Any thoughts?
1
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5answers
79 views

How to prove $\sin3θ=3\sinθ-4\sin^3θ$

I was solving an A Level past paper (November 2014 P32) when I stumbled upon this question. It first asks us to expand $\sin(2θ+θ)$ which is easy using the identity $\sin(A+B)=\sin A\cos B+\cos A\sin ...
1
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1answer
83 views

Which function to kill: Sine or Cos?

I got an equation which was a solution to some familiar Differential Equation I am solving, the solution takes the form of: $$V=Ce^{-ix}$$ but $$Ce^{-ix}=A\cos(x)+B\sin(x)$$ so ...
0
votes
2answers
22 views

Largest possible sphere is inscribed in a cube. What percentage is the volume of the sphere smaller than the volume of the cube?

Largest possible sphere is inscribed in a cube. What percentage is the volume of the sphere smaller than the volume of the cube? I have already found out: volume of the cube is $X^3$ volume of the ...
2
votes
3answers
55 views

Checking the result of integration, $\int\sinh^3x\cosh xdx$

I integrated $\int\sinh^3x\cosh xdx$ in the following way: \begin{align*} \int\sinh^3x\cosh xdx ={}& \int\sinh^2x\sinh x\cosh xdx = \frac12\int\sinh^2x\sinh(2x)dx ={} \\ {}={}& ...
1
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5answers
56 views

How to solve this limit involving sine and log?

I've tried L'Hopital's Rule but the differentiated numerator involves cos(1/x) which does not exist when x approaches 0. $$ \lim_{x\to 0^+} \frac{x^2sin\frac{1}{x}}{\ln(1+2x)}$$
0
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2answers
39 views

Value of ratio of inverse trigonometric values

How to prove that the value of the following expression is $2$ manually? Please tell me the quickest method to deal with such problems. ...
0
votes
2answers
49 views

Find the value of $\left(\cos 52^{\circ}+\cos 68^{\circ}+\cos 172^{\circ}\right)$

Find $\left(\cos 52^{\circ}+\cos 68^{\circ}+\cos 172^{\circ}\right)$ $\color{green}{a.)\ 0 }\\ b.)\ 1 \\ c.)\ 2 \\ d.)\ \text{none of these} $ In exam I often fail to remember the formula's ...
3
votes
1answer
59 views

Summation of the given series

Is there anyway to find the sum of: $\cos(A)+\cos^2(2A)+\cos^3(3A)+....$ upto 'n' terms. Actually original question was to find sum of : $\cos(A)+\cos(2A)+\cos(3A)+...$ upto 'n' terms and I found it ...
0
votes
1answer
20 views

Calculating rotations required to

I have a situation similar to a question asked on StackOverflow with the following image: I have an object that rests at the tip of the green arrow (point XYZ), and I'm using the following formulas ...
2
votes
3answers
37 views

Find the maximum value of $(12\sin x-9\sin^{2} x)$

The maximum value of $(12\sin x-9\sin^{2} x)$ is equal to $a.)\ 3 \\ \color{green}{b.)\ 4} \\ c.)\ 5 \\ d.)\ \text{none of these}$ As $-1\leq \sin x\leq 1 ,\\ 12\sin x-9\sin^{2} x \\ ...
2
votes
3answers
52 views

If $\sin x+\sin^{2} x=1$ , Find $\cos^{12} x+3\cos^{10} x+3\cos^{8} x+\cos^{6} x+2\cos^{4} x+\cos^{2} x-2 $

If $\sin x+\sin^{2} x=1$, then the value of $\cos^{12} x+3\cos^{10} x+3\cos^{8} x+\cos^{6} x+2\cos^{4} x+\cos^{2} x-2 $ is equal to $a.)\ 0 \\ b.)\ 1 \\ c.)\ 2 \\ \color{green}{d.)\ ...
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2answers
29 views

Limit problem involving trig functions and radical

I am stuck on this limit and have no idea how to solve it and which trig identity to use. Any help would be appreciated. Thanks! $\lim\limits_{x \to 0^-} \frac{\sqrt{1+2\sin^2 ...
2
votes
3answers
49 views

Find the two other sides in a 15-30-135 triangle

A triangle has angle measures of 15, 30, and 135 degrees. The side opposite the 15 angle is x feet, the side opposite the 30 angle is y feet, and the side opposite the 135 angle is 2 feet. Find x and ...
0
votes
4answers
61 views

Integral $\int_0^{\pi/2} \sin(ax)\cos(x)\,dx$

I have to evaluate an integral $I(a) = \sin(ax)\cos(x)$ from $0$ to $\pi/2$.The variable of $a$ is not is greater than $1$: $$\int_0^{\pi/2} \sin(ax)\cos(x)\,dx$$ I attempted to change the function ...
1
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1answer
35 views

Page 72 of Courant and Hilbert's Methods of Mathematical Physics, Vol 1.

We have the following identities: $$ \beta_\nu = b_\nu -\frac{1}{2}(b_{\nu-1}+b_{\nu+1}),\ \ \ \ (\nu=2,3,4,\ldots)\\ \beta_1=b_1-1/2 b_2 $$ $$s_n(x)=\sum_{\nu=1}^n b_\nu \sin(\nu x) \\ ...
0
votes
1answer
22 views

In a $\triangle ABC,$ Evaluation of minimum value of $\cot^2 A+\cot^2 B+\cot^2 C$

In a $\triangle ABC,$ Evaluation of minimum value of $\cot^2 A+\cot^2 B+\cot^2 C$, Given $A+B+C = \pi$ $\bf{My\; Try::}$ Using $\bf{A.M\geq G.M}$ $$\frac{\cot^2 A+\cot^2 B}{2}\geq \cot ...
11
votes
1answer
229 views

What is the geometry behind $\frac{\tan 10^\circ}{\tan 20^\circ}=\frac{\tan 30^\circ}{\tan 50^\circ}$?

This identity is solvable by help of trigonometry identities , but I think there is an interesting and simple geometry interpretation behind this identity and I can't find it. I found it when I ...
0
votes
2answers
35 views

How to differentiate $ y=\sin^2(2x)\cos(x) $?

I was solving some A Level past papers and I came across this question. We have the equation of the line $ y=\sin^2(2x)\cos(x) $ for $ 0\leq x \leq \frac{\pi}{2} $ and there is a maximum point M. We ...
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vote
4answers
53 views

If $\boxed{\tan A+\tan B+\tan C=6, \\ \tan A\tan B=2} $ in $\triangle ABC$ then find the type of triangle.

In $\triangle ABC$, $\tan A+\tan B+\tan C=6 \\ \tan A\tan B=2 $ Then the triangle is $a.)\text{Right-angled isosceles} \\ b.) \text{Acute-angled isosceles}\\ ...
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3answers
54 views

If $A=\sin^{20}\theta +\cos^{48}\theta $ then identify the correct option.

If $A=\sin^{20}\theta +\cos^{48}\theta $, then for all values $\theta$ a) $A\geq 1$ b) $ 0< A\leq 1$ c) $1<A< 3$ d) None of these $0 \leq \sin^{20}\theta \leq 1$ $0 \leq ...
0
votes
3answers
57 views

Integration of $\arctan$

This question seems so silly, yet I want to know if this is true! I know for a fact that the following is true$$\int \frac{1}{1+x^2} \,\text{d}x = \arctan(x) + C$$ But does this mean this is true ...