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

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51 views

Roots of $f(x)=a_0+a_1\cos x+a_2\cos 2x+\dots+a_n\cos nx$

If $a_i$'s are nonzero real numbers such that $a_n > {\sum^{n-1}_{i=0}}|a_i|$ prove that the number of roots of $f(x)=a_0+a_1\cos x + a_2\cos 2x+\dots+a_n\cos nx$ is at least 2n.
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2answers
53 views

Using induction to prove a formula for $\sin x+\sin 3x+\dots+\sin (2n-1)x$

I'm working from the text "Intro To Real Analysis" by William Trench. Here is what I have thus far. I will prove using Mathematical Induction that $\sin x+\sin 3x+...+\sin (2n-1)x=\frac{1-\cos ...
2
votes
1answer
73 views

If $\sin \phi$ and $\tan \phi$ are the roots of the equation $ax^2+bx+c=0$, compute $b^2-c^2$

If $\sin \phi$ and $\tan \phi$ are the roots of the equation $ax^2+bx+c=0$. Then $(b^2-c^2) = $ $\bf{Options::}$ $(a)\;\; 4ac\;\;\;\;\;\;(b)\;\; a^2\;\;\;\;\;\;(c)\;\; 4bc\;\;\;\;\;\;(d)\;\; ...
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0answers
35 views

Proving the trigonometric identity with angles in GP [duplicate]

Prove $$\sin\frac{2\pi}{7}+\sin\frac{4\pi}{7}+\sin\frac{8\pi}{7}=\frac{\sqrt{7}}{2}$$ Attempt- I tried to use trignometric identities but couldn't get the result
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2answers
92 views

What is $\frac{2x}{1-x^2}$ when $x=\sqrt{\frac{1-\cos\theta}{1+\cos\theta}}$?

If $$x=\sqrt{\frac{1-\cos\theta}{1+\cos\theta}}$$ Find $$\frac{2x}{1-x^2}$$ I got till here by simplification by taking the previous value of x, ie, ...
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1answer
43 views

Trigonometric Inequalities

If $$a\sin A+b\sin B+c\sin C=k$$ then the minimum value of $$\sin^2A + \sin^2B + \sin^2C =?$$ ATTEMPT- I tried to use A.M-R.M.S inequality but it didn't help
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0answers
49 views

Find $\int \tan(\tan x)\hspace{1mm}dx$

Find $\int \tan(\tan x)\hspace{1mm}dx$ This is an Interesting problem, which I have been trying from different directions, nothing seems to work, its been a day on this one. Can anyone figure out ...
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2answers
33 views

Trigonometric idnetity

IF $\sin \alpha = 3 \sin (\alpha+2\beta)$, then the value of $\tan (\alpha+\beta)+2 \tan \beta=$? ATTEMPT: $\sin \alpha = 3 (\sin (\alpha+\beta) \cos \beta + \cos (\alpha+\beta) \sin \beta)$ ...
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1answer
47 views

how to prove that $\cos n\pi=(-1)^n$?

I'm asked to prove that $$\cos n\pi=(-1)^n\qquad n\in\mathbb {Z}$$ I'm not sure how to approach the problem, I want to know if there is a different way to use induction
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1answer
33 views

Under what condition on f is this parametrized curve regular?

Consider a parametrized curve in $\mathbb R^2$ given by $$ \gamma (t)=(f(t)\cos(t), f(t)\sin(t)) $$ where $f$ is a smooth function of $t$. Under what condition on $f$ is $\gamma$ regular? I took the ...
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1answer
36 views

Finding area between two cosine curves

I must to find the area between these two curves: $$y = 2 \cos 7x, y = 2 − 2 \cos 7x$$ $$0 ≤ x ≤ π/7$$ And this is all I have so far: $$ 2\cos7x=2-2\cos7x $$ $$4\cos7x=2$$ $$\cos7x=1/2$$
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3answers
1k views

Ambiguity of notation: sin(x)^2

Several people have told me that $\sin(x)^2 = \sin(x^2)$. However, on several computing platforms, such as the TI-84 and Wolfram|Alpha, $\sin(x)^2 = \sin^2(x)$. Can I safely conclude that the notation ...
0
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1answer
22 views

Geometry Find the Radius of a circumcircle given the area of the triangle

Ok so here is what I know, the circumcircle of an equilateral triangle with an area of $4\sqrt{3}$ is drawn, calculate the radius lenght of the circumcircle. I also know that to find the radius I ...
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1answer
27 views

How to prove that this equality is the development of a fourier series?

how can I show that this identity is a development of a fourier series? $$f(x)=\sin^3 x=\frac{3}4 \sin x-\frac{1}4 \sin 3x$$ I tried this: obtain the Fourier coefficients whih $$b_n=\frac{2}\pi ...
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3answers
15 views

A right triangle has one leg twice as long as the other. Find a function that models its perimeter P in terms of the length x of the shorter leg.

A right triangle has one leg twice as long as the other. Find a function that models its perimeter P in terms of the length x of the shorter leg. I have tried adding 2x + x for the two legs, however ...
0
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1answer
10 views

Angle of Elevation and slope

Standing on top of a gentle 5degree slope I see the top of a tall building at an angle of elevation of 35degree 15’. I am 160 cm tall and it is 12 m from where I am standing to the foot of the ...
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1answer
39 views

Trigonometry question from “Quick Calculus” book [closed]

I'm working through a book "Quick Calculus" by Daniel Kleppner and Norman Ramsey. I don't understand one of the questions (frame 55, pg.29) being asked in the book see below... In the figure both ...
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0answers
28 views

Ordinary Differential Equation with a trigonometric function: radius of convergence?

For the equation $$x^2y'' + y' + \tan(x)\,y = 0$$ establish lower bounds for the radius of convergence about the point $$x_0 = 1.$$
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1answer
22 views

Trigonometry ramp problem [closed]

To load a war tank onto a transport trailer it has to go from ground level to $5$m high. The tank is limited to climbing ramps no more than $25^\circ$. Calculate the shortest ramp lenght that could ...
2
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1answer
35 views

Given a set of sequences, compute a corresponding set of functions

Consider the following set of sequences: $ S_k(n)= \begin{cases} 1 & \text{$n \equiv0\pmod{k}$}\\ 0 & \text{$n\not\equiv0\pmod{k}$}\\ \end{cases} $ I want to compute a set of ...
2
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1answer
49 views

Real life math to explore/solve

What are some examples of mathematics application in the real life that is interesting to explore about? And not too complicated but not too easy, something that exist around us. I'm interested in ...
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0answers
16 views

when you draw 1 altitude/ perpendicular bisector of an equilateral triangle, what can you form?

when you draw 1 altitude/ perpendicular bisector of an equilateral triangle, what can you form such that when you draw 4 equilateral triangles the foot of the perpendicular of the equilateral ...
2
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3answers
55 views

Why does $y = x\sin(\frac{180}{x})$ approach $\pi$?

A few days ago I was playing on my scientific calculator and I ran over an interesting little equation: $180\sin(1)$ is extremely close to $\pi$. At first I thought it was a coincidence, but then I ...
3
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1answer
61 views

Finding $\int_{0}^{1} \frac{\log(1+x)}{1+x^2} {\rm d}x$ by differentiating under the integral sign.

I've tried to find this integral by the method already outlined in the title. I decided to let $$ \displaystyle I(\alpha) = \int_{0}^{1} \dfrac{\log(1+\alpha x)}{1+x^2} \text{ d}x. $$ From this ...
3
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2answers
58 views

$\tan \left(\sec ^{-1}(x)\right)$

$$\tan \left(\sec ^{-1}(x)\right)$$ I know that sec(?)=$\frac{x}{1}$ and that sec=hyp/adj, therefore I conclude that hyp=x and adj=1 and that op=$\sqrt{x^2-1}$ Since Tan = opp/adj I thought the ...
0
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1answer
48 views

Using complex analysis to convert $b\cos \theta +a \sin \theta$ to a single trigonometric function

Using product $(a+bi)(\cos \theta+i \sin \theta) $ show that $$b\cos \theta +a \sin \theta=\sqrt{a^2 + b^2}\sin(\theta+\arctan(b/a))$$ and using this result show by induction that $$ ...
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0answers
29 views

How can I count logarithm of sinus?

I've red a book when was the information about logarithm of sinus/cosine etc. There was description of the logarithm sinus/cosine tables and how to using that. But there wasn't description how to ...
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4answers
23 views

Range of inverse trigonometric function

Find the range of $y$. $$y=\tan^{-1}\left(\frac{2x}{1+x^2}\right)$$ I used the following approach: Let $$x=\tan\theta$$ $$\therefore \theta=\tan^{-1}x$$ Since the principal solution of $\tan^{-1}$ ...
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0answers
22 views

Can a complex number have two arguments

Now, the reason why I wrote two $\theta$s is because my answer is the answer we get from $\theta$2 and the answer in the book is given the value of $\theta$1. So, I was just wondering whether both ...
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2answers
65 views

Solve the equation $a+b+c=abc$ for $a,b,c\in\mathbb{Z}$

Solve for $a,b,c$ (where $a$, $b$, and $c$ are integers) the equation $$a+b+c=abc.$$ I would prefer a solution using trigonometry and I think that it might use the formula $\tan A + \tan B + \tan ...
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2answers
34 views

Alternative of finding theta when sin $\theta$ and cos $\theta$ are given

For example, we're given a problem in which sin $\theta = \sqrt3/2$ and cos $\theta = -1/2$. To find out the angle $\theta$, I look at the unit circle and I get the answer. However, I was just curious ...
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1answer
40 views

trigonometric function and identities [closed]

If sin(x) = 1/3 and sec(y) = 17/15, where x and y lie between 0 and π/2, evaluate the expression using trigonometric identities. (Enter an exact answer.) sin(x − y) I have absolute no idea what this ...
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0answers
29 views

trigonometric functions and identities [closed]

Find all values of $x$ in the interval $[0, 2\pi]$ that satisfy the equation. $6 + 3 \cos(2x) = 9 \cos(x)$ I got $x = \dfrac \pi 3, \dfrac {5\pi}3$ and $2\pi$... but I probably did something wrong ...
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2answers
36 views

Determinant of a matrix with trigonometry functions.

Prove that the matrix is invertible for any value of $\beta$. I've done several exercises of this type. But I'm not sure with this one: $$\begin{bmatrix}\cos \beta & \sin \beta & 0\\ ...
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3answers
92 views

Find out $\theta$ when sin $\theta$'s and cos $\theta$'s value are given

Given: $\sin \theta = \frac12$, $\cos \theta = \frac{\sqrt{3}}{2}$. What I have tried: It is very easy looking at the angles' table and figuring out the value when the values of cos $\theta$ and sin ...
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4answers
98 views

Minimizing $\tan^2 x+\frac{\tan^2 y}{4}+\frac{\tan^2 z}{9}$

Given that $\tan x+2\tan y+3\tan z=40 , \ \ \ x,y,z \in \left(\dfrac{\pi}{2},\dfrac{3\pi}{2}\right),$ We need to find the minimum value of $ \tan^2 x+\dfrac{\tan^2 y}{4}+\dfrac{\tan^2 z}{9}$ ...
3
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1answer
56 views

Show that $\max(\mathrm{Re} (\exp(it)\cdot z) = |z| $

I need to show that $\max(\mathrm{Re} (\exp(it)z) = |z| $, with $t\in \mathbb{R}$ and $z\in \mathbb{C}$. Therefore I have calculated $\exp(it) = \cos(t) + i \sin(t)$. If we write $z= a+bi$, then $$ ...
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2answers
33 views

Basic Triangle Question [on hold]

Please help me out guys.I am new to trigonometry.. Plus do tell me that which two sides would be equal AC or BC? Thanks
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2answers
39 views

Trignometric problem (using De Movier's Theorem)

Ok so this question, I started out writing tan as sin and cos in the right side of the equation, simplified as much as possible and ended up with a very (sort of) fascinating equation which is ...
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0answers
28 views

For a boat to float in a tidal wave the water must be 2.5 meters deep… (trig questions)

$$y=5+4.6\sin\left(\frac{t}{2}\right)$$ What is the period in hours? Simply $p=2\frac{\pi}{n}=4\pi$ which is $\pi$ per hour If the boat leaves the bay at midday what is the latest time it can return ...
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0answers
36 views

period of cubic trigonometric functions

Can anybody explain how you would find the period of cubic trigonometric function. so I need to find the period of $f(x)=\sin^2\left(\frac{x}{3}\right)$. So I have began the question by finding the ...
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1answer
37 views

How to find coordinates of 3rd vertex of a right angles triangle when everything else is known?

I want to locate precisely the 3rd coordinate of a right angled triangle. I have: the length of three sides The three angles The other two coordinates of the triangle The triangle can lie in any ...
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5answers
221 views

Cosine of the sum of two solutions of trigonometric equation $a\cos \theta + b\sin \theta = c$

Question: If $\alpha$ and $\beta$ are the solutions of $a\cos \theta + b\sin \theta = c$, then show that: $$\cos (\alpha + \beta) = \frac{a^2 - b^2}{a^2 + b^2}$$ No idea how to even approach the ...
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5answers
53 views

Find the value of $\sec x$ using knowing that $9\sin x + 40\cos x = 41$.

I am trying to find the value of $\sec x$ using equation $9\sin x + 40\cos x = 41$. I have tried to solve but I failed.
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1answer
22 views

Find the domain to this function [closed]

Find the domain to the follwoing function $f(x)=[sin(π(x−1))]^{0.5} + (4-x^2)^{0.5}$ is$ [-1,0]\cup [1,2]$ the right answer
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1answer
33 views

Find the domain of the function [closed]

$f(x) = [\sin(\pi(x-1))]^{0.5} + \sqrt{4-x^2}$ What is the domain to this function? Would I have to solve for x?
2
votes
4answers
107 views

Solve system of equations with $\sin$ and $\cos$

Solve system of equations $\begin{cases} 3x^2 + \sin 2y - \cos y - 3 = 0 \\ x^3 - 3x - \sin y - \cos 2y + 3 = 0 \end{cases}$ I tried to use substitution $x = \cos t$ or sth, but I get literally ...
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3answers
40 views

Solve equation with two unknowns

I have these equations. $2\pi r_1+2\pi r_2=24$ and $\pi r_1^2+\pi r_2^2=20$ and to solve them I did the following steps Step 1 : $\frac {2\pi r_1+2\pi r_2}{2}= \frac {24}{2}$ Step 2 : $\pi ...
4
votes
1answer
77 views

Finding the maximum of $5\sin x+4\sin 2x$

How does one find the maximum value of $$ 5\sin(x)+4\sin(2x) $$ without using calculus?
4
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3answers
87 views

Find $\int \sinh^{-1}x\hspace{1mm}dx$

Find $\int \sinh^{-1}x\hspace{1mm}dx$ $ $ I am asked to use the following Equation: $$\int \tan^{-1}x\hspace{1mm}dx= x\tan^{-1}x-\ln(\sec(\tan^{-1}x))+C$$ $ $ The confusing part is : What has ...