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

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6
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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
votes
1answer
1k 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 ...
1
vote
1answer
38 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 ...
1
vote
3answers
326 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
votes
1answer
194 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 ...
1
vote
0answers
49 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 $$x_0 = 1.$$
2
votes
1answer
51 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
votes
1answer
495 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 ...
0
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0answers
36 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 ...
3
votes
3answers
81 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
votes
1answer
99 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
votes
2answers
66 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
votes
1answer
64 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 $$ ...
1
vote
4answers
43 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 $\tan^{-1}$ ...
0
votes
2answers
123 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 ...
0
votes
2answers
77 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 ...
1
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2answers
198 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\\ ...
1
vote
3answers
1k 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 ...
2
votes
4answers
116 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}$ ...
0
votes
1answer
98 views

determine shortest distance between circle intersections

I have three circles positioned shown in the fig. Each of them has the same radius. I know the distance between each of them (A-B, B-C, A-C). My goal is to find the shortest path between B and C. The ...
3
votes
1answer
91 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 $$ ...
1
vote
2answers
59 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 ...
0
votes
0answers
60 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 ...
1
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1answer
5k views

How to find coordinates of 3rd vertex of a right angled 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 ...
2
votes
5answers
297 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 ...
0
votes
5answers
71 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.
2
votes
4answers
137 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 ...
1
vote
3answers
47 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
89 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?
5
votes
3answers
152 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 ...
0
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2answers
55 views

is there a way on how i can find the area of the octagon formed by the equilateral triangles?

is there a way on how i can find the area of the octagon formed by the equilateral triangles?
1
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2answers
105 views

$\sin 2a + \sin 2b + \sin 2c - \sin 2(a+b+c) = 4 \sin (a+b) \sin (b+c) \sin (c+a)$

I am clueless about this homework question. Looking at it, I see I could use the compound angle and and sum formulas, and tried using them. Unfortunately, couldn't proceed beyond that. Help? $$\sin ...
0
votes
3answers
35 views

Question regarding in periodic function

I have question I know that $\cos(x+2\pi)=\cos x$ and $\sin(x+2\pi)=\sin x$ but if we have $\cos(x+\pi)=?$ and $\sin(x+\pi)=?$ with explaination thanks
1
vote
2answers
24 views

Trigonometric equation $\tan(\frac{\sqrt{3}x}{2})=-\sqrt{3}$

I want to solve a trigonometric equation below: $$\tan(\frac{\sqrt{3}x}{2})=-\sqrt{3}$$ What is the value of $x$ for $x>0$ Thank you for your help.
0
votes
1answer
15 views

How do I find and list compositions for (f) and (g)?

Ok, I've literally just spent the last 2 hours just to figure out two compositions problems for homework, and I've about had it. Anyone here that can help? Problem 1 $$ f(x) = 2x(2) - x -3 $$ $$ ...
2
votes
2answers
51 views

Solve the trigonometric equation/inequality

$(1).\quad\cos^2(2x) + \sin^4(x) = 2$ $($solve the equation$)$ $(2).\quad2\cos^2(3x) + 5\cos(3x) - 3 < 0$. For this question, I tried letting $t=\cos(3x)$. Thus, $2t^2 + 5t - 3< 0$, ...
1
vote
2answers
55 views

evaluating trig without a calculator

Evaluate $\csc^{-1} (\sec 5)$ without using a calculator. I have no idea where to begin on this problem, I've looked over trig identities, and cannot find one that I think applies. Any help would be ...
0
votes
1answer
266 views

Vector word problem

I'm having real trouble understanding exactly what's going on in this word problem. Here is the problem word for word: A submarine descends at an angle of $30^{\circ}$ below the horizontal with a ...
0
votes
1answer
46 views

symbolic solution to trig equation for a variable

Is it possible to solve the following singular transcendental equation in $a$ for the variable $a$? Any symbolic solution will do. $$\sqrt{s^2 - v^2} = 2a \, \sinh \left( \frac{h}{2a} \right)\,\,\,$$ ...
2
votes
2answers
486 views

Bilinear interpolation of angles

Is their a solution to do a bilinear interpolation in x,y of angles in [0°-360°[ ? The elementary formula of bilinear interpolation don't work on angles due to the discontinuity at 360°-0°. ...
2
votes
4answers
120 views

Limit involving sin

The question is $$\lim_{t\to0}{{1\over 2+ \sin (t)}-{1\over 2}\over \sin (t)}$$ since I cannot directly substitute, how would I go about factoring the $\sin t$. Any help is appreciated!
2
votes
3answers
46 views

Applying angle addition formulas for subtraction

The angle addition formula says that: $\sin(\phi + \theta) = \sin(\phi) \cdot \cos(\theta) + \cos(\phi) \cdot \sin(\theta)$ Why are the following steps valid?: $\sin(\phi − \theta) = \sin(\phi) ...
1
vote
1answer
43 views

Parabolic asymptote of $n\cot\frac\pi{2n}$

I have determined that $$\lim_{n\to\infty}\frac{n\cot\frac\pi{2n}}{n^2}=\lim_{n\to0^+}n\cot{\frac{\pi n}2}=\lim_{n\to0^+}\frac{n\cos{\frac{\pi n}2}}{\sin{\frac{\pi n}2}}=\frac2\pi$$ So that the ...
0
votes
1answer
93 views

A,B,P are three points on a circle having centre O. If angle OAP=25 and angle OBP=35 , then the measure of angle AOB is???

A,B,P are three points on a circle having centre O. If angle OAP=25 and angle OBP=35 , then the measure of angle AOB is???
1
vote
1answer
173 views

Limits involving theta, cos, sin

Question: $$\displaystyle\lim_{\theta \to 0^-} θ^3 \cos\left(\frac 2\theta \right)$$ also $$\displaystyle\lim_{\theta \to 0^+} θ^3 \cos\left(\frac 2\theta \right)$$ I have no idea where to begin ...
1
vote
2answers
165 views

Bound for $\left|\sin(x) +\cos(x)\right|$

I'm taking a numerical analysis class and i'm needing to bound $\left|\sin(x) + \cos(x)\right|$ quite often. So far i've been putting that this is always $\leq |1 + 1| = 2$. Is this the minimal bound? ...
2
votes
4answers
119 views

When is $\cos (x) \geq \frac{1}{2}$?

When is $\cos (x) \geq \frac{1}{2}$? I know the function repeats, so I know I should end up with an interval that allows for integer multiples. e.g. something like this (but obviously not this ...
1
vote
0answers
91 views

How to find the period of a exponential function? $5\cdot(-1)^k$

Currently in class we are learning Euler's formula and how it relates to imaginary numbers, exponentials, and trig functions. I know the left hand of the equation is $e^{jwt}$ and I could easily use ...
2
votes
3answers
57 views

How was this approximation of transcendent equation solution found?

I have an equation for $\xi$: $$\xi\gamma=\cos\xi,$$ where $\gamma\gg1$. I've tried solving it assuming that $\xi\approx0$ and approximating $\cos$ by Taylor's second order formula: ...
2
votes
2answers
58 views

Limit with trigonometric function

I have this limit, I have resolved it until a part but I'm stucked now. $$\lim_{x \to \frac{\pi}{4}} \frac{\tan^2(x)-1}{\cos(x)-\sin(x)}$$ $$ \lim_{x \to \frac{\pi}{4}} ...