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

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4
votes
1answer
139 views

Solve for sin(x)+sin(2x)+sin(3x)+…+sin(nx)=x

The question is $\sin (x)+\sin(2x)+.....+\sin(nx)=x$ where $n$ is any natural number. I used de Moivre formula to obtain the sum of $\sin(x)+\sin(2x)+...+\sin(nx)$, and I differentiated it to get $$ ...
0
votes
0answers
16 views

Workout line segment inside expanding circle

I have what is probably a fairly basic math problem for a game I'm creating. On each frame I need to work out how much a sub segment of a line passing though a circle will expand when the circle ...
0
votes
1answer
29 views

Simple trig question.

There exists an isosceles triangle. The sides are 6, 6, and 8. I am to find what all of the angles equal. My method to do this was to split the triangle in half giving me a right triangle where side ...
-1
votes
2answers
44 views

Exact values of $x$ for $2\cos^2x=1+\sin x$ [duplicate]

This question involves finding the exact values of $x$ such that $0 \leqslant x \leqslant 2\pi$. So far I have subtracted everything to the left side of the equation and then used the pythagorean ...
0
votes
3answers
31 views

Determine the exact value of equations involving more two trig variables

$2\cos^2x=1+\sin x$. Determine the exact values of $x$ such that $0 \leq x \leq 2\pi$. I am experiencing problems with factoring this question. First I started by getting everything on to the ...
4
votes
2answers
131 views

Is there a way to simplify a sum of cosecants?

A problem I have been working on recently results in a sum of cosecant terms. Specifically, $f(n) = \sum_{k=1}^n \csc \frac{\pi k}{2n+1}$ $g(n) = \sum_{k=1}^n [(-1)^{k+1}(\csc \frac{\pi k}{2n+1})]$ ...
1
vote
1answer
36 views

Determining the exact value of trigonometric functions using tan

Use the special triangles to give exact solutions where possible. Find all values of x such that $0 \le x \le 2\pi$. The question I have is $\tan^2x=1$. What I have done so far (it appears that ...
3
votes
1answer
22 views

Special Triangles and their related acute angles

So I've been working on some questions involving having to find the exact value of trig. functions involving a particular interval. I have worked through the question but now I have something I am ...
1
vote
1answer
34 views

$ \sum_{m=1}^{6}\frac{1}{\sin \left\{\theta+\left(m-1\right)\cdot \frac{\pi}{4}\right\} \sin \left\{\theta+m\cdot \frac{\pi}{4}\right\}} = 4\sqrt{2}$

If $\displaystyle 0 < \theta < \frac{\pi}{2}$ and $\displaystyle \sum_{m=1}^{6}\frac{1}{\sin \left\{\theta+\left(m-1\right)\cdot \frac{\pi}{4}\right\}\cdot \sin \left\{\theta+m\cdot ...
0
votes
1answer
27 views

Find all values of $x$, linear and quadratic functions

Use special triangles to give exact solutions where possible. Find all values of $x$ such that $0 \leq x \leq 2 \pi$. 1) $\cos^2 x + \cos x - 1 = 0$ For this question, I have factored in which the ...
1
vote
1answer
43 views

Why is the principal value of $\tan^{−1}θ$ always in $(\frac{\pi}{2}, \frac{\pi}{2})$? And, why is the principal value of $\arg z$ always in $(−π,π]$?

Why is the principal value of $\tan^{−1}θ$ always in $(\frac{\pi}{2}, \frac{\pi}{2})$? Also, why is the principal value of $\arg z$ always in the interval $(−π,π]$? Is this a convention? Why these ...
1
vote
2answers
16 views

Merge Velocities of Two Aircraft

INTRO Scroll down to skip to question with all data provided. Picture of aircraft at end of post. Simple question here guys if y'all are willing to help. I'm looking for the formulas and possible a ...
1
vote
2answers
35 views

Normal Vector of plane for Rotation

I reading a code where Normal vector to a plane is given. then a,b,c are taken (what I guess is direction ratio values). ...
0
votes
3answers
74 views

Prove $\sin^4\theta+\cos^4 \theta= 1-2\sin^2 \theta cos^2 \theta $

Is it possible to prove: $$\sin^4\theta+\cos^4 \theta= 1-2\sin^2 \theta cos^2 \theta ?$$
-1
votes
4answers
53 views

Solving Trigonometry Equation with powers

I am having difficulty understanding how to solve $3\tan^2{\theta} - 1 = 0$ Putting $ t = \tan\theta $, I get $ t = \pm \frac{1}{\sqrt3} $. Is that done ? How to find angle $\theta$ now? Any help ...
0
votes
5answers
36 views

Find radius of two concentric arcs

Consider arcs AB and A'B' in this diagram: Given the lengths of the arcs and the radial distance between them, how would you find their radii?
0
votes
1answer
24 views

If $|\sin^2 x+ 17 - x^2|=|16-x^2|+2 \sin ^2 x + \cos ^2x$ then $x$ lies in what interval

If $|\sin^2 x+ 17 - x^2|=|16-x^2|+2 \sin ^2 x + \cos ^2x~$ then $x$ lies in what interval? Hints Please? I dont know how to approach this problem.
-1
votes
1answer
53 views

Tough Trigonometric Identity Problem

I am having trouble solving a trigonometry identity problem. Specifically, I need to solve: $\dfrac{\sec(x)\sin(x)}{\tan(x)\cot(x)} = \sin^2x$ I tried solving this by turning $\sin^2x$ into it's ...
-1
votes
3answers
46 views

Verify this identity: $1-\frac{\sin^2x}{1−\cos x}=\cos x$ [closed]

This doesn't look like a though one but still i'm having a rough time with it, i guess i'm just doing something wrong but i can't find what i'm doing wrong. I got as close as $1-\cos x$ but i can't ...
0
votes
0answers
6 views

Solutions of $\sum_{n=1}^N a_n n\sin{(n x+\theta_n)}=\sum_{n=1}^N a_n n^2\cos{(n x+\theta_n)}=0$

Is there a solution for the equation $\sum_{n=1}^N a_n n\sin{(n x+\theta_n)}=\sum_{n=1}^N a_n n^2\cos{(n x+\theta_n)}=0$ in terms of the variable $x$, for some choice of coefficients $a_n$ and ...
0
votes
2answers
32 views

Find $\alpha$ from trigonometric equation…

I have this equation I don't know how to get quadriatic equation with tangent... $\ 1.77 = 4 \tan[\alpha] - 0.7848/\cos[\alpha]^2$ I should get this $\ 157 \tan[\alpha]^2- 800\tan[\alpha]+511= 0$
1
vote
1answer
34 views

How to calculate true lengths from perspective projection?

Suppose that I have a single point perspective drawing like . and suppose also that I know some of the real horizontal distances and distances along lines converging to vanishing point. E.g if i know ...
1
vote
2answers
29 views

Modular Arithmetic with Sines

Given $$\sin(10^{100})+\sin(n)=0$$ find $n$. I wrote so far that $$\sin(10^{100})=\sin(10^{100} \mod 360)$$ and I noticed that $10^3 \mod 360=280$ and $10^4 \mod 360=280$ so I (correctly) assumed ...
0
votes
1answer
22 views

How to proof $\arcsin(\sqrt{2*b*(\sqrt{1 + b^2} - b})) = \arccos(\sqrt{1 + b^2} - b)$

In a publication [1, p. 89] this equality is stated. Unfortunately, I seem unable to proof it myself $\arcsin(\sqrt{2*b*(\sqrt{1 + b^2} - b})) = \arccos(\sqrt{1 + b^2} - b)$ The original formulation ...
3
votes
2answers
75 views

How to evaluate $\left(\cos{\frac{5\pi}{9}}\right)^{11}+\left(\cos{\frac{7\pi}{9}}\right)^{11}+\left(\cos{\frac{11\pi}{9}}\right)^{11}$

How to evaluate $$\left(\cos{\frac{5\pi}{9}}\right)^{11}+\left(\cos{\frac{7\pi}{9}}\right)^{11}+\left(\cos{\frac{11\pi}{9}}\right)^{11}?$$ I found the problem on this page.
2
votes
1answer
68 views

Lowering powers of $\cos^2x \sin^4x$

First, I will be straight up, this is a homework question. I need to write $\cos^2 x \sin^4 x$ in terms of cosine to the first power. I know that $\sin^4x$ = $$ \frac{3-4\cos 2x+\cos 4x}{8}$$ from ...
0
votes
5answers
93 views

Range of $\cos(2 \sin x)$ [closed]

Actually, I am confused determining the range of the function given below. Could anyone tell me what the range of $ f(x) = \cos (2 \sin x)$ is ?
1
vote
2answers
47 views

Problem when $x=\cos (a) +i\sin(a),\ y=\cos (b) +i\sin(b),\ z=\cos (c) +i\sin(c),\ x+y+z=0$

Problem : If $$x=\cos (a) +i\sin(a),\ y=\cos (b) +i\sin(b),\ z=\cos (c) +i\sin(c),\ x+y+z=0$$ then which of the following can be true: 1)$\cos 3a + \cos 3b + \cos 3c = 3 \cos (a+b+c)$ 2)$1+\cos ...
0
votes
0answers
34 views

What is the expansion of $(a\sin A +b\sin B)^n$ in terms of $\sin$ or $\cos$ by power of zero and one?

What is the expansion of: $(a\sin A+b\sin B)^n$ in terms of $\sin$ or $\cos$ by power of zero or one? For example the second expansion is: $(a\sin A+b\sin B)^2=$ $a^2+b^2+$ ...
6
votes
3answers
51 views

Using the Mean Value Theorem, prove that $|\sin{a} - \sin{b}| \leq |a - b|$ $\forall a, b \in \mathbb{R}$

Using the Mean Value Theorem, prove that $|\sin{a} - \sin{b}| \leq |a - b|$ $\forall a, b \in \mathbb{R}$. I'm working towards figuring out an approach for finding that $|\sin{a} - \sin{b}| \leq ...
0
votes
2answers
49 views

Sum of the area of infinite similar equilateral triangles

How would I solve for the side depicted in the picture?
0
votes
2answers
43 views

Is the following trigonometric identity true?

I can't prove the following trigonometry identity: $$\tan(2x)\tan(30^\circ-x)+\tan(2x)\tan(60^\circ-x)+\tan(60^\circ-x)\tan(30^\circ-x)=1$$ Is it true and if yes how should I proceed?
0
votes
0answers
22 views

Finding a closed form for $\sum_{k=1}^n \sin(k)$ [duplicate]

I'm trying to prove by induction that $$\sin1+\cdots+\sin(n)=\frac{\sin\left(\frac{n+1}{2}\right)\sin\frac{n}{2}}{\sin\frac{1}{2}}$$ My only thought so far is to use the identity $\sin x+\sin ...
-1
votes
1answer
20 views

How to shift sum of cosines and constants by 90 degrees

Suppose that $f(t) = a+ cos(\omega_1 t) + cos(\omega_2 t) +...$, where every $\omega_i$ is multiples of $\omega_1$ and $a \in \mathbb{R}$. We want to shift this sum by 90 degrees to right. Without ...
1
vote
1answer
35 views

evaluation of some integral

I'm looking for a direct proof of the following identity: $$\frac{1}{\pi}\int_1^x\frac{dt}{t\sqrt{t-1}}\arcsin\left(\sqrt{\frac{x-t}{y-t}}\right)=\arcsin\sqrt{\frac{x}{y}}-\arcsin\sqrt{\frac{1}{y}},$$ ...
-1
votes
0answers
13 views

where will all cosines with frequency multiples of harmonic frequency be zero?

Let every cosine be of form $x_i(t) = a_i\cos(2\pi f_i t)$ with $a,t \in \mathbb{R}$ and $f$ is labelled as frequency. Suppose we have cosines with frequencies $f_1,f_2,..,f_n$ and every frequency is ...
1
vote
3answers
33 views

Alternative form of a Trigonometrics Expression

Express $8\sin\theta \cos\theta - 6 \sin^2 \theta$ in the form $R \sin(2\theta + \alpha) + k$ Edit: I am sorry, I thought it was a somewhat interesting question. I shall let you know of the progress ...
1
vote
2answers
61 views

How to show whether this limit exists or not?

It's my intuition that $$\lim_{x\to+\infty} \frac{\sin(x+\frac1x)}{\sin(x)}$$ does not exist. And I have been working on proving it. I have tried Heine's Theorem but for this moment I get stuck and ...
1
vote
3answers
43 views

Limit of two variable

Suppose that I have equation: $$\tan(a) = \dfrac{b-c}{bc + 1},\;\text{ where }\,a, b, c\, \text{ are variables.}$$ How can I show that, as $a \to \pi/2, \; bc+1\to 0\;$ mathematically, not ...
0
votes
3answers
43 views

Differentiability of $f(x)=sin(x)/x$ if $x\ne0$ and $1$ if $x=0$

I am trying to see if $$f(x)= \begin{cases} \frac{\sin(x)}x &\text{ if x}\neq0\\ 1 &\text{ if x}=0. \end{cases} $$ is differentiable more than once. This is what I did: $$f'(0)= \begin{cases} ...
0
votes
1answer
30 views

noob question on arithmetic with proving identities

how does $$\sin^2x-\cos^2x+\cos^4x$$ simplify to $$\sin^2x\times\sin^2x=\sin^4x \,\,\,\,\ ?$$ I would appreciate how the steps are done to arrive to the final answer. Thanks!
0
votes
1answer
36 views

Determining Exact Values of Trignometric Equations

Use the special triangles to give exact solutions where possible. Find all values of $x$ such that $0\le x \le 2\pi$ . (a) $\tan^2 x=1$ $\,$ (b)$\, \, 2\cos x + \sqrt{3}=0 \, \,$ (c) $\, \, ...
2
votes
1answer
40 views

Construct triangle from three points on base and difference in distances to third vertex

Imagine such a triangle: We know the differences in distances: $\overline{OA} - \overline{BO}$ and $\overline{CO} - \overline{BO}$, as well as the distances between the points on the base: ...
2
votes
2answers
43 views

Proving an identity, cos and sin, two variables

$$\frac{\cos(2x)+\cos(2y)}{\sin(x)+\cos(y)} = 2\cos(y)-2\sin(x)$$ The question asks to prove the identity. I tried simplifying the first half, thought maybe I could expand and simplify with the ...
0
votes
1answer
29 views

Finding intervals using local min and max (in interval notation form)

I am having some trouble with the following question: Find the critical points of the function and use the First Derivative Test to determine whether the critical point is a local minimum or maximum ...
1
vote
0answers
39 views

Hints to find analytical solution to integral

I have to evaluate the expression $$f(|\vec{c}|) = \int_0^\infty \int_0^{2\pi} (z(\vec{a})+z(\vec{a}+\vec{c})) \frac{(1-\cos(\theta_{\vec{a}+\vec{c}} - ...
1
vote
1answer
46 views

Is there any meaningful way in which the tangent function relate to e?

Is there any meaningful value for which $\tan(x)$ relates to $e$? What I mean to say is $\tan(1.21828\ldots)=e$, but is there any significance to this number? i.e. could it be expressed as some ...
0
votes
1answer
47 views

Are these trigonometry problems correct?

I have to prove the following trigonometric identities. However, since I can't prove them, I am starting to think they are not correctly stated. Problem 1: ...
4
votes
2answers
67 views

Integral of $\big((1+\cos(x))\sin(x)\big)^2$

What is $$\int \big((1+\cos(x))\sin(x)\big)^2dx$$ ?
2
votes
2answers
47 views

$\sin(\pi - a) = \sin (a)$. How/why? [closed]

Can someone please explain how and why $\sin(\pi - a) = \sin (a)$? The handout from class doesn't really explain it. I tried asking the teach but there's a bit of a language barrier and I'm not ...