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
votes
1answer
48 views

Rotation of complex numbers in a complex plane. Check my work?

Say that $c_1 = -i$ and $c_2 = 3$. For this problem, let $z_0$ be an arbitrary complex number. We can rotate $z_0$ around $c_1$ by $\pi/4$ counterclockwise to get $z_1$. Next, we canrotate $z_1$ ...
1
vote
1answer
75 views

Prove that $(n-1) \sum_1^n \cot(\theta_i) \leq \sum_1^n \tan(\theta_i) $

n is a positive integer and $\theta_i$ is such that $ 0^\circ \leq \theta_i \leq 90^\circ $ for all positive integers $i \leq n$ and $\sum_1^n \cos^2(\theta_i) = 1$. Prove that $(n-1) \sum_1^n ...
0
votes
2answers
19 views

calculating rotation direction between two angles

Consider the following scenario: Say I have a robot positioned at (0,0) and his current angle is 70. I need an algorithm that given two angles - the current angle and the target angle, will give ...
4
votes
0answers
59 views

If $x_1, x_2,…,x_{10}$ are such that $\sum_{i=1}^{10} \sin^2(x_i) = 1$, prove that $3 \sum_{i=1}^{10} \sin(x_i) \leq \sum_{i=1}^{10} \cos(x_i)$ [duplicate]

Take $x_1, x_2,...,x_{10}$ such that $\sum_{i=1}^{10} \sin^2(x_i) = 1$ with $x_1, x_2,...,x_{10}$ on $\left[0,\frac{\pi}{2}\right]$, prove that $3 \sum_{i=1}^{10} \sin(x_i) \leq \sum_{i=1}^{10} ...
5
votes
4answers
70 views

Find $\frac{d^2y}{dx^2}$ as a function of $x$ if $\sin y+\cos y=x$

Find $\frac{d^2y}{dx^2}$ as a function of $x$ if $\sin y+\cos y=x$ Ok bit confused as my textbook gives the answer to this problem as: $$\frac{d^2y}{dx^2}=\pm\frac{x}{\sqrt{(2-x^2)^3}}$$ So I ...
1
vote
1answer
59 views

Find $a$ when $\sin a= \cfrac{3}{5}$ where $0<a<\cfrac{\pi}{2}$ without a calculator

I have been trying to find $a$ when $\sin a= \cfrac{3}{5}$ where $0<a<\cfrac{\pi}{2}$ by using exact values but I can't seem to find a particular method to evaluate it. My original question is ...
2
votes
0answers
47 views

Prove that a complicated parametric equation is a portion of a circle

In a nuthshell, I would like to prove that some complicated parametric equation is that of a portion of a circle. Let $ P = \left[ \begin{array}{ccc}X \\Y \\Z \end{array} \right] $ be a 3D point ...
4
votes
5answers
106 views

How to solve $\sin78^\circ-\sin66^\circ-\sin42^\circ+\sin6^\circ$

Question: $ \sin78^\circ-\sin66^\circ-\sin42^\circ+\sin6° $ I have partially solved this:- $$ \sin78^\circ-\sin42^\circ +\sin6^\circ-\sin66^\circ $$ $$ ...
0
votes
1answer
33 views

Trouble with series question from STEP past paper

I have answered all parts of this question but the last part. By using the identity, $\cot x - \tan x = 2\cot 2x$ ...
7
votes
2answers
44 views

Find the values of $\cos(\alpha+\beta) $ if the roots of an equation are given in terms of tan

It is given that $ \tan\frac{\alpha}{2} $ and $ \tan\frac{\beta}{2} $ are the zeroes of the equation $ 8x^2-26x+15=0$ then find the value of $\cos(\alpha+\beta$). I attempted to solve this but I ...
2
votes
2answers
68 views

Trig functions of complex numbers

I was studying complex numbers with the help of Boas textbook. I came about certain problems, which I solved only to find that the answers provided in the solution manual to be different. ...
1
vote
1answer
77 views

Inequality problem: $\tanh(\pi x)\sin(\pi x)\geq x^2(1-x^3)$ on $[0,1]$.

How to show that $$\tanh(\pi x)\sin(\pi x)\geq x^2(1-x^3)$$ on $[0,1]$? I tried to expand $\tanh(\pi x)\sin(\pi x)$ in a Taylor expansion: $$\tanh(\pi x)\sin(\pi x)=\pi^2 x^2 - \frac{\pi^4 ...
2
votes
3answers
90 views

Integrating using half angle formula

I am reading through my textbook and there is a part of the solution to an example that I do not understand... $$\int\sin^4x\cos^2x\,dx = \int(\sin^2x)^2\cos^2x\,dx$$ ...
0
votes
2answers
44 views

Find value of $t$ between the difference of 3D vectors.

Hint: The distance between $2$ vectors equals the magnitude of their difference. What is the value of $t$ for which the vector $\mathbf v = \begin{pmatrix} 2 \\ -3 \\ -3 \end{pmatrix} + ...
6
votes
3answers
95 views

Integrating $\frac{\sec^2\theta}{1+\tan^2\theta \cos^2(2\alpha)}$ with respect to $\theta$

I'm having some issues with the following integral $$\int_{\frac{-\pi}{2}}^\frac{\pi}{2}\frac{\sec^2\theta}{1+\tan^2\theta \cos^2(2\alpha)}d\theta$$ My attempt is as follows, substitute ...
5
votes
5answers
54 views

Length of hypotenuse v/s change in height of the opposite

I have always struggled to understand mathematical concepts, and have a very different way of thinking about problems. I suspect this is a very simple problem, but its confusing me a great deal. I ...
1
vote
2answers
43 views

Given that $\tan x=\sum_{i=0}^{\infty}a_nx^n$, Show that $a_n=0$, for even n

Given that $\tan x=\sum_{i=0}^{\infty}a_nx^n$, Show that $a_n=0$, for even n. from the series expansions of $\sin x$ and $\cos x$, I get that $\tan ...
1
vote
1answer
30 views

Solving $\cos(2x+\frac{\pi}{4})= -1/2 $

My suggestion: $$\cos\left(2x+\frac{\pi}{4}\right)= -\frac{1}{2}$$ $$ 2x+\frac{\pi}{4} = \frac{2\pi}{3} \pm 2\pi n, n\in\mathbb{Z}$$ $$ x= \frac{\left( \frac{2\pi}{3} - \frac{\pi}{4} \right)}{2} \pm ...
-2
votes
0answers
44 views

Overlap between two angles

Imagine that we have $4$ line segments in a plane. The starting point of all segments is the same and end point of them could be any arbitrary value . We call the angle between the segments $1,2 ...
2
votes
2answers
69 views

Trigonometric Integrals $\int \frac{1}{1+\sin^2(x)}\mathrm{d}x$ and $\int \frac{1-\tan(x)}{1+\tan(x)} \mathrm{d}x$

Any idea of calculating this two integrals $\int \frac{1}{1+\sin^2(x)}\,dx$ and $\int \frac{1-\tan(x)}{1+\tan(x)} \mathrm{d}x$? I found a solution online for the first one but it requires complex ...
0
votes
3answers
216 views

Trigonometry sine and cos problem

Knowing that $$ \sin a - \sin b = \frac{1}{2} \quad\quad\text{and}\quad\quad \cos a + \cos b = \frac{3}{2} $$ calculate $\cos (a+b)$. I have tried various methods but I can't seem to get ...
3
votes
7answers
84 views

Prove that $\cos^2(\theta) + \cos^2(\theta +120^{\circ}) + \cos^2(\theta-120^{\circ})=3/2$

Prove that $$\cos^2(\theta) + \cos^2(\theta +120^{\circ}) + \cos^2(\theta-120^{\circ})=\frac{3}{2}$$ I thought of rewriting $$\cos^2(\theta +120^{\circ}) + \cos^2(\theta-120^{\circ})$$ as ...
3
votes
4answers
278 views

Integration of $\frac{\sin x}{\sin 4x}$

Question: Solve the following integral: $$\int \frac{\sin x}{\sin4x}dx$$ Attempt: Using trigonometric identities to expand $\sin4x$, I obtained the integral: $$\int \frac{1}{4\cos x \cos2x}dx$$ ...
0
votes
0answers
26 views

Get the coordinate offsets with known rotation angles (i.e. Yaw, Pitch, Roll)

I'm working on correcting an tilted object to its vertically placed position. Below is my drawing illustrating my situation: http://i.stack.imgur.com/0XotT.png Assuming: I have a stick stood on a ...
4
votes
3answers
182 views

The maximum and minimum values of the expression

Here is the question:find the difference between maximum and minimum values of $u^2$ where $$u=\sqrt{a^2\cos^2x+b^2\sin^2x} + \sqrt{a^2\sin^2x+b^2\cos^2x}$$ My try:I have just normally squared the ...
-1
votes
2answers
61 views

TRIG: What is the solution set of $\frac{1}{2}\ 4^{\sin^2(x)}=2^{\sin(x)}$ with the interval $[0, 2\pi]$? [closed]

My brother is currently in college-level Trig and sent me this problem: (P.S. -- Sorry about the awkward formatting, the parenthesis encase powers, not multipliers) "Find all real numbers in the ...
0
votes
1answer
22 views

Projection of a vector's reflection. Find the value of the matrix.

For the vector v, Let r be the reflection of v in the line x $= t \begin{pmatrix} 2 \\ -1 \end{pmatrix}$. There exists a $2 \times 2$ matrix R such that r = R v for all 2D vectors v. Find R. -- ...
0
votes
3answers
34 views

find limit points for the given trig function

$$-5 \sin\left(\frac{n\pi}{3}+\frac{1}{n}\right); \;n\in \mathbb{N}$$ The limit points are given as $$\pm 5(\sqrt{3}/2) , 0$$ I don't understand how these values come up. If anyone has a clue ...
0
votes
2answers
39 views

Indentifying $\sin(mx) = 2\cos(x)\sin\left[(m-1)x\right] - \sin\left[(m-2)x\right]$

I encountered in a work of Joseph Fourier's the identity: $$\sin(mx) = 2\cos(x)\sin\left[(m-1)x\right] - \sin\left[(m-2)x\right]$$ which holds for all real $m$ and $x$. I had trouble, however, ...
-2
votes
0answers
22 views

What is the relationship of these numbers?

I have two problems that closely relate to each other. I am working with angles. When the angle of Y is 90 degrees the answer to the first problem 360 degrees, while answer to the second is 180 ...
2
votes
3answers
84 views

General Solution of $\sin\theta=3\cos\theta$

I'm a high school level maths student currently working through some exercises for the general solution of trigonometric equations and have come across this one that I am stuck on. Any hints would be ...
0
votes
1answer
20 views

Remove scale transformation from a complex transform matrix 4x4?

My common task is I have a rect with coordinates of its $2$ points: $(x, y, z), (x + a, y + b, z)$. I applied a $4\times4$ transform matrix to it and it became a quadrilateral. Now for some reasons I ...
-1
votes
3answers
199 views

Solve a trigonometric equation ($\sin(β/2)=β/4$) [closed]

How can we solve the equation: $$\sin(\beta/2)=\beta/4$$ where $\beta $ is in radians?
8
votes
10answers
156 views

Find $\lim_{x\to \frac\pi2}\frac{\tan2x}{x-\frac\pi2}$ without l'hopital's rule.

I'm required to find $$\lim_{x\to\frac\pi2}\frac{\tan2x}{x-\frac\pi2}$$ without l'hopital's rule. Identity of $\tan2x$ has not worked. Kindly help.
0
votes
2answers
27 views

Find point(s) of intersection between a line and a circle whose radius is parameterized by the same variable as the line

Let's assume we have a line: $$\begin{align} x&: x_0 + v_xt, \\ y&: y_0 + v_yt \end{align}$$ and a circle $$\begin{align} x&: X_0 + kt\cos(s), \\ y&: Y_0 + kt\sin(s).\end{align}$$ ...
0
votes
3answers
40 views

eliminating variable from a pair of trig relations

If $$\operatorname{cosec} A - \sin A=m$$ and $$ \sec A - \cos A=n,$$ please show how to eliminate $A$; I have tried that and it came: $$\sin A \cdot \cos A = m\cdot n$$
3
votes
3answers
64 views

Solving for trigonometric values

Find $a+2b$ if $\tan a = 1/7$ and $\sin b=\frac{1}{\sqrt{10}}$. I had tried to solve it by trigonometric ratios but i could not. Please solve it by a method of class10 standards.
0
votes
1answer
47 views

Connection between the expression $4 \sin x + 3 \cos x$ and the equation $\cos 3x = \cos 2x$?

Express $4\sin x+3\cos x$ in the form $r\sin(x+\alpha)$. Hence find all the values of $x$ in the range $0\leq x \leq 360^{\circ}$ for which $\cos 3x=\cos 2x$. Okay so I managed to put the ...
3
votes
2answers
66 views

How do you find the value of $f(x)$ for this trig function satisfying all values of $x$?

If $ f(x) = 3[\sin^4(\frac{3\pi}{2} - x) + \sin^4(3\pi+x)] -2[\sin^6(\frac{\pi}{2} + x) + \sin^6(5\pi-x)] $ then, for all permissible values of $x$, $f(x)$ is:- Here's how I attempted it- $ f(x) = ...
4
votes
3answers
133 views

Evaluating a Summation with a binomial

Problem: Evaluate for $n=11$$$\begin{align} \sin^{4n}\left(\frac{\pi}{4n}\right) + \cos^{4n}\left( \frac{\pi}{4n}\right) = \frac{1}{4^{2n-1}} \left[ \sum_{r=0}^{n-1} \binom{4n}{2r} \cos\left(1 - ...
0
votes
1answer
33 views

Speed of light moving on a wall.

While studying, I came upon this word problem: "A police car is 20 feet away from a long straight wall. Its beacon, rotating 1 revolution per second, shines a beam of light on the wall. How fast is ...
0
votes
2answers
44 views

Number of solutions of a trigonometric equation involving sine

How to prove that $$\sqrt {2}\sin (\sqrt {2}x)=\sin (-x) $$ has more than one solution (at $x=0$) ?
6
votes
4answers
89 views

Find $\lim_{x \to 0}\frac{\cos 2x-1}{\cos x-1}$ without L'Hopital's rule.

$$\lim_{x \to 0}\frac{\cos 2x-1}{\cos x-1}$$ I have found the above limit using L'Hopital's rule but since this rule is not given in the book so I'm supposed to do it without using this rule. I know ...
0
votes
3answers
48 views

Find $LK_1^2 + LK_2^2 + \dots + LK_{11}^2$. [closed]

$K_1 K_2 \dotsb K_{11}$ is a regular $11$-gon inscribed in a circle, which has a radius of $2$. Let $L$ be a point, where the distance from $L$ to the circle's center is $3$. Find $LK_1^2 + LK_2^2 + ...
1
vote
1answer
42 views

Trigometry Equation Solving

How would one solve an equation of the form $\sin(ax) = \cos(bx)$ on the interval $[0, 2\pi]$? I understand that there are multiple solutions, however I only know how to arrive at the first one by ...
2
votes
1answer
97 views

How to prove $\cos{\frac{\pi}{11}}$ is a root of

I want to show that $x=\cos{\frac{\pi}{11}}$ is a solution of equation : $$8x^2-4x+\frac{1}{x}-4=4\sqrt{\frac{1-x}{2}}$$ Thanks in advance.
1
vote
3answers
44 views

Given that $2\cos(x + 50) = \sin(x + 40)$ show that $\tan x = \frac{1}{3}\tan 40$

Given that: $$ 2\cos(x + 50) = \sin(x + 40) $$ Show, without using a calculator, that: $$ \tan x = \frac{1}{3}\tan 40 $$ I've got the majority of it: $$ 2\cos x\cos50-2\sin x\sin50=\sin ...
4
votes
3answers
112 views

How to show $ \Big\vert \frac{\sin(x)}{x} \Big\vert $ is bounded by $1$?

This may be a silly question, but I cannot figure it out. I want to prove that $ \Big\vert \frac{\sin(x)}{x} \Big\vert \leq 1 $ for $x\in[-1,0)\cup(0,1]$, but I don't even know where to start.
1
vote
2answers
40 views

Specific resources to self-learn algebra

I'm attempting to self-learn algebra/trig from the ground up and looking for good resources that will help. Currently my goal is to learn enough to be able to take: MIT OCW Sing Var Calc MIT OCW ...
0
votes
2answers
50 views

trigonometry solution [duplicate]

$$\frac{\sin A - \cos A}{\sin A+\cos A}=\frac{X}{Y}$$ Then prove that $X^2 + Y^2 = 2$. Answer : $\frac{\sin A-\cos A}{\sin A+\cos A}=\frac{x}{y}$ $\implies x=k(\sin A-\cos A)$ and $y=k(\sin ...