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

learn more… | top users | synonyms (1)

2
votes
2answers
39 views

Deriving the cosine formula using vectors?

How did they go from 2b⋅c to -2bccosA? Where did they get the negative sign from?
1
vote
2answers
81 views

Integrating an equation with sin squared on top and cos plus constant on bottom

I am trying to integrate an equation of the form \begin{equation} \int_0^\pi \frac{\sin^2(x)}{\cos(x)+C}dx \end{equation} I can't think of a way to do it and Mathematica tells me that it is ...
2
votes
1answer
72 views

Calculating certain functions if only certain buttons on a calculator are permitted

A calculator is broken. The only keys that work are $\sin, \cos, \tan, \cot, \arcsin, \arccos$, and $\arctan$ buttons. The original display is $0$. In this problem, we will prove that ...
6
votes
1answer
221 views

If such $1+4\sin{10^\circ}=a+b\sin{c^\circ}$ How find this integer $a,b$

Interesting problem: Assmue that: $$1+4\sin{10^\circ}=a+b\sin{c^\circ}$$ where $a>0$,and $a,b,c$ are integers and $0<c<90^\circ$, show that $$a=1,b= 4,c=10$$ is unique solution ...
0
votes
1answer
34 views

Can $a \cos A + b \cos B + c \cos C$ equal $4 \sin A \sin B \sin C$?

This is a follow-up to my previous question about an identity with the sides and angles of a triangle. Can $a \cos A + b \cos B + c \cos C$ equal $4R \sin A \sin B \sin C$? I'm not sure if ...
2
votes
1answer
65 views

Prove $ \frac{a\sin A+b\sin B+c\sin C}{a\cos A+b\cos B+c\cos C}=R\left(\frac{a^2+b^2+c^2}{abc}\right) $

$ \frac{a\sin A+b\sin B+c\sin C}{a\cos A+b\cos B+c\cos C}=R\left(\frac{a^2+b^2+c^2}{abc}\right) $ This is what I have so far: I know that $A + B + C = 180^\circ$, so $C = 180^\circ - (A+B)$. ...
0
votes
5answers
43 views

Limit of sequence of sum of sinuses: $\lim \limits_{x \to \infty}{(\sin(1+x) - \sin(x))}$

How to find the limit $\lim \limits_{x \to \infty}{(\sin(1+x) - \sin(x))}$? I don't even know how to begin.
3
votes
1answer
30 views

How can I improve my explanation that the ratio $\theta=\frac{s}{r}$ that defines the radian measure holds for all circles?

I'm trying to demonstrate why the ratio $\theta=\frac{s}{r}$ (where $s$ is an arc measuring some $s$-units in length and $r$ is the radius of the circle) which defines the radian measure holds for all ...
1
vote
3answers
37 views

Finding Cartesian coordinates of remaining vertices of triangle, given a vertex and angle from y-axis

I have an isosceles triangle ABC, where the height h and angle at vertex A are known. The Cartesian coordinates of vertex A are also known to be (x,y). If the angle between the y-axis and the line ...
1
vote
2answers
44 views

Area of the overlap between a triangle and a square [closed]

$ABC$ is an equilateral triangle, each side has length 4. $M$ is the midpoint of $\overline{BC}$, and $\overline{AM}$ is a diagonal of square $ALMN$. Find the area of the region common to both ...
1
vote
1answer
64 views

How prove this idenity this $mv-3nu=m-3u$ with unit circle

Assmue the $m,n,u,v$ be real numbers,and such $$m^2+n^2=1,u^2+v^2=1,nv>0,m>0,u>0$$ and $$5mu=3(1-nv)$$ show that $$mv-3nu=m-3u$$ Following is My methods: let ...
1
vote
2answers
33 views

Find the tan A if the triangle is inside the square?

ABCD is a square. The problem asks for me to find $\tan(\angle QAP)$ if I am given the fact that $CP = CQ = \frac{AB}{4}$. This is what I have so far: I drew a line from $Q$ to $P$ to make ...
1
vote
1answer
42 views

Check my work on a problem involving Law of Cosines?

The problem is this: Jane walks North for 3 miles, then turns $45^\circ$ to the right. After that, she walks another 4 miles. How many miles will Jane be from her starting point? Give your answer ...
0
votes
2answers
68 views

Prove that $\frac{1}{90} \sum_{n=1}^{90} 2n \cdot \sin((2n)^\circ) = \cot (1^{\circ})$

Show that $$\frac{(2\sin(2^\circ)) + (4\sin(4^\circ))+ (6\sin(6^\circ)) + \ldots +(180\sin(180^\circ))}{90} = \cot(1^\circ).$$ I used a lot of steps, and typing it all down on here would take me an ...
1
vote
1answer
80 views

simplify cos 1 degree + cos 3 degree +…+cos 43 degree?

I am currently working on a problem and reduced part of the equations down to $\cos(1^\circ)+\cos(3^\circ)+.....+\cos(39^\circ)+\cos(41^\circ)+\cos(43^\circ)$ How can I calculate this easily using ...
1
vote
1answer
27 views

check my work on this problem: given tan(2x), find sin x + cos x?

$\tan 2x = - 24/7$ $90^\circ < x < 180^\circ$. Find the value of $\sin x+\cos x$. What I have so far: $\tan(2x) = -\frac{24}{7} \Rightarrow \frac{2\tan(x)}{1-\tan^{2}x} = -\frac{24}{7}$. ...
2
votes
2answers
74 views

Prove that $\cos(2a) + \cos(2b) + \cos(2c) \geq -\frac{3}{2}$ for angles of a triangle

Let the three internal angles of a triangle are $a,b,c$. Prove that $$\cos(2a) + \cos(2b) + \cos(2c) \geq -\frac{3}{2}.$$ I'm looking for an elementary, geometric proof. So avoid derivatives and ...
1
vote
1answer
33 views

Determine the range of f(x)=(sinx)/x

I am having trouble understanding the solution to this question. ''Determine the range of the following function: $f(x)$ = $(1$ $if$ $x=0)$ or (${\sin x\over x}$ if $x$$\neq$$0$) where the domain ...
2
votes
2answers
37 views

How to decompose this matrix exponential?

I would some help with the steps to decompose the below matrix exponential. $\exp\left[ \zeta \left ( \begin{matrix} -\cos(x) & i \sin(x) \\ -i \sin(x) & \cos(x) \end{matrix} \right ) ...
0
votes
0answers
24 views

System of Trigonometric Equations

Could someone please help me with the following system of equations
1
vote
2answers
157 views

How can I improve my explanation of why the ratio $\pi=\frac{C}{d}$ holds for all circles?

I'm trying to informally explain why $\pi$ holds for all circles. I would like to know if there is anything pertinent that I can add, or that is wrong with this explanation. It's an explanation, not ...
0
votes
4answers
67 views

Trigonometric function problem

Given: $f(x)=2\arctan(x) +\arcsin(2x/(1+x^2))$ prove that for every $x \ge 1, f(x)=\pi.$ any idea how to approach this question? thanks
0
votes
0answers
28 views

Singularities and trigonometric functions

For $f(z)=tan(z)/z$, I have found the singularities to be $z=0, z=\pi/2+2k\pi, z=3\pi/2+2k\pi$. k is an integer. I am trying to find the removable singularities. I have shown z=0 is a removable ...
0
votes
2answers
57 views

Simple Trigonometric Equation

If $\cos 2\theta=(\sqrt2+1)(\cos\theta-1/\sqrt2)$, then what is the value of $\theta$ ? I don't know how to solve this question. can someone tell me the required steps to solve this question.
0
votes
1answer
28 views

Find the magnitude of the $x$ and $y$ components to the nearest whole number?

Find the magnitude of the $x$ and $y$ components of $V$ to the nearest whole number: $||V||=27$ and $V$ has a direction of $60°$ $||V||=12$ and $V$ has a direction of $107°$ Please help.
0
votes
0answers
35 views

Will it clear the fence? Projectile motion

I'm probably being very dense, but I'm having a lot of trouble with this. The top of a vertical tower is 20m above ground level. When a ball is thrown horizontally from the top of this tower, it ...
0
votes
2answers
83 views

How do we make sense of angles which take irrational measures such as $\sqrt 2 ^\circ$?

If you were asked to draw such an angle how would you do so? Would you take it to a limit? Can the degree measure take the value of all real numbers?
2
votes
2answers
41 views

Integrate via substitution and derivation rule

i have to solve this integral $$\int_{-r}^{+r}\int_{-\sqrt{r^2-x^2}}^{+\sqrt{r^2-x^2}} \sqrt{1-\frac{x^2+y^2}{x^2+y^2-r^2}} \operatorname d y \operatorname d x$$ with substitution and then the ...
0
votes
2answers
39 views

Why is this trig function never undefined

$$f(x) = \frac{\left(\cos x\space +\space 0.5\right)}{\left(1\space +\space 0.5\cos x\right)^2}$$ Looking at the graph I know that the function is never undefined, but how would I show this or ...
1
vote
2answers
70 views

Atan2 Faster Approximation

I am using atan2(y, x) for finding the polar angle from the x-axis and a vector which contains the point (x,y) for converting Cartesian coordinates to polar coordinates. But, in my program which will ...
1
vote
4answers
134 views

Number of iterations to reach cosine's fixed point

I was messing around with my calculator the other day when I saw something interesting happen. Whenever I repetitively took the cosine of any number, it always ended up on a particular number ...
2
votes
1answer
51 views

Calculate Angle between Two Intersecting Line Segments

Need some help/direction, haven't had trig in several decades. On a 2 dimensional grid, I have two line segments. The first line Segment always starts at the origin (0,0), and extends to (1,0) along ...
9
votes
3answers
150 views

Is there a reason of $\cos(11x)+\sin(11(x+1))\approx 0$

Is there a reason of $$\cos(11x)+\sin(11(x+1))\approx 0$$
2
votes
4answers
94 views

Why are angles in “degrees” dimensionless?

Angles measured in radians are considered to be dimensionless because the radian measure of angles is defined as the ratio of two lengths $\theta=\frac{s}{r}$ (where $s$ is some arc measuring ...
2
votes
1answer
39 views

Solve for r where $\tan^{-1}\frac{60}{r} = \sin^{-1}\frac{60}{r - 10}$?

I'm trying to find the value of r where $$\tan^{-1}\frac{60}{r} = \sin^{-1}\frac{60}{r - 10}$$ It's taken me a few hours to get to this point (my trig skills are pretty bad), and I'm not sure where ...
0
votes
1answer
51 views

Help Figuring out Period of Trig Function

I'm working on a javascript project for hobby that I'm trying to get some neat effects working in. The issue is its been a long time since I was in trig and I'm having a hard time grasping exactly how ...
2
votes
1answer
56 views

Find $r$ knowing that $r=\frac{60}{\sin^{-1}\frac{60}{r}}$

I'm trying to find the value of $r$ knowing that: $$r=\frac{60}{\sin^{-1}\frac{60}{r}}$$ I'm not really sure how to approach finding the solution. Can anyone help me out? I've spent well over an ...
6
votes
1answer
210 views

Exact arctangent of product of tangents

Calculate $x$, if $$\tan(x)=\tan9\tan69\tan33$$ (Using sexagesimal degrees) Since $\tan3x=\tan(60-x)\tan x \tan(60+x)$: \begin{align*} \tan27&=\tan69\tan9\tan51\\ ...
0
votes
0answers
19 views

Proving that sum of (possibly) phase shifted sinusoids of frequency $w$ gives a sine wave with that frequency

I've been trying to get a proof of it WITHOUT using Euler's formula. The only thing I found was this link. The proof itself is quite long for a theorem it proves, I got a feeling it could be done in a ...
5
votes
1answer
88 views

Evaluate $\int \frac{\mathrm dx}{1+\cos^2 x}$

$$\int \frac{1}{1+\cos ^2x} \,\mathrm dx$$ I have to integrate the expression above: I tried with substitutions $\cos x=t$ and $1+(\cos x)^2=t$, but those didn't work, and I couldn't find any useful ...
24
votes
2answers
420 views

Can both $x$ and $\sin(x)$ be rational at the same time?

Except, of course, trivial $x=0$ case ($\sin0=0$); $x$ is measured in radians. The question turned out to be more complicated than it seemed to me at the first sight. All I came up with, that posed ...
1
vote
1answer
21 views

How to solve this integral with trigonometric functions?

How can I compute this integral manually? $\int_{1}^{t} sin2(t-\tau) cos2\tau d\tau$ I've tried some substitutions, trigonometric manipulations, but still cannot reach a reasonable next step. Any ...
1
vote
2answers
33 views

Squeeze Theorem Question

My question is from this Video In the last example He says that $$\lim_{x \to 0} x^2 \cos(\frac{1}{x^2}) = 0$$ Squeeze Theorem: $$g(x) \leq f(x) \leq h(x)$$ Given: $$-1 \leq \cos(x) \leq 1$$ he ...
-1
votes
2answers
21 views

Having trouble with application

We are doing application problems in trigonometry and I am having trouble drawing a sketch from the words provided The question is: A captain knows that his ship is due south of a lighthouse. His ...
3
votes
2answers
31 views

Confusion to use formula $l = r\theta$

I have been teaching my brother some trignometry. There is a formula as arc length of circumference of a circle. The basic formula is $$l = r\theta.$$ But sometimes for length they use $l = 2r$ and ...
0
votes
0answers
32 views

Infinity and the complex infinity

What's the difference between infinity and the complex infinity, and why is $\tan 90^{\circ}$, according to Wolfram Alpha, equal to the complex infinity and not undefined? Please see the following ...
1
vote
2answers
69 views

Simulating simultaneous rotation of an object about a fixed origin given limited resources.

Sorry if the title is a bit cryptic. It's the best I could come up with. First of all, this question is related to another question I posted here, but that question wasn't posed correctly and ended ...
0
votes
2answers
52 views

Trigonometric inequality $\frac{\cos x -\tan^2(x/2)}{e^{1/(1+\cos x)}}>0$

How can I solve the following inequality? $$\frac{\cos x -\tan^2(x/2)}{e^{1/(1+\cos x)}}>0$$
1
vote
2answers
47 views

Describing a point trigonometrically

Let us show that the transformation that reflects a point through a line through the origin is linear. This is the transformation that takes a point on one side of the line and moves it ...
3
votes
2answers
59 views

How to evaluate $\cos(22^\circ)\cos(38^\circ) - \sin(22^\circ)\sin(38^\circ)$?

How does one evaluate this? Does this generalize to $\cos(x)\cos(y) - \sin(x)\sin(y)$?