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 (2)

0
votes
0answers
121 views

smaller circle into larger circle : find length of common arc

Let a circle of radius $r$ be contained in a larger circle of radius $R$ such that the two circles touch. What is the length, in radians of the common arc, in blue? I think the solution is ...
3
votes
6answers
198 views

How do I verify that $\sin (\theta)$ and $\cos (\theta)$ are functions?

I am studying pre-calculus mathematics at the moment, and I need help in verifying if $\sin (\theta)$ and $\cos (\theta)$ are functions? I want to demonstrate that for any angle $\theta$ that there ...
3
votes
3answers
108 views

How do you actually calculate inverse $\sin, \cos, $ etc. ?

I started to wonder, how does one actually calculate the $\arcsin, \arccos, $ etc. without a calculator? For example I know that: $$\arccos(0.3) = 72.54239688^{\circ}$$ by an online calculator, but ...
0
votes
5answers
251 views

Prove that $\frac{\sin3A}{\sin2A-\sin A} = 2\cos A+1$.

Prove $$\frac{\sin3A}{\sin2A-\sin A} = 2\cos A+1$$ I am confused on what to do after changing $\sin 3A$ into $\sin(2A+A)$ and then applying the compound angle rule to make $\sin2A\cos A - ...
2
votes
1answer
62 views

Trigonometric regression

What methods are performed for regression with trigonometric functions? E.g. : Sequence: $$-1, 0, 1, -1, 0, 1, \text{.....}$$ Regression: ...
0
votes
3answers
58 views

Permutation Formula Question

The problem that I am having trouble figuring out is: In how many ways can give five men and five girls be seated at a round table so that each girl is between two men? I know that the formula for ...
4
votes
1answer
108 views

Proving $\cos x < 1 - \frac{x^2}{2} +\frac{x^4}{24}$

I wish to prove the following inequality for $x\ne 0$: $$\cos x < 1 - \frac{x^2}{2} +\frac{x^4}{24}$$ Using the fact that I already prove: $$\cos x > 1 - \frac{x^2}{2}$$ My try: $\cos x = 1 - ...
1
vote
1answer
41 views

Reciprocal Identities Help.

$\displaystyle\frac{\sin^2A + 2\cos A - 1}{2 + \cos A - \cos^2A}$ = $\displaystyle\frac{1}{\sec(A)+1}$ Can someone help me with this one? I can't seem to get it right, I get lost, right side has me ...
1
vote
1answer
53 views

How to trace the graphic of $\cos(x) + \cosh(y) = k$?

Is there some systematic way to trace the graphic of $\cos(x) + \cosh(y) = k$ given a fixed value for $k$? Suppose $k = 1$: if I choose empirically $y = 1.2$, I know that should be $\cos(x) = - ...
1
vote
0answers
140 views

How to find Latitudes and Longitudes of projections of the vertices of a rectangular plane below earth's surface?

I want to find out the latitudes and longitudes of projections of the vertices of a rectangular plane inside the earth's surface. I know dimensions of rectangle, angles of orientation and latitude and ...
7
votes
4answers
382 views

Trigo Problem : Find the value of $\sin\frac{2\pi}{7}+\sin\frac{4\pi}{7}+\sin\frac{8\pi}{7}$

Find the value of $\sin\frac{2\pi}{7}+\sin\frac{4\pi}{7}+\sin\frac{8\pi}{7}$ My approach : I used $\sin A +\sin B = 2\sin(A+B)/2\times\cos(A-B)/2 $ $\Rightarrow ...
1
vote
3answers
304 views

Moments at which moving points on a circle coincide

Points A $(0,1)$ and B $(1,0)$ start moving along the circumference of a unit circle with center $(0,0)$ in the same, positive (that is, counterclockwise) direction. Every minute, points A and B ...
1
vote
2answers
65 views

How to apply the law of sines in a non right triangle?

From http://www.sparknotes.com/testprep/books/sat2/math2c/chapter9section9.rhtml, I saw that you can apply the law of sines to solve the measures of all the variable values of a non-right triangle ...
1
vote
1answer
41 views

Factoring Trigonometric Equations

And so begins another day in my quest to pass Calculus I. I have a question about factoring trigonometric expressions. I'm sorry I can't be more accurate than that, but I'm not sure of the names of ...
2
votes
1answer
81 views

Taylor expansion and trigonometric functions

I've seen a proof including this claim: $$x-\frac{x^3}{6} \le \sin x$$ Now, for my understanding, the series $x-\frac{x^3}{6} +\frac{x^5}{120} -... $ is converging to $\sin x$ in an alternating way. ...
2
votes
1answer
56 views

Trigonometry equation solution

I have to solve this equation for all solutions $$\sin(2x) = -\cos(2x)$$ Here are my steps $$\sin(2x) + \cos(2x) = 0$$ $$\cos(2x)(\tan(2x) + 1) = 0$$ Upon solving these two equations, I find: For ...
2
votes
3answers
84 views

Limits of trig functions

How can I find the following problems using elementary trigonometry? $$\lim_{x\to 0}\frac{1−\cos x}{x^2}.$$ $$\lim_{x\to0}\frac{\tan x−\sin x}{x^3}. $$ Have attempted trig identities, didn't help. ...
0
votes
1answer
41 views

how to solve this equation $z=yb\cot(bx/2)$?

how to solve this equation $z=yb\cot(bx/2)$? $b$ is unknown, the $x,y,z$ are known numbers, and $x\ne0,b>0$, we want to have the solution for $b$. Until now I have no idea about this.
0
votes
2answers
341 views

Domain of arctan(1/x)

I had this as part of a question in an exam. And, I reasoned, even when it's arctan(1/0) (undefined), it is pi/2. And, so I said, domain belongs to all Real Numbers. Why isn't it this
0
votes
1answer
47 views

If $0 \lt x \lt \frac{\pi}{2}$, prove that $\cos(x) \le \frac{\sin(x)}{x} \le \frac{1}{\cos(x)}$

How can I find the following product using elementary trigonometry? Suppose $0 \lt x \lt \frac{\pi}{2}$ is an angle measured in radians. Use the trigonometric circle and show that $\cos(x) \le ...
0
votes
1answer
54 views

Trignometric Indentities

Prove the following identity: $4\sin^2x + 7\cos^2x = 4 + 3\cos^2x$ I have no idea where to start on these type of trigonometric identities with the numbers in front of the sin and cos.
2
votes
2answers
85 views

A little Problem in Trigonometry (Multiple Angle)

If $\tan^2 \theta = 1 + 2\tan^2 \phi$, show that $\cos 2\phi = 1 + 2\cos2\theta$. What I have done.. $$\implies \tan^2 \theta = 1 + 2\tan^2 \phi\\ \implies 1 + \tan^2 \theta = 2 + 2\tan^2 \phi\\ ...
1
vote
1answer
21 views

How to find the Fourier components

I've got a sinusodial signal: \begin{equation} \Delta I=\cos(\Delta \omega t + \varphi)). \end{equation} and I would like to rewrite it as the sum of a cosine and a sine signal(without a phaseterm): ...
1
vote
1answer
281 views

Summation of cos (2n-1) theta

By considering $\sum\limits_{n=1}^N z^{2n-1}$, where $z=e^{i\theta},$ show that $$ \sum\limits_{n=1}^N \cos{(2n-1)} \theta = \frac{\sin(2N\theta)}{2\sin\theta}, $$ where $\sin\theta\neq0$ I ...
1
vote
1answer
66 views

Rationality in Triangle

How can I justify this answer? I think the answer is infinite, but cannot justify it///
0
votes
1answer
61 views

Trigonometric Calculus

If $$f(x) = 2 \sin x \cos x + \sin x,$$ evaluate $f(-\frac{\pi}{4})$. The answer suggests $$-\frac{{2+\sqrt{2}}}{2}$$ but I am clueless on the procedures to get to the answer. Help please!
0
votes
2answers
78 views

Trigonometric identity problem

$$\frac{\sin(2x)}{2 \sin (x)}-\frac{\cos(2x)}{\cos(x)+\sin(x)}=\sin(x).$$ I got it on a test and want an answer. I always hit a dead end with the identities I learned.
6
votes
1answer
85 views

Link between a cubic polynomial and a trig identity

Alright, so I am told to prove that: $$\tan (3A) = \frac{3\tan(A)-\tan^3(A)}{1-3\tan^2(A)}$$ This can be pretty easily done by applying the $\tan$ addition formula, taking the angles $2A$ and $A$, ...
1
vote
3answers
101 views

How does $\arccos$ actually work? [duplicate]

$$\arccos\left(\frac{\text{adjacent}}{\text{hypotenuse}}\right)=θ$$ I regard the above as witchcraft. How would I work this out if I didn't have a calculator? Once I know how to workout arcos without ...
-1
votes
2answers
80 views

How is arccos derived?

Stupid question but I need to understand: If $ \cos = \dfrac{\text{adjacent}}{\text{hypotenuse}} $ What is arcosine? $ \text{adjacent} \cdot \text{hypotenuse}$? Is this the same for arcsine and ...
3
votes
3answers
186 views

$4 \sin 72^\circ \sin 36^\circ = \sqrt 5$

How do I establish this and similar values of trigonometric functions? $$ 4 \sin 72^\circ \sin 36^\circ = \sqrt 5 $$
0
votes
1answer
86 views

Proove that cos(x / 2) + cos(y / 2) - cos(z / 2) = 4 * sin((pi - x) / 4) * sin((pi - y) / 4) * sin ((pi + z) / 4

Help me proove that cos(x / 2) + cos(y / 2) - cos(z / 2) = 4 * sin((pi - x) / 4) * sin((pi - y) / 4) * sin ((pi + z) / 4 where x + y + z = pi I've reached 2 * sin((x + z) / 4) * (cos((x + z) / 4) - ...
1
vote
2answers
28 views

Prove $a_n$ is within a specific range for all $n\in\mathbb{N}$

Let $a_n = \cos (a_{n-1})$ and $a_1 = {\pi\over 4}$. How to prove the range of $a_n$ is within the closed interval: $\left[ {{1 \over {\sqrt 2 }},{\pi \over 4}} \right]$? I thought about ...
1
vote
0answers
73 views

Logarithm and “basic” functions.

To express the antiderivatives of $\frac{1}{x}$, we cannot apply the formula $\int x^n dx=\frac{x^{n+1}}{n+1}+C$ and we need to introduce a new function, the logarithm. But how can we prove that ...
3
votes
2answers
357 views

Determining the angles of a triangle given the ratio between its edges

Given that a triangle has edges of ratio 2 : 3 : 4, the task is to determine the three angles, say in degrees. I started by drawing 4 cm segment on the paper, then drew perpendicular segments of ...
5
votes
3answers
88 views

How to calculate the range of $x\sin\frac{1}{x}$?

I want to find the range of $f(x)=x\sin\frac{1}{x}$. It is clearly that its upper boundary is $$\lim_{x\to\infty}x\sin\frac{1}{x}=1$$ but what is its lower boundary? I used software to obtain the ...
0
votes
1answer
304 views

Equation of tangent on Cartesian plane given center and radius of a circle

If I have a generic circle with radius $r$ and center $(h, k)$, and a tangent line with point of tangency $(x, y)$, can you give me the equation of the tangent line? Thanks!
3
votes
2answers
269 views

How to integrate this formula with secant, exponential, and tangent?

How to integrate this? $$\int \sec^2(3x)\ e^{\large\tan (3x)}\ dx$$
0
votes
2answers
174 views

What do the trigonometric ratios actually represent

First I had the simple notion that the trignometric ratios are ratios of sides of a triangle. But when $\sin(0) = 0 , \cos(0) = 1 $ what do they mean? I had read about the unit circle definition ...
0
votes
1answer
58 views

Exact calulation of trigonometric functions

My question is a bit related to computer science and I am not quit sure if this place is a good one for it. Let me know if I should move it. So as you know values of functions like sin or cosine can ...
2
votes
3answers
419 views

How do I evaluate integrals that involve the signum ($\text{sgn}$) function?

For example, I want to evaluate $$ \displaystyle \int_{0}^{2\pi} \left| \sin x \right| \text{ d}x $$ and I already know that: $$ \displaystyle \begin{aligned} \int \left| \sin x \right| \text{ d}x ...
0
votes
0answers
120 views

Frequency of a trigonometric function: $g(x) = \sin(2\pi(x-a)b \ \cos(2\pi(x-a)c)) + d.$

I have a trigonometric function $g(x)$ and I'm interested in how its frequency changes as $|x|$ increases. When I plot the function I can see that the frequency increases with $|x|$, but is there a ...
2
votes
0answers
1k views

Angular velocity of the minute hand

The exercise is to calculate the angular velocity (in radians per hour) of the rotation of: the hour hand, and the minute hand (of the clock). Neither of my answers coincides with the answers in ...
5
votes
0answers
127 views

A trigonometric integral with sin(cos(x)) in exponent

Evaluate: $$\int_0^{\pi} x\csc^{\sin(\cos x)}(x)\,dx$$ I honestly don't know how to deal with this case. If I apply the property $\int_a^b f(x)\,dx=\int_a^b f(a+b-x)\,dx$, I get: $$\int_0^{\pi} ...
4
votes
4answers
339 views

Convolution integral $\int_0^t \cos(t-s)\sin(s)\ ds$

How can I calculate the following integral? $$\int_0^t \cos(t-s)\sin(s)\ ds$$ I can't get the integral by any substitutions, maybe it is easy but I can't get it.
1
vote
1answer
62 views

$\int_{-\pi}^{\pi} \cos nx \, dx$ is always 0 when actually integrated.

It's pretty apparent $\int_{-\pi}^{\pi} \cos nx \, dx$ should be 2$\pi$ when n = 0. However it seems $\int_{-\pi}^{\pi} \cos nx \, dx$ is always 0 when actually integrated given that n is integer. I ...
1
vote
2answers
68 views

$\frac{1}{\sin\theta\cdot \sin2\theta} + \frac{1}{\sin2\theta\cdot \sin3\theta} + \cdots + \frac{1}{\sin n \theta \sin (n+1)\theta}$ [closed]

$$\sum_{k=1}^n \frac{1}{\sin k\theta \sin (k+1)\theta} = \dfrac{1}{\sin\theta\cdot \sin2\theta} + \dfrac{1}{\sin2\theta\cdot \sin3\theta} + \cdots + \frac{1}{\sin n \theta \sin (n+1)\theta}$$ up to ...
1
vote
1answer
47 views

Is the given triangle unique?

I was reading Polya's How to Solve It when I came across the following problem. Construct a triangle with an angle, the length of altitude through that angle and the perimeter of the triangle given. I ...
1
vote
2answers
75 views

Trigonometrical Solve

There are 2 different values of $ \ \theta \ $. They are $ \ a \ $ and $ \ b \ $, such that $ \ 0 \ < \ a,b \ < \ 360^\circ \ $. If $ \ \sin(\theta+\phi) = \frac{1}{2} \sin2\phi \ $ , prove ...
0
votes
1answer
74 views

De Moivre and trignometry question

I showed this first result and after that for $x^4-10x^2+5=0$, I solved for $\tan 5\theta=0$, I understand all this , but then I get $\theta=\pi/5$. I know I have to multiply by $n$ to get 5 ...