What is Trig functions: Definition and 218 Discussions
In mathematics, the trigonometric functions (also called circular functions, angle functions or goniometric functions) are real functions which relate an angle of a right-angled triangle to ratios of two side lengths. They are widely used in all sciences that are related to geometry, such as navigation, solid mechanics, celestial mechanics, geodesy, and many others. They are among the simplest periodic functions, and as such are also widely used for studying periodic phenomena through Fourier analysis.
The trigonometric functions most widely used in modern mathematics are the sine, the cosine, and the tangent. Their reciprocals are respectively the cosecant, the secant, and the cotangent, which are less used. Each of these six trigonometric functions has a corresponding inverse function (called inverse trigonometric function), and an equivalent in the hyperbolic functions as well.The oldest definitions of trigonometric functions, related to right-angle triangles, define them only for acute angles. To extend these definitions to functions whose domain is the whole projectively extended real line, geometrical definitions using the standard unit circle (i.e., a circle with radius 1 unit) are often used. Modern definitions express trigonometric functions as infinite series or as solutions of differential equations. This allows extending the domain of sine and cosine functions to the whole complex plane, and the domain of the other trigonometric functions to the complex plane from which some isolated points are removed.
Initially, I was attempting to find the function which expresses the area under enclosed between the function ##\arcsin(\sin(x))## and the ##x##-axis (so technically I am looking for ##\int_{0}^{x} \arcsin(\sin(t)) dt## specifically, but got caught up on finding the general antiderivative)...
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As I continue to study the gyroscope with Tait-Bryan angles or Euler angles, and work out relationships to develop steady precession, I notice that the trig functions cancel.
I stumble on terms like:
1. sin(theta)cos(theta) - cos(theta)sin(theta)
2. Cos_squared +...
The trig identities for adding trig functions can be seen:
But here the amplitudes are identical (i.e. A = 1). However, what do I do if I have two arbitrary, real amplitudes for each term? How would the identity change?
Analysis: If the amplitudes do show up on the RHS, we would expect them...
This was the question,
The above solution is the one that I got originally by the question setters,
Below are my attempts (I don't know why is the size of image automatically reduced but hope that its clear enough to understand),
As you can see that both these methods give different answers...
Hello.
Sin and cos separately oscillates between [-1,1] so the limit of each as x approach infinity does not exist.
But can a quotient of the two acutally approach a certain value?
lim x→∞ sin(ln(x))/cos(√x) has to be rewritten if L'hôp. is to be applied but i can't seem to find a way to...
Take any trig function, say, arcsin (x). Why is the answer x when taking the inverse of sin (x)?
Why does arcsin (sin x) = x?
Can it be that trig functions and their inverse undo each other?
This is the form of the function above:
I started by equating (1) to 1/2:
$$T(\varphi)=\frac{r^{2}+\tau^{2}-2\tau\cos\varphi}{1+\tau^{2}r^{2}-2\tau r\cos\varphi} = \frac{1}{2},$$
which can be rearranged to:
$$2r^{2}+2\tau^{2}-1-\tau^{2}r^{2}=2\tau\left[2-r\right]\cos\varphi$$
using...
I have the statement \sin[\sin^{-1}(x)] = x \hspace{7pt} if -1 \leq x \leq 1. How can I tell if plugging in x will return x for \cos[\cos^{-1}(x)] and \tan[\tan^{-1}(x)] ? What if the positions of the regular and inverse functions were reversed? For example, \cos^{-1}[\cos(x)].
I am only...
I have been wondering, what is conceptual physics?
I remember taking a class in high school that was physics oriented, for example two trains leave a station at different speeds, and arrive at a central point, where do they overlap. Also there were trig functions on how to find the height of a...
Homework Statement
Evaluate and express your answer in radians:
$$cot^{-1}\left(1\right)$$
Homework EquationsThe Attempt at a Solution
I start by identifying that the domain of Arccotangent is all real numbers. So 1 is in the domain.
From here, I looked at the unit circle and saw that...
Homework Statement
hello, I'm currently studying simpsons rule (unrelated) however the method requires the answer to cos^2(1^2) the answer given by my tutor is 0.2919, I have been unable to get this answer after inputting cos in various ways I always get 0.9997, which is right and if 0.2919 is...
Homework Statement
why this formula works ?
Homework EquationsThe Attempt at a Solution
when i take the derivative of the right side ,,, there is an additional "a" in the numerator in place of 1,, why the derivative of arcsine of (u/a) not exactly same with the expression under the integral sign
Hi there! I haven't yet taken a trigonometry course (I'm in High-school), but I have an amateur interest in surveying. Recently I began thinking about how I could calculate the height of a point relative to me, or the distance of the object from me. Naturally, I immediately thought of the...
Not a particular problem to wonder about but more of a general question, when one has a free body diagram, when is it best to use sine and when is it best to use cosine?
I am reviewing some of my tests for a final, and having previously re-read my forces chapter, I thought that angles rising...
I'm trying to make an approximation to a series I'm generating; the series is constructed as follows:
Term 1:
\left[\frac{cos(x/2)}{cos(y/2)}\right]
Term 2:
\left[\frac{cos(x/2)}{cos(y/2)}-\frac{sin(x/2)}{sin(y/2)}\right]
I'm not sure yet if the series repeats itself or forms a pattern...
Homework Statement
Show that ##e^x = x## does not have any solutions, and show that ##\sec x = e^{-x^2}## has only one solution.
Homework EquationsThe Attempt at a Solution
Here is my proof of the first proposition: Since ##e^x## is concave up on ##\Bbb{R}##, it must lie above all of its...
Homework Statement
what's the best way to solve this equation: 3cos(θ) + 1.595*sin(θ) = 3.114
Homework Equations
(sinθ)^2 + (cosθ)^2 = 1
The Attempt at a Solution
I tried using the identity above to solve this equation and ended up with cosθ = +/- 1.0526.
Is there a way to solve for W in the below equation. There has to be multiple solution for W, but I am at a loss as to how to solve this.
2*C*sin(W)+P*cos(N*W)=P or 2*C/P*sin(W)+cos(N*W)=1
C and P are constants
N is an integer
Mentor note: moved to homework section
y = sin(x)
y = cos(x)
y = tan(x)
y = csc(x)
y = sec(x)
y = cot(x)
(a) 0 (b) 4 (c) 6 (d) 2
I thought it was (c) because i graphed all the trig functions and they passed the vertical line test but the answer sheet is saying (d) 2
Homework Statement
I have a series of 12 values that I need to calculate the Theoretical Intensity, I, using the formula below.
I have found values for all variables and their uncertainties, and have calculated the I value for each set using the formula. Now I need to calculate the...
Here is the question:
This is the step I came to after taking the derivatives and doing some simplification:
^ I did the work myself on paper, I just couldn't type out the whole thing clearly so that anyone else can see what I'm referring too... so I used some online tool to show that...
Homework Statement
A highway is to be built between two towns, one of which lies 35.0km south and 72.0km west of the other. What is the shortest length of highway that can be built between the two towns, and at what angle would this highway be directed with respect to due west.
Homework...
Homework Statement
Find the domain of this function and check with your graphing calculator:
f(x)=(1+cosx)/(1-cos2x)
Homework EquationsThe Attempt at a Solution
i get to (1+cosx)/(1+cosx)(1-cosx) which is factored. so then setting each one to zero one at a time i figure out that
cosx = -1 and...
On the paper I'm reading the arctan of 35 over 65 is approx. 28.30degrees.
When I use the Google calculator "arctan(35/65)" gives me 0.493941369 rad.
What am I doing wrong?
I'm realizing now how much I need to know the exact values of various trigonometric functions, as shown in various trig tables. Memorizing is pretty arduous, and I'd prefer to understand it, so how can I learn all of these?
How do you know when to use exponentials and trig functions when solving for the wave function in a finite square well? I know you can do both, but is there some way to tell before hand which method will make the problem easier? Does it have something to do with parity?
Can somebody please explain to me why the integral of, for instance, cos((2*pi*x)/a)*cos((4*pi*x)/a) vanishes over the interval 0 to a? As I understand it, this is generally the case when integrating sines and cosines with different arguments "over the interval of a period." But I'm confused...
Homework Statement
A spectator is standing 50 ft from the freight elevator shaft of a building which is under construction. The elevator is ascending at a constant rate of 20 ft/sec. How fast is the angle of elevation of the spectator's line of sight to the elevator increasing when the elevator...
What can be said about the arguments of the exponential functions and trig functions ? Can the argument be a vector or must it be a scalar ? If it can only be a scalar must it be dimensionless ?
There are 2 trig functions on the same set of axis.
f(x)=600sin(2π3(x−0.25))+1000 and f(x)=600sin(2π7(x))+500
How do I go about finding the points of intersections of the two graphs?
This was from a test I had recently and didn't do too well on,so any help would be much appreciated.
I started...
As part of a personal musicology project I found myself with the mathematical model of a geometry which utilizes the equation
a*(a/b)sin(pi*x)
The only problem with this is that I need to take the integral from -1/2 <= x <= 1/2, and according to Wolfram Alpha no such integral exists. I can...
I work a good deal better when the equation is in x and y form, is it possible to set up a trig expression like 5Cos(x)/(Sin(x)-1)and substitute the proper x or y equivalent so long as I remember to replace the trig identities later when the problem is finished? Or can you just not solve these...
Homework Statement
f(x) = sin5x ; [π/5,2π/5] finding the point c which f'(x) =0. I understand the theorem and how to complete it, my issue is using the triq functions
Homework Equations
f'(x) = 5cos5x
The Attempt at a Solution
5cos5x=0
cos5x=0
5x=π/3
x=π/15
my answer is not correct, I am...
Homework Statement
[23/4, 2] 4/(x√(x4-4))
Homework Equations
∫ du/(u√(u2 - a2)) = 1/a(sec-1(u/a) + c
The Attempt at a Solution
I first multiplied the whole thing by x/x. This made the problem:
4x/(x2√(x4 - 4))
Then I did a u substitution making u = x2. Therefore, du = 2xdx. I multiplied by...
Homework Statement
Find d^2/dx^2 and both complex number forms for the complex number equation (1+icos(x))/(1-icos(y))[/B]
Homework Equations
1. z=a+bi
2. re^itheta
The Attempt at a Solution
I have multiplied both sides by 1+icosy and gotten as far as (1+icosx+icosy-cosxcosy)/(1+cos^2y) but...
$$\int_{0}^{\pi/2}\d{}{x} \left(\sin\left({\frac{x}{2}}\right)\cos\left({\frac{x}{3}}\right)\right)\,dx$$
the ans the TI gave me was $\frac{\sqrt{6}}{4}$
the derivative can by found by the product rule. but really expands the problem
so not sure how the $\frac{d}{dx}$ played in this.
Find a formula for g(x)= sin(arccos(4x-1)) without using any trigonometric functions.
I have the answer key right in front of me, but i still get how to start it off or the steps in solving these kind of questions or how to do it at all :/
Thanks!
I know if we set
x = \cosh \theta , y = \sinh \theta
and graph for all \theta 's, we get a hyperbolic curve since then
x^2 - y^2 = 1.
But — unlike the case of making a circle by setting
x = \cos \theta , y = \sin \theta
and graphing all the \theta 's — in the hyperbolic graph the angle...
So I have several problems that ask me to find all points of intersection algebraically, but I haven't been able to make much headway on most of them.
The first problem
Homework Statement
Find all the points of intersection algebraically of the graphs of ... on the interval [0, 4π]...
I just wanted to clear a couple of things up in terms of strict mathematical definition...
Is the correct definition of the trigonometric ratios:
cos\varphi=\frac{|x|}{r}, sin\varphi=\frac{|y|}{r}
as opposed to:
cos\varphi=\frac{x}{r}, sin\varphi=\frac{y}{r}
(note the lack of...
Homework Statement
ln(sec^-1(3x^2 +1))
Homework Equations
The Attempt at a Solution
1/sec-1(3x2+1) * 1/(3x2+1)(sqrt(3x2+1)2-1) * 6x
Is this correct ?, do I just simplify from here ?
Homework Statement
Given u(x,t) = sum( e^(-at/2)*cos(n*pi*x/2L) * Re[A_n*e^(i*w_n*t)+B_n*e^(-i*w_n*t)], and the boundary conditions u(-L)=u(L)=0 for all t;
du/dt = 0 for all x at t = 0;
u(x,t=0) = e^(-|x|/l)
Find A_n and B_n.
Homework Equations
N/A
The Attempt at a Solution
I have...
so my given: s(t)=cos(pie8*t/4)
took the derivative= velocity function
then, v(t)= -pie/4 *sin(pie*t/4)
When is the particle at rest? v(t)=0
now, 0= -pie/4 *sin(pie*t/4)
im lost here. I know it's very simple I am just over thinking. What do I do from here?
thanks
This is for Calculus II. I've found most of the integrations on inverse trig functions to be pretty simple, but for some reason this one is throwing me off.
Homework Statement
\int\frac{x+5}{\sqrt{9-(x-3)^2}}dx
The Attempt at a Solution
I started by breaking the integral up...