What is Trigonometric functions: Definition and 163 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.
Let z=x+iy, and w=u+iv. I am looking for a formula to find the arctangent of z, or w=arctan(z). I want the results of u and v to be in terms of trigonometric and hyperbolic functions (and their inverses) and not in terms of logarithms. The values u and v should be functions of x and y.
R is the triangle which area is enclosed by the line x=2, y=0 and y=x.
Let us try the substitution ##u = \frac{x+y}{2}, v=\frac{x-y}{2}, \rightarrow x=2u-y , y= x-2v \rightarrow x= 2u-x + 2v \therefore x= u +v##
## y=x-2v \rightarrow y=2u-y-2v, \therefore y=u- v## The sketch of triangle is as...
Hi,
I have given the following, which I would like to show that this estimation is correct, where ##|\theta| \leq \frac{\pi}{^2}## and ##M \geq 1##:
$$\frac{1}{M^2}\frac{\sin^2(M\theta)}{\sin^2(\theta)} \geq \frac{4}{\pi^2}$$
I would approach an estimation of the denominator via ##\sin(x)...
Create one equation of a reciprocal trigonometric function that has the following:
Domain: ##x\neq \frac{5\pi}{6}+\frac{\pi}{3}n##
Range: ##y\le1## or ##y\ge9##
I think the solution has to be in the form of ##y=4sec( )+5## OR ##y=4csc( )+5##, but I am not sure on what to include...
Is tan^2 (x) the same as tan(x)^2?
Note: I could have used any trig function.
I know that tan^2 (x) means (tan x)^2.
What does tan (x)^2 mean? Is it proper notation?
So far I've got the real part and imaginary part of this complex number. Assume: ##z=\sin (x+iy)##, then
1. Real part: ##\sin x \cosh y##
2. Imaginary part: ##\cos x \sinh y##
If I use the absolute value formula, I got ##|z|=\sqrt{\sin^2 {x}.\cosh^2 {y}+\cos^2 {x}.\sinh^2 {y} }##
How to...
Hey everyone .
So I've started reading in depth Fourier transforms , trying to understand what they really are(i was familiar with them,but as a tool mostly) . The connection of FT and linear algebra is the least mind blowing for me 🤯! It really changed the way I'm thinking !
So i was...
Hello,
It has been a long time since I first looked at this, so thought I might ask for some help in clarifying this problem:
Is an equation of the form --> Velocity = (Distance) * (Trigonometric function) a valid one in physics?
If so, what is the relationship of trigonometric functions...
Hello everyone,
I have noticed a striking similarity between expressions for creation/annihilation operators in terms position and momentum operators and trigonometric expressions in terms of exponentials. In the treatment by T. Lancaster and S. Blundell, "Quantum Field Theory for the Gifted...
The answer for derivative of y=5tanx+4cotx is y'=-5cscx^2. But how come on math help the answer is 5sec^2x-4csc^2x? I have a calculus test coming up and I really would appreciate if someone could explain!
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Oh nvm I see my mistake!
Homework Statement
Homework Equations
General Formula for Tan(a)=Tan(b)
The Attempt at a Solution
See the question I have uploaded.
I have tried solving it this way,
Firstly I applied the Quadratic Formula to get,
Now we have two cases,
CASE-1
When
So General Formula here will...
The circle ## x^n+y^n=1 ##, for n integer >2 in a metric space with distance function: ## \sqrt[n] {dx^n+dy^n} ## has corresponding trigonometric Sine and Cosine functions defined in the usual way.
Finding the sine or cosine of the sum of two angles, derivatives and curvature of a line in such...
I am in the trigonometry section of my precalculus textbook by David Cohen. In Section 6.2, David explains how to evaluate trig functions without using a calculator but it is not clear to me.
Sample:
Is cos 3 positive or negative?
How do I determine if cos 3 is positive or negative without...
I found in a book that the domain of tan x was {(2n+1)π/2 , n∈I}
The graph however shows that for every value of x , the function takes on a value .So, why is the domain like this?
Homework Statement
A transverse wave that is propagated through a wire, is described through this function: y(x,t) = 0.350sin(1.25x + 99.6t) SI
Consider the point of the wire that is found at x= 0:
a) What's the time difference between the two first arrivals of x = 0 at the height y =...
Homework Statement :[/B]
Solve for ##x ##: $$ \sin ^{-1} {x} +\sin ^{-1} {(1-x)} =\cos ^{-1} {x} $$
Answer given: ##0## or ##\frac {1}{2}##.
Homework Equations :[/B]
All relevant formulae on inverse circular functions may be used.
The Attempt at a Solution :[/B]
Please see the pic below...
Homework Statement
Hello!
I am doing exercises on sinusoid functions from the beginning of Trigonometry.
I hoped I understood the topic, but it seems not quite, because I don't get the results authors show as examples for one of possible answers, as there can be a few answers to the same...
Homework Statement
Hello!
I am at the topic on graphing trigonometric functions. Exercises are rather easy at this point, but I have a problem deciphering how authors of the book choose points for x values. Please, take a look at few examples (including screen shots I attach), and, please...
Hi all,
I have a trigonometric function series
$$f(x)={1 \over 2}{\Lambda _0} + \sum\limits_{l = 1}^\infty {{\Lambda _l}\cos \left( {lx} \right)} $$
with the normalization condition
$$\Lambda_0 + 2\sum\limits_{l = 1}^\infty {{\Lambda _l} = 1} $$
and ##\Lambda_l## being monotonic decrescent...
Homework Statement
I'm searching for the integral that gives arcosu
Homework Equations
as we know : ∫u'/[1-u^2]^0.5 dx = arcsinu
derivative of arccosu = -u'/[1-u^2]^0.5 + C
derivative of arcsinu= u'/[1-u^2]^0.5
The Attempt at a Solution
when I type the -u'/[1-u^2]^0.5 on the online integral...
Homework Statement
Show that sin 600° . cos 330° + cos 120° . sin 150° = - 1
Homework Equations
I know that sinΘ = opposite/hypotenuse and cosΘ = adjacent/hypotenuse.
The Attempt at a Solution
I am equipped with knowledge about what sinΘ and cosΘ is from right angled triangle.
I stand in...
This is not a homework question but a general doubt.
Suppose we have a function y = pcosx, where 'p' is an arbitrary constant. So my question is how will the graph of this function change with different values of 'p'?
This doubt can also be extended for other functions like y = pex, y = p...
Evaluate \left\lfloor{\tan^4 \frac{3\pi}{7}+\tan^4 \frac{2\pi}{7}+2\left(\tan^2 \frac{3\pi}{7}+\tan^2 \frac{2\pi}{7}\right)}\right\rfloor.
Hi MHB,
I don't know how to solve the above problem, as I have exhausted all possible methods that I could think of, and I firmly believe there got to be...
First part of the question was to work out the integral 1/(y+cos(x)) between x=0 and x=pi/2 by using the substitution t=tan(x/2).
Got this to be \frac{2}{\sqrt{y^2-1}}arctan(\sqrt{\frac{y-1}{y+1}})
The next question says HENCE find integral with the same limits of \frac{1}{(y+cos(x))^2}
Ive...
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?
Im trying to solve for a constant in an equation and it involves taking the arctanh(6.55) and my calculator is giving me an error, is there a way around this?
What's $1. ~ \displaystyle \arccos(\cos\frac{4\pi}{3})?$ Is this correct?
The range is $[0, \pi]$ so I need to write $\cos\frac{4\pi}{3}$ as $\cos{t}$ where $t$ is in $[0, \pi]$
$\cos(\frac{4\pi}{3}) = \cos(2\pi-\frac{3\pi}{3}) = \cos(\frac{2\pi}{3}) $ so the answer is $\frac{2\pi}{3}$
Homework Statement
There's no reason to give you the problem from scratch. I just want to show that 5 trigonometric functions are linearly independent to prove what the problem wants. These 5 functions are sin2xcos2x. sin2x, cos2x, sin2x and cos2x.
Homework Equations...
Hello, I am a first year science teacher doing my best with teaching physics for the first time (my degree is chemistry but I am in a very small school).
I am teaching projectile motion. I was creating a worksheet and trying to solve a problem I made up when I realized something wasn't working...
Can someone please show me, step-by-step, how
cos(θ)sin(θ+φ)-sinθcos(θ+φ)
simplifies to sinφ?
I know I have to use trig identities, and I got to cos2θsinφ-sin2θcosφ but I'm not sure where to go from there.
Thanks
I am familiar with the importance of the following inverse circular/hyperbolic functions:
##\sin^{-1}##, ##\cos^{-1}##, ##\tan^{-1}##, ##\sinh^{-1}##, ##\cosh^{-1}##, ##\tanh^{-1}##.
However, I don't really get the point of ##\csc^{-1}##, ##\coth^{-1}##, and so on.
Given any equation of the form...
1) Problem: given that x is an obtuse angle for which cos^2x/(1 + 5sin^2x) = 8/35, find the value of cosx/(1 - 5 sin x) without evaluating x.
2) Relevent equations:
sin(-x) = - sin x
cos(-x) = cos x
sin(180° - x) = sin x
cos(180° - x) = - cos x
sin^2x + cos^2x = 1
3) Attempt:
cos^2x/(1 +...
Homework Statement
On very hot days there sometimes can be a mirage seen hovering as you drive. Very close to the ground there is a temperature gradient which makes the refraction index rises with the height. Can we explain the mirage with it? Which unit do you need to extremalise? Writer the...
Hello,
I am trying to solve this. This material is not covered in my class, but I still want to know how to do it.
If cos(t)=$\frac{-9}{10}$ where $\pi$ <t<$\frac{3\pi}{2}$ find the values of
cos(2t)=
sin(2t)=
cos($\frac{t}{2}$)=
sin($\frac{t}{2}$)=
Give exact answers, do not use decimal...
I'm developing my own trigonometric function concerning a "real world" problem of my choosing. I decided to go with the orbit of Neptune around the sun. I just don't know how to develop the equation itself, like if it would be sine or cosine? I'm just lost as to where to begin. If anyone can...
I am looking for a rigorous (preferably HIGHLY rigorous) treatment of the trigonometric functions from their definitions through to basic relationships and inequalities through to their differentiation and integration ... and perhaps further ...
Can someone please suggest
(i) an online...
I've muddled my way through the majority of my weekend assignment and I'm stuck on a problem where I need to add two formulas together.
1.) 20-10cos(x*pi/4)
2.) 30+20sin(x*pi/4)
I end up with a sinusoidal function which I can then graph and determine the max, min, etc.
We recently went over...
So here's the question:
Suppose cos(θ) =x/4. Find expressions for the other five trigonometric functions in terms of x.
In our practice problems we never had a variable x used and we were able to use the pythagorean theorem to determine the final side of the triangle and simply figure out the...
Homework Statement
Suppose the function is y = a cot k(x−b)
Then (give exact answers; you can type pi for π):
a =
b =
k =
Suppose the function is y = a tan k(x−b), where b > 0.
Then:
a =
b =
k =
The Attempt at a Solution
Then (give exact answers; you can type pi for π):
a = 4...
Mod note: Fixed the LaTeX. The closing itex tag should be /itex, not \itex (in brackets).
I find it easier to use # # in place of itex, or $ $ in place of tex (without the extra space).
Homework Statement
Prove \lim_{x \to 0} \frac{x}{\sin^2(x) + 1} = 0
Homework Equations
Given below:
The...
Mod note: Moved from a technical math forum, so this post is missing the homework template
i am trying to prove that ##1/sec∅-tan∅ ≡ sec ∅ + tan∅##
this is how i attempted it, i tried to show that the left hand side is equal to the right...
## 1/ 1/(cos∅-sin∅)/cos∅##
where i end up with
##...
The problem is as the title says. This is an example we went through during the lecture and therefore I have the solution. However there is a particular step in the solution which I do not understand.
Using the Taylor series we will write sin(x) as:
sin(x) = x - (x^3)/6 + (x^5)B(x)
and...
I'm having a problem understanding exactly why trig functions are defined the way they are. Of course, the definition in terms of 0 to 90 degree angles within right triangles is easy: the functions just give the ratio of the sides given the angle. However, I don't understand how or why trig...