What is Trigonometry: Definition and 660 Discussions
Trigonometry (from Greek trigōnon, "triangle" and metron, "measure") is a branch of mathematics that studies relationships between side lengths and angles of triangles. The field emerged in the Hellenistic world during the 3rd century BC from applications of geometry to astronomical studies. The Greeks focused on the calculation of chords, while mathematicians in India created the earliest-known tables of values for trigonometric ratios (also called trigonometric functions) such as sine.Throughout history, trigonometry has been applied in areas such as geodesy, surveying, celestial mechanics, and navigation.Trigonometry is known for its many identities. These
trigonometric identities are commonly used for rewriting trigonometrical expressions with the aim to simplify an expression, to find a more useful form of an expression, or to solve an equation.
I was able to solve with a rather longer way; there could be a more straightforward approach;
My steps are along these lines;
##\sinh^{-1} x = 2 \ln (2+ \sqrt{3})##
##\sinh^{-1} x = \ln (7+ 4\sqrt{3})##
##x = \sinh[ \ln (7+ 4\sqrt{3})]##
##x = \dfrac {e^{\ln (7+ 4 \sqrt{3})} - e^{-[\ln 7+ 4...
So I am studying precalculus along with some basic calculus (I am not very patient but I feel relatively confident about my precalculus knowledge). Do you think there’s any real use of memorizing all identities for tangent and cotangent?
My question is on the highlighted part (circled in red);
Why is it wrong to pre-multiply each term by ##e^x##? to realize ,
##5e^{2x} -2-9e^x=0## as opposed to factorising by ##e^{-x} ## ?
The other steps to required solution ##x=\ln 2## is quite clear and straightforward to me.
Four points lie on the surface of a sphere. Given the six distances between the points, calculate the radius of the sphere.
This is (allegedly) an advanced high school level problem. However, it is a remembered problem, so it is possibly misremembered (i.e. there might have been some “bice...
Hi all,
I am starting with the following equation: ##2\cot\left(\frac{\theta}{2}\right) = \cot\left(\frac{k_{1}}{2}\right) - \cot\left(\frac{k_{2}}{2}\right)##
with the following definitions: ##k_{1} = \frac{K}{2} + ik, k_{2} = \frac{K}{2}-ik, \theta = \pi(I_{2}-I_{1}) + iNk##, where...
I'm not sure how to go about solving this mathematically? In just using what seems obvious, I know the angle pi would work, because pi = -1, and 2pi = 1. However, as far as manipulating the equations in a way where it can solve itself without me having to look at a chart where cos for both x...
I was trying to show that ##sin(x-y) = sin(x)cos(y)-cos(x)sin(y)## using Pythagoras' theorem and ##cos(x-y)=cos(x)cos(y)+sin(x)sin(y)##.
I have:
$$sin^2(x-y)=1-cos^2(x-y)$$
$$sin^2(x-y)=1-(cos(x)cos(y)+sin(x)sin(y))^2$$...
I proceeded as follows
$$\int\frac{2(\sqrt3-1)(cosx-sinx)}{2(\sqrt3+2sin2x)}dx$$
$$\int\frac{(cos(\pi/6)-sin(\pi/6))(cosx-sinx)}{(sin(\pi/3)+sin2x)}dx$$
$$\frac{1}{2}\int\frac{cos(\pi/6-x)-sin(\pi/6+x)}{sin(\pi/6+x)cos(\pi/6-x)}dx$$
$$\frac{1}{2}\int cosec(\pi/6+x)-sec(\pi/6-x)dx$$
Which leads...
Problem Statement : I draw a picture of the given problem alongside. P is the location of the man and Q that of his friend at a height ##h## above. If the sun is at a position ##\text{S}_1## at 6 pm, at what time is the sun at position ##\text{S}_2##?
Attempt : If ##\text{S}_2Q## is inclined to...
Was solving a problem in mathematics and came across the following integration. Unable to move further. Can somebody provide answer for the following ( a and b are constants ).
In circuit analysis, everything seems to work out when you set i*cos = sin. But thats not a legitimate equation, so why does that work? Is there a proof that this is a real equation?
The problem is based on a similar thread. In fact, the first question is extremely similar. However, the second question is the one I consider more interesting but I posted the first one too for context.
If this was just 1 pulley and two masses, then equilibrium is only possible if both masses...
In my working i have,
...
##\cos C = 2\cos^2 \dfrac{1}{2} C -1##
##c^2= a^2+b^2-2ab(2\cos^2 \dfrac{1}{2} C-1)##
##c^2= a^2+b^2+2ab(1-2\cos^2 \dfrac{1}{2} C)##
##c^2= (a+b)^2 (1-2\cos^2 \dfrac{1}{2} C)##
Now from here, ##k^2 =2## but text gives different solution. I am still checking...
I let,
## 4\tan^{-1}\left[\dfrac{1}{5}\right]- \tan^{-1}\left[\dfrac{1}{239}\right]= \dfrac{π}{4}##
##\tan^{-1}\left[\dfrac{1}{5}\right]- \dfrac{1}{4}\tan^{-1}\left[\dfrac{1}{239}\right]= \dfrac{π}{16}##
Then i let, ##\tan^{-1}\left[\dfrac{1}{5}\right] = α ...
In my approach (using a right angled triangle) i let,
##\cos^{-1} x = C## ⇒##\cos C = \sqrt{1-y^2}##
and
##\cos^{-1} y= A## ⇒ ##\cos A= \sqrt{1-x^2}##
Also, ##A+C = \dfrac{π}{2}##
and ##\cos \dfrac{π}{2}= 0##
##xy - \sqrt{(y^2) ⋅(x^2)}=xy-xy=0##
It follows that,
##\cos^{-1} [xy -...
Ok in my approach i have,
##2 \tan^{-1} \left(\dfrac{1}{5}\right)= \sin^{-1} \left(\dfrac{3}{5}\right) - \cos^{-1} \left(\dfrac{63}{65}\right)##Consider the rhs,
Let
##\sin^{-1} \left(\dfrac{3}{5}\right)= m## then ##\tan m =\dfrac{3}{4}##
also
let
##\cos^{-1} \left(\dfrac{63}{65}\right)=...
Hi.
I have two trigonometric equations whose graphs I am trying to understand.
Here are the equations:
1. a sin(x) - b cos(y) = y; a = 2, b = 2
2. a sin(x) + b cos(y) = 1; a = 1, b = 1
My question is why the graphs are the way they are.
What should I do to understand them?
Can anyone...
Is there an existing ray trace program that can trace planar light rays through this monocentric, model lens? Parameter values are given above. Input ray angles are all zero. Does some program give the output ray angle values at the second surface? How about for any arbitrary ray incoming to...
My take:
I got ##BC=10.25## cm, using cosine rule...no issue there. For part (b)
##BK=3cm## using sine rule i.e ##\sin 30^0 =\dfrac{BK}{6}##
Thus it follows that ##∠BDK=48.59^0## ...⇒##∠ADB=131.4^0## correct...any other approach?
Also:
##∠ADB=48.59^0## when BD is on the other side of the...
Hi all!
I've never been studied the identities and such of secant, cosecant and cotangent. Yet I think, it would be useful to have them in my toolbox. Thus I'm asking, if anyone would know a reasonable book or other kind of material (paper or pdf) about trigonometry that has brief theory...
my notebook says that we can rewrite the integral
$$\int {75\sin^3(x) \cos^2(x)dx}$$
as
$$\int {75 \cos^2(x)\sin(x)dx} - \int {75\sin(x)\cos^4(x)dx}$$
however, i have literally no idea how it got to this point, and i unfortunately can't really provide an "attempt at a solution" for this...
text solution here;
I was solving this today...got stuck and wanted to consult here...but i eventually found the solution...any insight/alternative approach is welcome...
My approach;
...
##\sin^2y+ cos^2 y= 2a^2-2a \sin x - 2a\cos x+1##
It follows that,##2a(\sin x + \cos x)=2a^2##
##\sin...
The official solution says ±25.4°, but I'm having trouble reproducing it. Here is my solution:
1) The components of the velocity of firework F with respect to the ground G in the moment of explosion are the following (Notice, I'm using sin, because the statement says 30.0° from vertical.)...
Refreshing on trig. today...a good day it is...ok find the text problem here; With maths i realize one has to keep on refreshing at all times... my target is to solve 5 questions from a collection of 10 textbooks i.e 50 questions on a day-day basis...motivation from late Erdos...
I am looking for good textbooks in physics, algebra, and trigonometry textbooks that are up to date and a good read. I heard that Feynman’s Lectures was really good. Is it still up to date enough?
Any opinions?
I can solve this by using the double-angle formula but the teacher expects another method not involving the double-angle formula.
Is there a way to solve this without using double-angle formula?
Thanks
This is the problem. The question is simple i just need some clarification as indicated on the part highlighted below in red.
Now from my understanding tangent repeats on a cycle of ##π## radians...why do we have 2 the part circled in red below i.e ##2##? This is the part that i need clarity...
Some textbooks I found online ( open source )
College Trigonometry 3rd Corrected Edition - STITZ ZEAGER OPEN SOURCE MATHEMATICS
Precalculus 3rd, Corrected Edition - Lakeland Community College, Lorain County Community College
A First Course in Linear Algebra - Robert A. Beezer
Cheers.
I just simply used the formula to solve. Note the "x" represents multiplication in this case
0.5 x a x c Sin B
This is based on the conditions given in the textbook I am using which quotes "Use this formula to find the area of any triangle when you know 2 sides and an angle between them"
So I...
This is the question...
My attempt on part (i),
##b=\dfrac {16π}{2π}=8##
##11=a sin 32π+c##
##c=11##
##5=-a\frac {\sqrt 3}{2} +11##
##10=-a\sqrt 3+22##
##12=a\sqrt 3##
##a=\dfrac {12}{\sqrt 3}##
Is this correct? Thanks...
My interest is on finding the value of ##A## only. From my calculations, ##A=1##and not ##2## as indicated on textbook solution.
In my working we have; i.e ##4=A +3.##
The values of ##B##and ##C## are correct though. Kindly advise.
Find the question and textbook solution.
Problem Statement : I copy and paste the statement of the problem directly from the text.
Attempt : I wasn't able to go far into the solution. Below is a rough attempt.
##\begin{equation*}
\begin{split}
\sin^2A-\sin A\sin B+\sin^2B-\sin B\sin C+\sin^2C-\sin C\sin A & = 0\\
\sin A(\sin A -...
It is about that the rznge of 60 degrees = R of 30 degrees, but how would I prove that?
Sin(120) needs to equal sin(60)
How can i prove that theyll be the same range(without air resistance?)
My take: (only looking at the sin(alpha) part as that neefs to be equal)
using trig identity
-...
The correct solution uses angles and trigonometry. My solution is as following:
- Suppose the forces exerted by friends 1 and 2 are F1 and F2 respectively.
- There are no net force in the x-direction, so F(total x) = 0.
- F(total y) = F1 + F2 - mg = 0 (initially). Rearranging gives g =...
I am teaching myself math and wondering if any of you have recommendations on trigonometry books for beginners and self study. Any help is appreciated!
Use the data from the website sunrise-sunset . org / us / lowell-ma to build a model (a sinusoidal function) whose output is the number of hours of daylight in Lowell when the input is the ordinal date (1 though 366) of the year 2020. Find (and show your calculations for finding): Amplitude...