# 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.

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1. ### Solve the given trigonometry 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...
2. ### B Memorizing trigonometric identities

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?
3. ### Solve the given trigonometry problem

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.

5. ### Prove the given hyperbolic trigonometry equation

I have, Using ##\ cosh 2x = 2 \cosh^2 x - 1## ##\cosh x = 2 \cosh^2\dfrac{x}{2} -1## Therefore, ##\cosh x -1 = 2 \cosh^2\dfrac{x}{2} -1 - 1## ##\cosh x -1 = 2 \cosh^2\dfrac{x}{2} -2## ##=2\left[ \cosh^2 \dfrac{x}{2}...
6. ### I Spherical trig - sphere radius from 6 lengths

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...
7. ### I Trig Manipulations I'm Not Getting

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...
8. ### Solve for all angles x: cos(2x) + cos(x) = 0, where 0<x<2pi

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...
9. ### Solve the given trigonometry equation

In my approach i have the following lines ##\ln (x + \sqrt{x^2+1}) = 2\ln (2+\sqrt 3)## ##\ln (x + \sqrt{x^2+1} = \ln (2+\sqrt 3)^2## ##⇒x+ \sqrt{x^2+1} =(2+\sqrt 3)^2## ##\sqrt{x^2+1}=-x +7+4\sqrt{3}## ##x^2+1 = x^2-14x-8\sqrt 3 x + 56\sqrt 3 +97## ##1 = -14x-8\sqrt 3 x + 56\sqrt 3 +97##...
10. ### Calculate the value of ##θ## and ##X##

My take, ##5 \cos 0 = 10 \cos θ## ##\cos θ = 0.5## ##⇒θ = 60^0## and ##X= 10 \cos (90^0-θ)=\cos 30^0= 8.66## (to two decimal places). ...insight welcome
11. ### B How Do You Derive the Formula for sin(x-y)?

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: $$sin^2(x-y)=1-cos^2(x-y)$$ $$sin^2(x-y)=1-(cos(x)cos(y)+sin(x)sin(y))^2$$...
12. ### Integrate [cosec(30°+x)-cosec(60°+x)] dx in terms of tan x

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...
13. ### Different sunset times due to elevation ##h## at a point on the Earth

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...
14. ### Taking the Limit of this fraction involving trig functions

Can't attempt to solve the task. I'd appreciate it a lot if you could help!
15. ### A Integration of trigonometric functions

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 ).
16. ### B Understanding the Relationship Between i*cos and sin in Circuit Analysis

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?
17. ### The Relationship Between Masses and Angles in a Pulley System

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...
18. ### Find the value of ##k^2## in the problem involving trigonometry

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...
19. ### Prove that ## 4\tan^{-1}\left[\dfrac{1}{5}\right]- \tan^{-1}\left[\dfrac{1}{239}\right]= \dfrac{π}{4}##

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 ...
20. ### Solve the given problem involving: ##\tan^{-1} (2x+1)+ \tan^{-1} (2x-1)##

I let ##\tan θ = 2x+1## and ##\tan β = 2x-1## ##θ + β = \tan^{-1} \left[\dfrac{(2x+1)+(2x-1)}{1- (2x+1)(2x-1)}\right]## ... ##θ + β = \tan^{-1} \left[\dfrac{4x}{1- 2x^2+1}\right]## ##θ + β = \tan^{-1} \left[\dfrac{4x}{2(1-x^2)}\right]## then ##\tan^{-1} \left[\dfrac{4x}{2(1-x^2)}\right]=...
21. ### Solve the problem that involves ##\cos^{-1} x + \cos^{-1}y##

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 -...
22. ### Find the smallest value of angle ##α + β ##

In my approach i have, ##α + β = \tan^{-1} \left[ \dfrac{\dfrac{a}{a+1} + \dfrac{1}{2a+1}}{1-\dfrac{a}{a+1} ⋅\dfrac{1}{2a+1}}\right]## ... ##α + β = \tan^{-1} \left[ \dfrac{2a^2+3a+1}{(a+1)(2a+1)}\right] \div \left[\dfrac{2a^2+2a+1}{(a+1)(2a+1)}\right]## ##α + β = \tan^{-1}...
23. ### Prove that the given inverse trigonometry equation is correct

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)=...
24. ### I Questions about these Trigonometry Graphs involving sin() and cos()

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...
25. ### Looking for trigonometric ray tracing software for optics

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...
26. ### Derivation or proof of derivative sin (x)

How do I do this from here without using the derivatives of sin or cos ?
27. ### Solve the given problem that involves Trigonometry

For part (a), We know that ##\cos (-θ)=\cos (θ)## and ##\sin (-θ)=-\sin (θ)## ##\cos (A-B)=\cos A\cos (-B) -\sin A\sin(-B)## ##\cos (A-B)=\cos A\cos (B) +\sin A\sin(B)## ##\cos (A-B)=\cos A\cos B+\sin A\sin B## For part (b) ... ##f(θ)=\cos 60^0- \sin (θ+30^0)\sin (θ-30^0)## ##f(θ)=\cos...
28. ### Find the possible values of angle ##∠ADB##

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...
29. ### Evaluating the angle theta using inverse sin

I just need to know how to find Θ in sin2Θ=0.51 I know I can use Θ = arcsin(0.51) but what about sin2Θ = 0.51
30. ### Geometry Looking for a good book about trigonometry

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...
31. ### Solve the given trigonometry problem

My take; ##x^2=\dfrac{(1+\sin θ)^2}{cos^2θ}=\dfrac{(1+\sin θ)^2}{1-\sin ^2θ}=\dfrac{1+\sin θ}{1-\sin θ}## we know that, ##x=\dfrac{1+\sin θ}{\cos θ}## ##⇒1+\sin θ=x\cos θ## therefore, ##x^2=\dfrac{x\cos θ}{1-\sin θ}##...
32. ### Can a human calculate this without a calculator?

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...
33. ### Solve the given trigonometry problem

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...
34. ### Finding the Distance to a Building with Trigonometry

I'm doing self-study out of a free .PDF book entitled, Trigonometry, by Richard W. Beveridge (©June 18, 2014). The problem I'm interested in is as follows: "A woman standing on a hill sees a building that she knows is 55 feet tall. The angle of depression to the bottom of the building is 27°...
35. ### Relative Velocity and Angles of Movement (Sears & Zemansky's Exercise)

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.)...
36. ### Prove the trigonometry identity and hence solve given problem

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...
37. ### Studying What should I learn first: Trigonometry or Geometry?

I’m teaching myself algebra right now so I’m not at that point, but I was wondering when I finish algebra what should I study next? Trig or Geometry?
38. ### Intro Physics Best Physics, Algebra, and Trigonometry Textbooks (Modern)

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?
39. ### Solving trigonometry equation involving half-angle

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
40. ### Solve the given trigonometry equation

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...
41. ### Algebra College Algebra, Mathematics & Trigonometry Textbooks

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.
42. ### Finding area of a non right angled triangle

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...
43. ### Find the unknown values in the problem involving trigonometry graphs

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...
44. ### Find the unknown values in the problem involving trigonometry graphs

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.
45. ### Proving three angles are equal if they satisfy two conditions

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 -...
46. ### Proving Sin(120) = Sin(60) with Trigonometry

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 -...
47. ### Solving Physics Problem with Angles and Trigonometry

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 =...
48. ### How to Solve a Trigonometry Equation Using Identities and Alternative Methods?

Find the Mark scheme solution here; Now find my approach; Using the trig. identities It follows that, ##\frac {1}{\sqrt 2}##⋅ ##sin ∅##+##\frac {1}{\sqrt 2}##⋅ ##cos ∅##=##{\sqrt 3}##⋅ ##cos ∅##+##sin ∅## →##sin ∅##[##\frac {1}{\sqrt 2}##-##1]##=##cos ∅##[##\frac {-1}{\sqrt 2}##+##{\sqrt...
49. ### Other Best trigonometry books for beginners and self study

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!
50. ### MHB Trigonometry Help: Model Daylight Hours in Lowell, MA 2020

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...