What is Trigonometery: Definition and 83 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. brotherbobby

    Two roads meeting a river at different angles (heights and distances)

    Problem statement : I copy and paste the problem as it appears in the text. I hope am understanding its wording correctly. Diagram : The river is shown in blue. The two roads start from the crossing C and end in A and B, making angles ##60^{\circ}## and ##30^{\circ}## resectively. The longer...
  2. C

    Why is this result different? (calculating the sides of a triangle)

    so basically, here is a photo from the textbook(in attachments) and I'll write here how I did it. In my opinion, results should have been the same, but for some reason, they differ. So, if anyone can tell me what I am doing wrong I would appreciate it since I can't find mistakes caused by wrong...
  3. brotherbobby

    A formula involving the sum of cosines of the angles of a triangle

    Problem Statement : The statement appeared on a website where a different problem was being solved. I got stuck at the (first) statement in the solution that I posted above 👆. Here I copy and paste that statement from the website, which I cannot show : Attempt : To save time typing, I write...
  4. MevsEinstein

    B What function satisfies this table?

    I have a table of values (from my own analysis, not from a textbook) that represents a portion of a periodic function: x y 0 30 1 20 2 10 3 10 4 20 5 30 What function satisfies the table? What I know is that the function is periodic. I was thinking I could use cosine because its...
  5. rudransh verma

    Trigonometric ratios of angles above 90 degrees

    I have been doing the resolutions of vectors on x and y-axis with making triangles and reference angles in all quadrants. But I want to calculate now how to find something like ##\sin 235## without the help of reference angles. I know we don’t need to. Calculator and Taylor theorem is handy here...
  6. Franklie001

    Excel Worksheet problems -- Piston and connecting rod connected to a crankshaft

  7. C

    Calculate dimensions of floating laptop stand

    I am planning to build my own laptop stand with an acrylic sheet. The final product should look like the one in the attached image (bottom right) Material: most laptop stands are made with 6mm acrylic thickness, hence this is what I will use. Also I already have this material at home. I am...
  8. D

    AntiNoise Amplitude and Phase

    Summary:: AmbientNoise + AntiNoise combined calculation I am having trouble with this question: Noise cancelling headphones use both passive (insulated earphones) and active (electronic “anti-noise”) methods to nullify ambient noise. One task of a sound engineer is to design low-energy...
  9. pairofstrings

    B Cosine of 1 degree and cosine of 60 degrees?

    Why is cos (1)° = 0.9998? cos(60)° = ½? Thanks.
  10. Z

    Comp Sci Solving the ballistic trajectory equation for Launch Angle

    In a process of writing a game. Effectively need to know how to angle the barrel for the projectile to hit the selected target. So for the equation y = h + x * tan(α) - g * x² / 2 * V₀² * cos²(α) Everything except α is known. Could anyone more wise in the ways of science than me help me solve...
  11. N

    Trigonometry - Horizon Related Word Problem

    Hello! I'm trying to solve this problem. Here's the diagram I tried to make. I have difficulty understanding this math problem.. I've tried to solve the problem using the symmetry of the triangles but I didn't get the right answer, and I can't seem to understand the "concept" of the horizon...
  12. goodOrBad

    Need to find forces in S1 and S2

  13. S

    B Trigonometry (Law of Cosines)

  14. Pushies

    CoM reduced by total body height when bending your knee X degrees

    Lets say that a man with a standing height of 185cm bent his knee 30 degrees, how many centimeters will be reduced from his standing height? Assume his femur length is 60cm and his tibia (shin) length is 50cm. Can anyone give me a hint? I've tried to use trigonometry but i don't think i fully...
  15. B

    Jenkins-White Optics: Relation between Prism/Deviation Angle and Rays

    I've tried to attempt the first part of the problem(spent over an hour on this) as second part could be easily optained with some calculus ,I asked my friend but alas nobody could conjure the solution to this dangerous trigonometric spell. It was just pages and pages of concoction of...
  16. MrDickinson

    Project Motion/Trigonometry Question

    My reasoning and answer is wrong, but I cannot figure out why. Perhaps it is strange, perhaps not, but I want to figure out why my initial method of solving this problem did yield an incorrect answer. I began by creating an equation and drawing a right triangle. x is the horizontal part of...
  17. S

    Half Angle Formula for Tangent

    Hi all, I am a self learner (graduated very long ago and rusty at math) working through the Riley, Hobson and Bence text, chapter 1. 1. Homework Statement Use the fact that ##sin(\pi/6) = 1/2## to prove that ##tan(π/12) = 2 − \sqrt{3}.## Homework Equations ##tan(2x) = \frac { 2 tan(x)} {1...
  18. M

    I Converting from spherical to cylindrical coordinates

    I have the coordinates of a hurricane at a particular point defined on the surface of a sphere i.e. longitude and latitude. Now I want to transform these coordinates into a axisymmetric representation cylindrical coordinate i.e. radial and azimuth angle. Is there a way to do the mathematical...
  19. H

    ##\int (\sin x + 2\cos x)^3\,dx##

    Homework Statement $$\int (sinx + 2cos x)^3dx$$ Homework Equations The Attempt at a Solution $$\int (sinx + 2cos x)^3dx$$ $$\int (sinx + 2cos x)((sinx + 2cos x)^2dx)$$ $$\int (sinx + 2cos x)(1 + 3cos^2x+2sin2x)dx$$ How to do this in simpler way?
  20. S

    Critical angle: find the depth of the fish looking up

    Homework Statement A fish floats in water with its eye at the centre of an opaque walled full tank of water of circular cross section. When the fish look upwards, it can see a fish-eye view of the surrounding scene i.e. it is able to view the hemisphere of the scene above the water surface, and...
  21. J

    A ladder against a wall problem

    Homework Statement A 70 kg window cleaner uses a 16 kg ladder that is 5.6 m long. He places one end on the ground 2.0 m from a wall, rests the upper end against a cracked window, and climbs the ladder. He is 3.5 m up along the ladder when the window breaks. Neglect friction between the ladder...
  22. S

    Fresnel Equations and Snell's Law

    Homework Statement From the Fresnel equations and Snell’s Law, prove that, when θ = θB where tanθB = nt/ni, (θB is the Brewster angle); (a) Reflection coefficient = 0 , and (b) transmission coefficient = n/n’ Homework Equations reflection coefficient = (ntcosθi - nicosθt) / (ntcosθi + nicosθt)...
  23. M

    I How to calculate angular rotation for a 2D line?

    For illustration purposes, I have attached an image of the line with the angle that I want to calculate. I am trying to determine the angle of rotation and the calculation that I am using currently is as below: angle = math.atan2(y,x) I use this formula to calculate the rotation for A and A'...
  24. navneet9431

    Problem in finding a general solution

    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...
  25. Dreezy

    I Help creating a coordinate system for a robotic arm

    Hey guys, I would appreciate some help with the math behind creating a working coordinate system for a robotic arm. I am currently trying to determine what servo angles are necessary to align a robotic arm's claw to the given coordinates. Geometrically simplified, the robotic arm is a...
  26. H

    How to solve this trigonometry equation

    Homework Statement ##\sin a + \cos b## = ##\frac{-1}{2}## ##\cos a + \sin b## = ##\frac{\sqrt 3}{2}## 0 < a < ##\pi/2## ##\pi/2## < b < ##\pi## a + b = ? By calculating sin (a+b) Homework Equations The Attempt at a Solution I tried : ##\sin a + \cos b =...
  27. rishi kesh

    I Why tan x=x as x approaches 0?

    Hi! In one of my textbook i saw the relation tan(x) = x where x is very small value and expressed in radians. I want to know why its true and how it actually works. I would appreciate someone's help :smile:
  28. H

    Find the limit as h --> 0 for this trigonometery equation

    Homework Statement ##\lim_{h \to 0} \frac{f(x - 2h) - f(x + h)}{g(x + 3h) - g(x-h)}## While f(x) = cos x g(x) = sin x Homework Equations The Attempt at a Solution Using L Hopital i couldn't make it more simple. I tried to divide it by cos and sin Can you give me clue?
  29. P

    Kinematics -- flying a plane in the wind to a destination

    Homework Statement A plane is taking off from an airport directly west of the airport it wishes to touch down in. The plane in still air can travel with a constant speed of 730 km/h. If a wind is blowing constantly at 92 km/h [45° S of E], at what angle must the plane fly to compensate for the...
  30. Wrichik Basu

    Intro Math Books on Spherical Trigonometry

    What are some good books to learn spherical trigonometry from basics to the advanced level?
  31. MrGoATi

    Trigonometry: count sin+cos when tg-ctg=-7/12

    Mentor note: Moved thread to homework section ok So I'm doing supposedly easy trigonometry problems. i did the easiest ones. now I have no idea how to solve 2. first one is count sin+cos When tg - (1/tg) = -(7/12) what i figured is that i probably need to use...
  32. shihab-kol

    B Why Does tan x Have This Domain?

    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?
  33. V

    Calulate the distance of objects in Earth's orbit

    Homework Statement I am working on a report dealing with the velocity and acceleration of objects in Earth's surface based on distance from the Earth and thus far I have used the orbital speed equation and the acceleration equation. To get dive deeper into the math I would like to attempt to...
  34. SciencyBoi

    Trigonometric inequality problem.

    Homework Statement Find the solution of the inequality ## \sqrt{5-2sin(x)}\geq6sin(x)-1 ## Answer: ## [\frac{\pi(12n-7)}{6} ,\frac{\pi(12n+1)}{6}]~~; n \in Z## Homework Equations None. The Attempt at a Solution There are two cases possible; Case-1: ##6sin(x)-1\geq0## or...
  35. M

    Finding an angle of a triangle as function of another angle

    1. Find α(β) given that the sum of the 2 sides= ##(x+y)## and its third, ##z## is a constant for 0<β<180. You can imagine that there's two pieces of string connected between two points. One string is as long as the distance between the two points while the other string is longer. If you...
  36. Wrichik Basu

    A problem in Inverse Circular Functions in Trigonometry

    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...
  37. Wrichik Basu

    Problem in finding the General Solution of a Trigonometric Equation v3

    Homework Statement :[/B] Find the general solution of the Trigonometric equation: $$3\sin ^2 {\theta} + 7\cos ^2 {\theta} =6$$ Given andwer: ##n\pi \pm \frac {\pi}{6}## Homework Equations :[/B] These equations may help: The Attempt at a Solution :[/B] Please see the pic below: It...
  38. Wrichik Basu

    Problem in finding the General Solution of a Trigonometric Equation v2

    Homework Statement :[/B] Find the general solution of the equation: $$\tan {x}+\tan {2x}+\tan {3x}=0$$ Answer given: ##x=## ##\frac {n\pi}{3}##, ##n\pi \pm \alpha## where ##\tan {\alpha} = \frac {1}{\sqrt {2}}##. Homework Equations :[/B] These equations may be used: The Attempt at a...
  39. Wrichik Basu

    A problem in finding the General Solution of a Trigonometric Equation

    Homework Statement :[/B] Find the general solution of the Trigonometric equation $$\sin {3x}+\sin {x}=\cos {6x}+\cos {4x} $$ Answers given are: ##(2n+1)\frac {\pi}{2}##, ##(4n+1)\frac {\pi}{14}## and ##(4n-1)\frac {\pi}{6}##. Homework Equations :[/B] Equations that may be used: The...
  40. Vital

    Graph r = 6 cos() issues with plotting on xy-plane

    Homework Statement Hello! Last week I have came here for the help related to this problem. I am creating a new thread to describe the issue more precisely. I will be grateful for your help and explanation. I post the explanation for the book first accompanied by attached pictures, and below I...
  41. Wrichik Basu

    A problem in Trigonometry (Properties of Triangles) v3

    Homework Statement In any triangle ABC, prove that $$a^2 b^2 c^2 \left (\sin {2A} +\sin {2B} + \sin {2C} \right) = 32 \Delta ^3$$ Here ##\Delta ## means the area of the triangle. Homework Equations The Attempt at a Solution
  42. Wrichik Basu

    A problem in Trigonometry (Properties of Triangles) v2

    Homework Statement In any triangle ABC, prove that $$ a^2 + b^2 +c^2 =4 \Delta (\cot {A}+\cot {B}+\cot {C}) $$ Homework Equations The Attempt at a Solution
  43. Vital

    Understanding the size of the angle

    Homework Statement Hello! Please, take a look at the exercise I post below. I have solved it correctly, and I understand how to solve it; so no problems here. But what I do have a problem with is the size of the angle between two points. Please, see details below. I will be grateful for your...
  44. Vital

    Find the height of the tree - understanding the task

    Homework Statement Hello! Please, help me to understand the task - I seem to fail to understand what goes where, and hence cannot proceed to solving the exercise. Please, take a look at the task, and then my questions. The task is on using the law of sines. Before trying to solve it I need to...
  45. Vital

    Simplify the equation - question about the angle

    Homework Statement Hello! Please, take a look at the following equation and help me to understand where the authors got the value of φ = π/3. I don't see where it is derived from as no additional conditions are given. Homework Equations x(t) = 5e(-t/5) cos(t) + 5e(-t/5) √3 sin(t) The...
  46. Vital

    Solving for x in the Inequality cos(x) ≤ 5/3

    Homework Statement Hello! The task is to express the exact answer in interval notation, restricting your attention to -2π ≤ x ≤ 2π. Homework Equations The given inequality: cos(x) ≤ 5/3 The Attempt at a Solution I have only one doubt here, and I don't see my mistake. I see that if cos(x)...
  47. Vital

    Solve the equation, exact solutions in [0, 2π)

    Homework Statement Hello! Please, let me know if I am heading towards a correct path in solving the equation. I get stuck in the middle, and obviously head away from the result presented in the book. Homework Equations cos(2x) = 2 - 5 cos(x) The Attempt at a Solution Gather all on one side...
  48. E

    Values of x for which a geometric series converges

    Need help with a homework question! The question gives: The first three terms of a geometric sequence are sin(x), sin(2x) and 4sin(x)cos^2(x) for -π/2 < x < π/2. First I had to find the common ratio which is 2cos(x) Then the question asks to find the values of x for which the geometric series...
  49. Toby_phys

    Special relativity - transformation of angle

    Homework Statement Homework Equations Gamma factor: $$\gamma = \frac{1}{\sqrt{1-\beta^2}} $$ Lorentz contraction $$l'=\frac{l}{\gamma}$$ Trig: $$ cos\theta = \frac{adjacent}{hypotenuse}$$ The Attempt at a Solution I have all the quantities but the algebra doesn't seem to work out...
  50. T

    Applied Mathematics Problem

    Hi! New to this forum and signed up because I've just started a foundation degree in mechanical engineering, and having been out of education for a very long time beforehand, getting back to grips with mathematics and the like! Anyway, I would like to ask for some advice on this problem, I know...