What is Trigonometric: Definition and 1000 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. Y

    MHB Solving Trigonometric Limit w/o L'Hospital Rule

    Hello, I need some help with this limit, I have no clue how to do this with the next constraint: No use of L'hopital rule... Thank you ! \lim_{x\to1}\frac{sin(x^{3}-1)}{x-1}
  2. H

    MHB Solving a Trigonometric Equation

    2Cos (3/10x) - 3Sin(1/2x)=0 How would I go about solving this equation? I do not even know where to begin.
  3. M

    MHB Solving a Trigonometric Equation for Cos (x)

    Hi, may you help me? I have a question I solved but couldn't ended. If cos(x)+5Sec(x)=1.1 then Cos(x)=? 2Cos(x)+5Sec(x) =11/10 2Cos(x)+5/Cos(x)=11/10 2Cos(x)*Cos(x)+5=11/10*Cos(x) 2Cos^2(x)+5=11/10*Cos(x) 2Cos^2(x)-11/10-Cos(x)-5=0 then i used delta=b^2-4ac then i found delta=-4.79...
  4. M

    MHB Finding the Range of a Trigonometric Function

    Hello, I have some questions and i couldn't solve them can you help me? If y=5cos(x)+3 then what is the heap of ? (a) All real numbers (b) alpha<= y <= alpha (c) -2 <= y <= 10 ( d)-2 <= y <= 8 What is the solution? Thank you.
  5. T

    Solving Equations Using Trigonometric Identities

    Hey PF! Homework Statement Find exact solutions for the following equations over the domain 0 ≤ x <2π 2sinx = 3 + 2cscx Homework Equations sin2+cos2=1 The Attempt at a Solution 2sinx = 3 + 2cscx 2sinx = 3 +2(1/sinx) sinx = 3/2 + 1/sinx sinx - 1/sinx = 3/2 (1-1-cos2x)/sinx = 3/2 -cos2x/sinx =...
  6. D

    MHB Simplification of Trigonometric Expression

    $$ \cos[k_n(L-\varepsilon)]-\cos k_nL $$ I just read this for small epsilon this result can be further simpli ed by using the trigonometric identity for $\cos[k_n(L-\varepsilon)]$. Doing so result is $$ 2\sin k_nL $$ I don't see this? Can someone explain?
  7. T

    MHB Solving a Trigonometric Equation

    solve for tan^4x + tan^2x = sec ^4x - sec^2x i solved and ended up with RIHS= tan^4x?
  8. A

    Trigonometric function and complex exponential

    1. Homework Statement - multiplication of trigonometric function and complex exponential 2. Homework Equations the question is, Akcos(ωt) × e-jωt 3. The Attempt at a Solution it is, Ak/2 + (Ak/2)e-j2ωt ? by using cos(ωt) = 1/2ejωt + 1/2e-jωt
  9. M

    Trigonometric substitutions

    Homework Statement Hi I need help with the following integral. \int_0^2 \! \frac{1}{(x^2+4)} \, dx Homework Equations I believe that these are both trigonometric substitutions. However, these are the simplest in my textbook and I can't even understand them. :-( The Attempt at a Solution...
  10. K

    Trigonometric Identity Problem

    Homework Statement Prove the Identity sinθ/(1+cosθ) = 1-cos(θ)/sinθ Homework Equations sinθ/cosθ = tanθ sin^2θ + cos^2θ = 1 The Attempt at a Solution sinθ/(1 + cosθ) = LS cosθtanθ/(1+cosθ) = LS cosθtanθ/(sin^2θ + cos^2θ + cosθ) = LS cosθtanθ/(tan^2θcos^2θ +...
  11. alane1994

    MHB Area between two trigonometric curves.

    http://www.mathhelpboards.com/f12/im-clueless-how-start-2322/My math class uses an online homework system. I got the answer wrong to the question, but I can get a similar question. Here is one that is similar to the earlier one. You have two functions that are graphed. y=\frac{\csc^2{x}}{4}...
  12. alane1994

    MHB Solving a Trigonometric Equation

    I have this as part of a calculus problem. I guess I am a little rusty on this. Any help would be appreciated. 4\cos^2{x}=\frac{\sec^2{x}}{4} I believe that I need to solve for cosx. And then determine what cosx is in radians. That should be the lower limit of the integration. I know how it...
  13. M

    Some trigonometric, exponential thing?

    How can we say: f(x)=A'sin(kx)+B'cos(kx) or equivalently f(x)=Ae^{ikx}+Be^{-ikx}?? How are these two equivalent knowing that e^{ix}=cosx+isinx I don't get this?
  14. F

    Solving Trigonometric integrals using cauchy residue theorem

    Homework Statement evaluate the given trigonometric integral ∫1/(cos(θ)+2sin(θ)+3) dθ where the lower limit is 0 and the upper limit is 2π Homework Equations z = e^(iθ) cosθ = (z+(z)^-1)/2 sinθ = (z-(z)^-1)/2i dθ = dz/iz The Attempt at a Solution after I substitute and...
  15. F

    Summation of a trigonometric series

    Homework Statement Find the limit of the series \lim_{n \rightarrow \infty} \sum_{i=1}^{n} cos (i \theta / n) , 0≤θ≤π/2 Homework Equations The Attempt at a Solution I know that the expansion looks like \cos \theta / n + cos 2 \theta / n + ... + cos \theta , but I couldn't begin...
  16. O

    Solving a general complex trigonometric equation

    Hello everyone! Homework Statement Solving a relatively general equation representing a constant combination of 2 trigonometric functions.Homework Equations a\cos{(vx+p)}+b\cos{(ux+q)}=cThe Attempt at a Solution I really don't have any idea for a general solution to this equation.. Best Regards
  17. Z

    Trigonometric Function Homework: Satellite Orbit Calculations

    Homework Statement A satellite follows a path that may be modeled by the function H(t)= 350 sin 28∏(t-25) + 400 Time is measured in days. Homework Equations The Attempt at a Solution A) Determine the period of the satellite: 2∏*1/28∏ = 1/14 days B) How many minutes will it...
  18. Feodalherren

    Trigonometric Identity Homework: Solving with Sin and Cos Formulas

    Homework Statement Homework Equations Any trig formulas The Attempt at a Solution The yellow paper is me switching everything to sin and cos to see if that helps but it doesn't. I'm completely stuck here.
  19. Feodalherren

    Trigonometric Equation: Solving Sin 2x = -2cos x in the Interval 0 ≤ x < 2∏

    Homework Statement All solutions in the interval 0 ≤ x < 2∏ Sin 2x = -2cos x The Attempt at a Solution I went about it in two different ways. The first way: I simply used the double angle formula to get 2sinx cosx + 2cos x = 0 From this we get that whatever makes cos x = 0...
  20. Z

    Dumbfounded by this trigonometric equation.

    Homework Statement A beach ball is riding the the waves near Tofino, BC. The ball goes up and down with the waves according to the formula h= 1.4 sin(∏t/3), where h is the height in meters above sea level, and t is the time in seconds. A) In the first 10s when is the ball @ sea level. B)...
  21. M

    Simplifying a trigonometric product

    Was just double checking if we can simplify this in the following way, ∫ sinx * √(1 + cos2x) dx = ∫ sinx * (1 + cosx) dx
  22. K

    How to write powers of inverse trigonometric functions?

    Does ##(\sin^{-1}\theta)^2 =\sin^{-2}\theta## ?
  23. J

    Is there a known formula for integrating cos^5x sin^5x?

    ∫cos^5 x sin^5 x dx I thought I would try to solve this by first doing: ∫(1 - sin^2 x) cos^3 x sin^5 x dx but would like to know if that's right. [Edit] Is the answer something like (sin^6 x) / 6 - (sin^8 x) / 4 + (sin^10 x) / 10 ?[Edit]
  24. R

    Trigonometric substitution. Pretty confused where constant comes from. (fixed)

    Find ∫from .6 to 0 x^2/ sqrt(9-25x^2) dx My teacher worked this on the board a little confused O obviously the trig sub is asintheta. But it isn't in the right form yet. So get it there you pull out a 25 --) sqrt(25(9/25) - x^2 ) 5sqrt((9/25)-x^2 so x= (3/5) sintheta so dx = 3/5costheta. so...
  25. M

    Signs of the trigonometric functions of the angle -200

    Determine the signs of the trigonometric functions of the angle -200° in standard position. sin(-200°) cos(-200°) tan(-200°) csc(-200°) sec(-200°) cot(-200°) Information: Q1= (All Positive) Q2= (sin csc) Q3= (tan cot) Q4= (cos sec) Answer: sin(-200°) = Positive cos(-200°) =...
  26. M

    Values of the six Trigonometric Functions

    Find the values of the six trigonometric functions of an angle θ in standard position whose terminal side is containing the points (-3,0) Sinθ= Cosθ= Tanθ= Cscθ= Secθ= Cotθ= I believe the following are correct But I am not sure, Please give insight Sinθ= 0/3 Cosθ= -3/3 Tanθ= 0/3...
  27. B

    Limits Involving Trigonometric Functions (identities)

    Does anyone know of websites where I can find many problems on the topic in the title line (my textbook has far too few)? Thanks!
  28. R

    Trigonometric Methods, using instantaneous value of current

    Homework Statement The instantaneous value of current, i amp, at t seconds is given by: i = 15 sin(100π.t + 0.6) Find the value of; a) amplitude b) period c) frequency d) initial phase angel e) value of i when t = 2.5s f) time when current first reaches maximum value Homework...
  29. Feodalherren

    Mastering Trigonometric Identities: A Comprehensive Guide

  30. P

    Trigonometric Equations Solution

    Hi, I had to solve without the aid of a calculator: 2cos(x) = sin(2x) What I did was perform the following substitution: 2cos(x) = 2sin(x)cos(x) I then canceled and got: sin (x) = 1 For 0 to 3pi, I got two answers, pi/2 and 5pi/2. I did not get one of the answers. Why...
  31. A

    Trigonometry Word Problem involving Inverse trigonometric functions

    Homework Statement A ship is d feet from a dock (horizontal distance). The dock is 40 feet above sea level. The angle of depression from the dock to the ship is θ. Write θ as a function of d. Homework Equations This question is accompanied with choices. The choices are (A) θ = 40arctan(d) (B)...
  32. A

    Solution of Trigonometric Equations

    Homework Statement Find the solution set of each of the following equations for the interval 0° ≤ x ≤ 360°. sin^2x = sinx Homework Equations Trigonometric Identities for Sine. The Attempt at a Solution This is my attempt so far: \sin^2x = sinx \sin^2x - sinx = 0 \sin^2x - sinx + \frac{1}{4}...
  33. A

    Finding x in an inverse trigonometric equation

    Homework Statement Given y = 2/3(arctan \frac{x}{4}), find x. Homework Equations Inverse trigonometric identities for tan. (I guess)The Attempt at a Solution I never attempted because it seems impossible for me. You think this is possible? Thank you!
  34. A

    A question about trigonometric identities

    Is this identity possible? cot 2x = \frac{cos3x + cosx}{sin 3x + sinx} Thanks!
  35. A

    Why Does arccos(x) - arcsin(1-x) Equal 90° Have No Solutions?

    Homework Statement Find the solution set of the equation below for the interval 0° ≤ x ≤ 360°: arccos(x) - arcsin(1-x) = 90° Homework Equations all given above. The Attempt at a Solution i tried to consider (1-x) as "complementary" but I'm not really sure about that and I would...
  36. A

    Proving Trigonometric Identities problem

    Homework Statement Verify that \frac{cosθ}{1-tanθ} + \frac{sinθ}{1-cotθ} = sinθ + cosθ is an identity.Homework Equations - Reciprocal Trigonometric Identities - Pythagorean Trig IdentitiesThe Attempt at a Solution Every time I try to manipulate the LHS of the equation I always get -1 and as far...
  37. A

    Proving Trigonometric Identities

    Homework Statement Prove that sin6 + cos6 = 1 - 3sin2cos2 Homework Equations (1) The Attempt at a Solution I tried to convert those all in terms of sine and cosine only but it didn't work.
  38. A

    Inverse Trigonometric Equation problem

    EDIT: 1.Homework Statement The problem asks to simplify this: cos(arcsin x)Homework Equations let arc sin x = θThe Attempt at a Solution EDIT: I have to edit because I received a warning for not posting for #3 and #2. I originally posted "cos(arcsin x)" for #2 because that was what I know...
  39. P

    Trigonometric Limit Problem: Finding the Limit as x Approaches 0

    Homework Statement find the limit as x approaches 0 of 3sin4x / sin3x Homework Equations sin2x = 2sinxcosx lim as x ->0 of sinx / x =1 The Attempt at a Solution sin4x = 2 sin2x cos2x
  40. D

    Proving trigonometric identities

    Homework Statement Prove that: (1-tanθ)/(1+tanθ)=(cotθ-1)/(cotθ+1) Homework Equations Trig Identities: tanθ= sinθ/cosθ cotθ= cosθ/sinθ 1+tanθ=secθ 1+cotθ=cosecθ The Attempt at a Solution These sorts of equations are coming up a lot and I am having trouble understanding...
  41. S

    Trigonometric integration question

    The question asks to find ∫secxtan2x I rewrote tan2x as (sec2x-1). Then I expanded the equation having sec3x-secx and I know the integral of secx which is 0.5ln|tanx+secx|, but my question is, is integrating sec3x by parts the correct path? or not? Thanks
  42. S

    Help with trigonometric integration problem

    The problem is to find \int(cos5x)/(\sqrt{sinx}) I rewrote cos5x as cosx(1-sin2x)2, finally getting: (cosx-2cosxsin2x+cosxsin4x)/\sqrt{sinx} and I don't even know if this is the correct path, so, any advice would be appreciated.
  43. D

    Trigonometric identities hard question

    Homework Statement Simplify the following: sin(b)/cos(b) + cos(b)/sin(b) Homework Equations Trigonometric identities The Attempt at a Solution Ok so i have no clue how to do this,I keep trying but can't seem to get the right answer, I have tried to do this: sin(b)/cos(b) +...
  44. L

    Solutions of trigonometric equations

    [b]1. The problem statement: QUESTION:the general solution of equation (√3 - 1)sin∅+(√3 + 1)c0s∅=2 is 1) ∅=2nπ+- (π/4)+(π/12) 2) ∅= nπ + (-1)^π (π/4)+(π/12) 3) ∅=2(n+1)π (π/4)+(π/12) The Attempt at a Solution ACTUALLY I DONT NEED THE ATTEMPT, i know the answer is option 1 , i don't...
  45. S

    MHB Help on Trigonometric sums. (Assorted type)

    I have some questions and doubts in trigonometry. I hope somebody can solve these questions. Q1) If for real values of x, cos\theta = x +\frac{1}{x}, then a) \theta is acute angle b) \theta is right angle c) \theta is an obtuse angle d) no value of \theta is possible I will post the following...
  46. J

    MHB Number of Real Solutions for Trigonometric Equation

    Number of real solution of the equation $\sin (\sin (\sin x)) = \cos (\cos (\cos x))$ where $ 0 \leq x\leq 2\pi$
  47. M

    Solving Trigonometric Equations: sin(θ) = 3/5

    Spending my summer studying on my own, and I have come across some things which I would like to see how they are solved. If you are capable of completing, please show work instead of just answers so that I can follow what you did, and take notes on it. Homework Statement If sin(θ) =...
  48. H

    Limit of a trigonometric integral

    Homework Statement Excersice of Big Rudin: ch. 4, prob. 9. If A\subset [0,2\pi] and A is measurable, prove that \lim_{n\to\infty}\int_{A}\cos\,nx\,dx=0 Homework Equations Bessel's inequality The Attempt at a Solution I give my solution, but I post because I think that my answer...
  49. J

    Proving the trigonometric identity

    Homework Statement To prove that \sum over m=1 to 15 of sin(4m-2) = 1/4sin2, where all angles are in degress Homework Equations The Attempt at a Solution Tried to solve it using identity sinx+siny=2sin((x+y)/2)cos((x-y)/2)..but all attempts failed..help
  50. D

    Limit proof of trigonometric function

    Homework Statement Prove that limh→0[(cos(h-1))/h]=0. Homework Equations Half angle formula From an example: limh→0[(cos(h-1))/h] = limh→0[-(2sin2(h/2))/h] They state that they use the half-angle formula in the following way: cosh= 1-2sin2(h/2) I'm not really sure how they are getting...
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