Trigonometric Definition and 1000 Threads

  1. C

    MHB Simple Trigonometric Identities

    If (sinΘ) = 2/3 with Θ in quadrant 1, find (secΘ)[/SIZE] Θ = theta Completely new at trigonometric identities, would be a great help!
  2. X

    (Question) Trigonometric substitution triangles

    [Prefix] When we do trigonometric substitutions (such as for the integral x^3/(a^2-x^2)^2), we say something like "let x = asinp for -pi/2 <= p <= pi/2" then we carry on and solve the integral. However, sometimes our answer is ugly and we get some term in our expression like "cosp"- so we draw...
  3. anemone

    MHB Can You Prove This Trigonometric Inequality?

    Prove $\tan 6^{\circ}+\tan 24^{\circ}+\tan 42^{\circ}+\tan 86^{\circ}\gt 4$.
  4. K

    Solving this set of trigonometric equations

    I have the following set of equations from which I need to find δ and φ uniquely (i.e without quadrant ambiguity). In other words I need to have expressions for tan δ and tan φ involving A,B,C and D which are known quantities. A=1-sin(squared) δ/2 x sin 4φ B=1+sin(squared) δ/2 x sin 4φ C=...
  5. B

    Sin^4Ө =3/8-3/8cos(2Ө) Prove the following trigonometric identity

    Homework Statement Prove the following trigonometric identity. The question is sin^4Ө =3/8-3/8cos(2Ө) Homework Equations I think I'm supposed to use the power reducing formulas for trigonometric identities which are sin^2(u)= (1- cos(2u))/2 cos^2(u)=(1+cos(2u))/2 *Let u represent any...
  6. manongistong

    Solve Trigonometric Problem for Dynamics: 4th Hint Equation

    This problem is same as the problem on this link https://www.physicsforums.com/threads/trigonometric-problem.76696/ . I would like to ask the number 4 hint which is "4) We therefore have, for example the equality: " the equations can't be seen on my pc as it will only outputs this...
  7. Stephanus

    Proof limit derivatave of sin(x)

    Dear PF Forum, Continuing our debate discussion in differential in slice of X. I read this particulare website. About proofing the derivative of sine(x). http://tutorial.math.lamar.edu/Classes/CalcI/ProofTrigDeriv.aspx In there, the web writes arc AC < |AB| + |BC| < |AB| + |BD|...
  8. karush

    MHB Simplifying a Trigonometric Expression

    $$\frac{\cot^3\left({y}\right)-\tan^3\left({y}\right)} {\sec^2\left({y}\right)+\cot^2\left({y}\right)} =2\cot\left({2y}\right)$$ I tried the LHS but could get it to reduce.
  9. enter

    Getting a triangle from trigonometric function

    How can I get a right triangle from the inputs and outputs of trigonometric functions? For example: sin(x) = y The triangle would have one angle as x and the opposite edge of the triangle would be y/hyp etc. How can I get all of these values from any trigonometric function? Please tell me if I...
  10. P

    Inverse trigonometric functions

    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...
  11. T

    Solving trigonometric equations using compound angle formula

    Hello. I find this whole topic of compound angle formula really confusing. I've been doing equations like cos(60+x) = sinx using the co function identities so far, yet this one seems to be impossible to do using cofunction identities so I need to know how to do it using compound angle formula...
  12. karush

    MHB Integral: I am considering a trigonometric substitution

    $\int\frac{3{x}^{3}}{\sqrt{4{x}^{2}-1}}dx $ I wasn't sure what substitution to use due to what is in radical? $x=\frac{1}{2}\sec^2 \left({\theta}\right)\ dx=\frac{\sin\left({\theta}\right)}{\cos^3\left({}\right)}$
  13. M

    MHB How to Solve Trigonometric Equation sin2x = sqrt(2)/2 Using Algebra?

    Solve sin2x= sqrt(2)/2 (using algebra) the interval is between 0 and 2pi i got the first two answers: pi/8 and 3pi/8, but i don t understand how to get the other two: 9pi/8 and 11pi/8. thanks!
  14. Byeonggon Lee

    Should I memorize all these trigonometric integrals?

    I only memorized these trigonometric differential identities : `sin(x) = cos(x) `cos(x) = -sin(x) because I can convert tan(x) to sin(x) / cos(x) and sec(x) to 1 / cos(x) .. etcAnd there is no need to memorize some integral identities such as : ∫ sin(x) dx = -cos(x) + C ∫...
  15. Cosmophile

    Generalized Trigonometric Intervals

    Homework Statement Identify the intervals of increase/decrease of ##f(x) = \sin x + \cos x## Homework Equations ##f(x) = \sin x + \cos x## ##f'(x) = \cos x - \sin x = \sqrt 2 \cos(x+\frac {\pi}{4})## The Attempt at a Solution ##f## is increasing when ##f'(x) > 0## ##\sqrt 2 \cos(x+\frac...
  16. S

    Summation of a trigonometric function

    Homework Statement By considering ∑z2n-1, where z=eiθ, show that Σcos(2n-1)θ=sin(2Nθ)/2sinθ. (Σ means summation from 1 to N)Homework Equations Just a guess. S=a(1-r^n)/(1-r) The Attempt at a Solution I was thinking this but it doesn't seem to work very well...
  17. M

    Master Trigonometric Identities with Double Angle Techniques

    I am doig trigonometric identities and i got this one, (all will be in the picture the solution and my work) i used the double angle for this but i am afraid i didn't get the exact idea, just guessing, good guessing, so i want to know how is the proper way to reach the solution
  18. terryds

    How Do You Solve This Trigonometric Limit Problem?

    Homework Statement [/B] ##\lim x\rightarrow \frac{\pi }{4} (\frac{1-\tan x}{\sin x - \cos x})## The Attempt at a Solution [/B] By assuming y = x-π/4 , the limit become : ## \lim y\rightarrow 0 (\frac{1- \tan (y+\frac{\pi}{4})}{\sin (y+\frac{\pi}{4}) - \cos (y+\frac{\pi}{4})}) = \lim...
  19. M

    Transforming Trigonometric Equation

    Homework Statement In 2001, Windsor, Ontario received its maximum amount of sunlight, 15.28 hrs, on June 21, and its least amount of sunlight, 9.08 hrs, on December 21 Due to the Earth's revolution about the sun, the hours of daylight function is periodic. Determine an equation that can...
  20. W

    Trigonometric functions (identity&equations)

    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 +...
  21. I

    Apparent trigonometric inconsistencies

    a is the opposite side b is the adjacent side c is the hypotenuse. x is the angle Problem: expression for sin x as a function of a and c. Solution: Identities taken from textbooks: sin x = a/b (1) tan x = a/c (2) sin x = sqrt(tan^2 x / (1 + tan^2 x)) (3) substituting (2)...
  22. terryds

    What Is the Limit of This Trigonometric Expression as x Approaches π/2?

    Homework Statement Find the limit of : lim x-> (π/2) (2-2sin x)/(6x-3π) 2. The attempt at a solution lim x-> (π/2) (2-2sin x)/(6x-3π) =lim x-> (π/2) 2-2 sin x / 6 (x- (1/2)pi) Assuming that y = x - (π/2) So, lim y->0 (2-2sin(y+pi/2))/6y lim y->0 (2-2 (sin y cos pi/2 + cos y sin pi/2)/6y...
  23. R

    Finding the sum of inverse trigonometric expression

    Homework Statement Find ## \sum_1^{23} tan^{-1}(\frac{1}{n^2+n+1}) ## Homework Equations ## tan^{-1}x + tan^{-1}y = tan^{-1}(\frac{x+y}{1-xy} )##The Attempt at a Solution I think we have to split the question in a form of relevant equation given above. First thing what should I do?
  24. H

    Trigonometric Circle : f(x) = cos(2x+1)

    Homework Statement I have the function f(x) = cos(2x+1) I need to resolve f(x) = 0.6 when -PI <= x <= PI then put my answers in a Trigonometric Circle Homework EquationsThe Attempt at a Solution First I do this: cos(2x+1) = 0.6 and use my TI calculator to solve for X and I find the following...
  25. iwantcalculus

    Questions on Trigonometric equation solving

    Hello reader, I have an exam really soon and it includes a good bit of trigonometry, but I'm having problems with the trig stuff because this exam does not allow calculators and since I was dependent on the calculator, I haven't memorized anything about the trigonometric functions. I don't know...
  26. A

    This is the question that we will explore in this webpage.

    Is there any way I can reduce or simplify expressions like cos(sin-1(x)), sin(cos-1(x)), cos(tan-1(x)), tan(cot-1(x)) etc.? (I refer to the arc functions, i.e. inverses, by the superscript -1, not the reciprocals)
  27. R

    A definite trigonometric absolute integral

    Homework Statement Find ## \int_0^{nπ+v} |sinx| dx ## Options are, 2n+ 1+ cosv 2n+1-cosv 2n+1 2n+cosv Homework Equations Integration of sinx is -cosx.The Attempt at a Solution Sin x is + ve from 0 to π, Negative from π to 2π We can make |sinx| as sinx in 0 to π, And - sinx in π to 2π But here...
  28. anemone

    MHB Simplify a trigonometric expression

    Simplify $\left(\sqrt{3}\sec \dfrac{\pi}{5}+\tan\dfrac{2\pi}{5}\right)\tan \dfrac{2\pi}{15}$.
  29. MarkFL

    MHB Solve Trigonometric Sum: Find S Value

    It can be shown that the following sum: S=\sum_{k=1}^{89}\left(\sin^6\left(k^{\circ}\right)\right) is rational. Find the value of $S$. (Callme)
  30. T

    Rewrite the trigonometric expression as an algebraic equation

    What is this asking for? Is it asking for the trig expression in the form of x and y? Or is it asking for something else?
  31. E

    Proving a Trigonometric Identity

    Homework Statement Prove that: \cos^6{(x)} + \sin^6{(x)} = \frac{5}{8} + \frac{3}{8} \cos{(4x)} Homework Equations I am not sure. I used factoring a sum of cubes. The Attempt at a Solution I tried \cos^6{(x)} + \sin^6{(x)} = \cos^4{(x)} - \cos^2{(x)} \sin^2{(x)} + \sin^4{(x)} . But I...
  32. R

    Mean value theorem for trigonometric function

    Homework Statement Verify Lagrange's MVT for ## f(x)= sinx - sin2x ## in [ 0, π ] Homework Equations ## f'(c) = \frac{f(b)-f(a)}{b-a} ## The Attempt at a Solution Got on solving cosx= 2cos2x How to find c lies in [0, π ]? Solved it using quadratic equation but it gives a complicated value...
  33. A

    Integration Using Trigonometric Substitution Help Needed

    Homework Statement Integral of $$ x^3\sqrt{x^2+16}dx $$ answer should give $$ 1/5(x^2+16)^{5/2} -16/3(x^2+16)^{1/2}+C $$ Homework Equations x=atanθ The Attempt at a Solution Mod note: The integral is ##\int x^3 \sqrt{x^2 + 16} dx## The published answer is ##1/5(x^2+16)^{5/2}...
  34. L

    MHB How to Solve Trigonometric Functions When \( \cos(t) = -\frac{9}{10} \)?

    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...
  35. W

    MHB Find the Smallest Angle for sin theta+5cos theta=4 | Trigonometric Equation Help

    Can anyone help with this trig question, Determine the smallest angle in degrees such that sin theta+5cos theta=4 Iknow i need to use the quadratic formula but really stuck on it.
  36. Math Amateur

    MHB HIGHLY Rigorous Treatment of the Trigonometric Functions

    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...
  37. Eclair_de_XII

    Can someone check my solution to a trigonometric equation?

    Homework Statement 4sec2θ tanθ = 8 tanθ Homework Equations sec2θ = tan2θ + 1 The Attempt at a Solution 4(tan2 + 1) tanθ = 8 tanθ (4tan2 + 4) tanθ = 8 tanθ 4 tan3 + 4 tanθ = 8 tanθ 4 tan3 - 4 tanθ = 0 4tanθ(tan2θ - 1) = 0 4tanθ = 0 tanθ = 0 tan-1(tanθ) = tan-1(0) tan-1θ = 0, π tan2θ - 1 = 0...
  38. B

    Center on mass trigonometric trick

    hey there I am doing a lot of center of mass problem most of them with collisions , and most of the solutions I see, have a big part of "trigonometric trciks", like the way u watch the vectors in cm frame and the way in the lab frame. does anyone met this kind of solutions and have some tricks...
  39. E

    MHB Proving a trigonometric identity II

    Hi, I need help proving the following trig identity, (2sinx)\overline{secxtan(2x)}=2cos^2x-csc^2x+cot^2x I have tried starting from the left hand side, the right hand side, and doing both together, but nothing seems to work. One of the ways I tried: LHS...
  40. E

    MHB Proving a trigonometric identity

    Hi, I need help proving the following trig identity: \frac{\cot^2(x)-\cot(x)+1}{1-2\tan(x)+\tan^2(x)}=\frac{1+\cot^2(x)}{1+\tan^2(x)} Me and my friend have spent several hours determined to figure this out, starting from the left hand side, the right hand side, and doing both together, but...
  41. W

    Trigonometric Identities and equations

    1) Problem statement: Solve the trigonometric equation for the domain is. 0°<x<360° 5(sinx - cosx) = 4sinx - 3cosx 2) Relevent equations: secx=1/cosx cosec x=1/sinx cot x= 1/tanx tan x=sinx/cosx cot x=cosx/sinx sin^2 x + cos^2 x=1 1 + tan^2 x= sec^2 x 1+ cot^2 x = cosec^2 x Template of answer...
  42. M

    MHB Proving a trigonometric identity

    How prove $\cos\frac{8\pi}{35}+\cos\frac{12\pi}{35}+\cos\frac{18\pi}{35}=\frac{1}{2}\cdot\left(\cos\frac{\pi}{5}+\sqrt7\cdot\sin\frac{\pi}{5}\right)$?
  43. A

    MHB Simplifying a trigonometric expression

    (tg 570* sin 1469) / (ctg 495 * cos 781) = Its in degrees
  44. Euler2718

    Determining Missing Trigonometric Ratios

    I'm not stuck per say, but I need to know if I have the right idea for solving the rest of these questions. 1. Homework Statement For the following given trigonometric ratio and domain, determine the missing trigonometric ratio. Homework Equations cos\theta = -\frac{1}{2} ...
  45. Ganesh Ujwal

    How do I prove this seemingly simple trigonometric identity

    Mod note: Fixed the LaTeX. ##a=sinθ+sinϕ## ##b=tanθ+tanϕ## ##c=secθ+secϕ##Show that, ##8bc=a[4b^2 + (b^2-c^2)^2]## I tried to solve this for hours and have gotten no-where. Here's what I've got so far : ##a= 2\sin(\frac{\theta+\phi}{2})\cos(\frac{\theta-\phi}{2}) ## ## b =...
  46. jacobi1

    MHB Trigonometric absolute value integral

    Evaluate \lim_{n \to \infty} \int_0^1 | \sin(nx)| \ dx.
  47. D

    HELP with trigonometric identites

    Okay so I'm solving this equation: 2sin(x)-3cos(x)=1 I'm solving it by equating the left hand side to a single trigonometric equation. What I did first: I equated the left hand side to a sine function. So 2sin(x)-3cos(x)=Rsin(x+a). Then I used the addition formula. Rsin(x+a)=...
  48. I

    Find the following fourier series in trigonometric form

    Homework Statement Find the following Fourier series in trigonometric form. Homework Equations $$y(t)=a_0+\sum\limits_{n=1}^{\infty} a_n cos(n\omega_{0}t)+b_n sin(n\omega_{0}t)$$ The Attempt at a Solution The graph above is represented by the function: $$ x(t) = \left\{ \begin{array}{ll}...
  49. V

    Complex trigonometric integral

    Homework Statement Calculate the complex integral along the closed path indicated: $$ \oint_C\frac{\sin{z}}{z^2+\pi^2}dz,\,\,|z-2i|=2.$$ Homework Equations $$ \sin{z}=\frac{e^{iz}-e^{-iz}}{2i} $$ $$ e^{iz}=e^{i(x+iy)}=e^{-y+ix}=e^{-y}(\cos{x}+i\sin{x}) $$ The Attempt at a Solution I really...
  50. DavideGenoa

    Is There an Error in Kolmogorov and Fomin's Trigonometric Inequality Proof?

    I read that, for ##\delta>0##, if ##\delta<z\leq\pi##, then ##\sin\frac{z}{2}\geq\frac{2\delta}{\pi}##. I cannot prove it. I know that ##\forall x\in\mathbb{R}\quad|\sin x|\leq |x|##, but that does not seem useful here... Thank you so much for any help!
Back
Top