What is identities: Definition and 422 Discussions

In trigonometry, trigonometric identities are equalities that involve trigonometric functions and are true for every value of the occurring variables for which both sides of the equality are defined. Geometrically, these are identities involving certain functions of one or more angles. They are distinct from triangle identities, which are identities potentially involving angles but also involving side lengths or other lengths of a triangle.
These identities are useful whenever expressions involving trigonometric functions need to be simplified. An important application is the integration of non-trigonometric functions: a common technique involves first using the substitution rule with a trigonometric function, and then simplifying the resulting integral with a trigonometric identity.

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  1. M

    Location of proofs of trigonometric identities

    Where would be the best place to find every trigonometric identity, from sin[2] + cos[2] = 1, to the matrix identities (and Euler's equation would be helpful, also) Also the location of mathematical analysis symbols would be helpful, also. Thank you very much in advance :)
  2. S

    Please, check my work on Trig Identities

    Please, check my work. Homework Statement a) Show that sqrt{[1+tan^2x]/[1+cot^2x]}=tanx b) Show that [cosx+sinx]/[cosx-sinx]=1+[2tanx]/[1- tanx] c)Show that cotxcosx+tanxsinx=(cosecx+ secx)(1-sinxcosx) d) Show that cosec^2x-cosecx=cot^2x/[1+sinx] e) Show that sin^3x-cos^3x=...
  3. N

    Proof the trig sum and difference identities

    My homework is to find the sin or cos value of angle that is not directly known on the unit circle. So of course we are given an equation which adds or subracts known values to get the desired one. The problem is that I don't think memorizing them is helping me learn. I want to know how the...
  4. C

    Solving Trigonometric Identities (A bit Hard)

    Alright, well I am having a difficult time getting these equations to equal out! I keep hitting trouble and have hit a road block. Here is the problem... csc(x)+cot(x) = cot(x)csc(x) tan(x)+sin(x) I am just going to work on the LEFT PART of the equation. 1 + cos(x) sin(x)...
  5. P

    Trigonometric Identities: How to Find Cos x and Tan 2 x given Sin x = 2/√13

    Hi, I am given that, for π/2 < x < π, sin x = 2/√13 a) Find Cos x b) Find tan 2 x So, what I did was: I drew a triangle and found that the missing side was equal to 3. From then, I deduced that cos x was equal to 3/√13 The problem was however that the angle must lie between the...
  6. S

    Learning Trig Identities: Deriving the Essentials

    Hello all, I'm wanting to learn how to derive all of the trig identities (well, not all, but the most common) rather than memorizing them. Perhaps someone here could provide me with a list of "essentials" that are the framework for deriving others. For example, I know there are a few that can...
  7. B

    Prove the Following Vector Identities Part 1

    Homework Statement \displaystyle \frac{d}{dt} (\vec u (t) \vec v (t))= \vec u (t)' \vec v(t)+\vec u(t) \vec v(t)' I know this is a product rule on the RHS but how does one prove it? Thanks
  8. N

    Ward-Takahashi identities from symmetry

    Does anyone know how to derive the Ward-Takahashi identity for a field starting from a known conserved Noether current (or equally helpful, from a known symmetry transformation of the Lagrangian)? It'll probably be enough to allow me to do it for myself if you could explain quantitatively what...
  9. L

    Solving Sin(2α)=-8/17 Using Identities

    Homework Statement Sin(2α)=-8/17 Using either double angle or half angle identities, sin,cos,tan,csc,sec and cot of alpha. Homework Equations The Attempt at a Solution I tried and tried BUT i am lost because in order to find cos of alpha using identies, you still have to have...
  10. R

    Vector calculus identities and maxwell equations

    so we have the identity \nabla\times\nabla\phi = 0 and from Maxwell's equations we have \nabla\times \textbf{E} = -\frac{d\textbf{B}}{dt} But we also have that \textbf{E} = -\nabla\phi So the problem I'm having is this -\textbf{E} = \nabla\phi which i substitute into the...
  11. P

    Evaluating trig identities without calculator

    How do you do stuff like sin(pi/4) in your head or sin(1) or sec(pi(x))? Thanks for your help. I'm in Calc II and fully understand the calculus but my trig foundation from high school isn't the best.
  12. P

    What are the essential trig identities for solving calculus problems?

    I am currently taking Calc II and we are doing trig substitutions at the moment. I seem to have forgotten almost all the important trig identities. I understand that this sounds like something I should google but I honestly cannot find a good list of the trig identities I should know at this...
  13. jfy4

    Deriving EFE from the Bianchi Identities

    Hi, I ran this by my friend awhile ago and I'm not sure how to feel about it still... Consider the following: We know \nabla_\alpha G^{\alpha\beta}=0\implies \partial_\alpha G^{\alpha\beta}+\Gamma^{\beta}_{\gamma\delta}G^{ \gamma \delta}+\Gamma^{\delta}_{\delta\gamma}G^{ \gamma...
  14. S

    Proof of inverse trigonometric identities

    Homework Statement Show that arcsin(1/sqrt(5)) + arcsine(2/sqrt(5)) = Pi/2 Homework Equations The Attempt at a Solution Can someone please give me so much as a hint?
  15. R

    Identities for atan(a+b) or atan(a*b)

    Hi, Does anyone know of any nice identities for the inverse or arc tangent of a sum or a product? Thanks, rhz
  16. H

    How to Find Maximum and Minimum Values of Sin^4theta + Cos^4theta

    Homework Statement prove that sin^4theta + cos^4 theta =1/4 (3+cos4theta).hence,find the greatest and least value of sin^4theta + cos^4 theta. please give hints to start . no idea at all...i start from the RHS and get 2cos^4 theta - 2 cos^2 theta +1 , then how sin^4theta + cos^4 theta...
  17. N

    Trouble with trig identities in solving integral.

    Homework Statement \int\frac{sec^{2}x}{tan^{4}x}dxHomework Equations The Attempt at a Solution I have the answer as -1/3 cot^3(x) + C listed. All the intermediate steps are given, but the first one is they have converted the sec/tan integral in to: \int{cot^{2}x}{csc^{2}x} dx I am a little...
  18. N

    Solving Trig Identities: tan^2x - sin^2x = sin^2x*tan^2x Explanation

    Homework Statement tan^2x - sin^2x = sin^2x*tan^2x Homework Equations The Attempt at a Solution This is how my teacher solved it. Sin^2x sin^2x ______ - ______ Cos^2x 1 sin^2x - cos^2x*sin^2x _________________________ cos^2x = sin^2x (1-cos^2x)...
  19. N

    How to Prove Trig Identity m) Sin^2 x / sin^2 x + cos x^2 = tan^2 x / 1 + tan^ x

    Homework Statement Prove this identity m) Sin^2 x / sin^2 x + cos x^2 = tan^2 x / 1 + tan^ x Homework Equations http://i52.tinypic.com/105tdtk.jpg Letter (m) on the top The Attempt at a Solution I don't know to much about trig identities, he barely taught anything. But...
  20. romsofia

    Identities between exponential and logarithmic functions?

    Hello! I was wondering if they're any identities between exponential and logarithmic functions? Maybe identities isn't the right word, but what I'm talking about is something like euler's formula. Other than than {e^x} and {ln(x)} are inverses, if that counts. Any help is very much appreciative!
  21. nukeman

    Identifying Triogonometric Identities - ?

    Homework Statement I am having issues with these. Does anyone know of a tutorial for these, like something I can print out that has sample problems with solutions on how to solve them? Thanks very much! (example problem question) (sinx + cosx)^2 = sin 2x + 1 I am just having a...
  22. nukeman

    Having Trouble: Solving a Trigonometric Equation using Identities ?

    Having Trouble: Solving a Trigonometric Equation using Identities ? Homework Statement I am having problems with these. Need some help please! Here is a sample question I am having troubles with: 3 cos θ + 3 = 2 sin^2 θ, 0 <= θ < 2pi I am really struggling with these. Any help...
  23. nukeman

    Pythagorean Identities - finding sin and cos ?

    Homework Statement For tan(theta) = /2/3 (theta in quadrant 2) find sin(pheta) and cos(pheta) using the pythagorean identities. Homework Equations The Attempt at a Solution Which 2 pythagorean identities would I use? My trouble is with finding sin and cos from tan.
  24. R

    Coulomb's Law from vector identities

    I can show that Coulomb's Law + superposition implies \nabla \cdot \mathcal {E} = \frac{\rho}{\epsilon_0} and \nabla \times \mathcal{E} = \mathbf{0}. I want to go the other way and derive Coulomb's law and superposition from the vector identities. I know that Gauss' Law implies Coulomb's law if...
  25. P

    Trigonometric Formulas, Identities and Equation

    Homework Statement t = tetha FInd sin 2t when cos t = 4/5 Homework Equations no idea The Attempt at a Solution no. . . . btw what should I do 1st? I don't understand this question
  26. Saitama

    Identities for solving log questions

    I was wondering if there could be more identities than i have read. I was doing questions on log and found many questions in this form:- (loga b)2 (loga b)(loga c) This is not my homework but i require these identities to solve the questions. Please someone tell me how to solve these...
  27. C

    Proving vector calculus identities using the levi-civitia symbol

    Homework Statement Prove \nabla \bullet (\textbf{A} \times \textbf{B}) = \textbf{B} \bullet (\nabla \times \textbf{A}) - \textbf{A} \bullet (\nabla \times \textbf{B}) I'd like to prove this using the levi-civitia symbol: \epsilon_{ijk} and einstein-summation convention as practice and...
  28. M

    Why Do Negative Angles Have Opposite Trig Identities?

    My book was proving that.. sin(-x) = -sin(x) tan(-x) = -tan(x) cos(-x) = cos(x) .. I posted an attachment of the books explanation.. I just don't understand how their explanation has anything to do with sin of a negative angle equaling sin of an angle times negative one, the same goes for...
  29. B

    Curl identities symbol question

    Hello, I would like to ask a question on curl. The wikipedia page http://en.wikipedia.org/wiki/Vector_calculus_identities" gives formulas of various operations, among which: \nabla \times (A \times B) = A(\nabla \cdot B) - B(\nabla \cdot A) + (\underbrace {B \cdot \nabla...
  30. D

    Need help proving vector identities

    Homework Statement i have to prove that ∇x(FxG)=(G⋅∇)F-(F⋅∇)G+F(∇⋅G)-G(∇⋅F) where F and G are vector fields with F=F1,F2,F3 and G=G1,G2,G3 ∇=d/dx,d/dy/d/dz Homework Equations The Attempt at a Solution i have tried applying scalar multiplication and the cross product to...
  31. L

    Verifying Trigonomic Identities

    Hey, Can anyone help me solve this equation please? Thanks Homework Statement cos(3θ)=cos3θ-3sin2θcosθ Homework Equations cos(u+v) = cosu cosv - sinu sinv sin2u = 2sinu cosu The Attempt at a Solution
  32. O

    Set Theory Proof(Using Identities)

    Hi, I've been trying for 3 hours to solve this proof using identities. I can't seem to get it. Can i get a little help please? Prove: A U B = (A ∩ B') U (A' ∩ B) U (A ∩ B) thanks
  33. M

    Is ArcTan(-1) equal to both 7pi/4 and 3pi/4?

    Homework Statement ArcTan(-1) = theta I figured that theta was both 7pi/4 and 3pi/4.. but the book has -pi/4 Because my two answers are both equivalent to -pi/4 would I be wrong? Homework Equations The Attempt at a Solution
  34. P

    What is the identity of a3 + b3 + c3 in math?

    Homework Statement what is the identity of a3 + b3 + c3 i am getting a bit confused please help! Homework Equations The Attempt at a Solution
  35. W

    Prove that (1+tanx)^2-2tanx=1/(1-sinx)(1+sinx)

    Homework Statement prove that: (1+tanx)^2-2tanx=1/(1-sinx)(1+sinx)Homework Equations Main question is does 1/cosx^2 = secx^2? More in my attempt at the solution.The Attempt at a Solution (1+tanx)^2-2tanx=1/(1-sinx)(1+sinx) (1+tanx)(1+tanx)= 1+2tanx+tanx^2 1+2tanx+tanx^2-2tanx=1+tanx^2...
  36. Q

    Showing the Hermitean Adjoint Property for Operators

    How would I go about showing: \hat{A}^{\dagger} + \hat{B}^{\dagger} = \left( \hat{A} + \hat{B} \right) ^{\dagger}
  37. C

    Rewrite polar equation as Cartesian (trig identities)

    Homework Statement x=2acos(theta- pi/3)cos(theta) y=2acos(theta- pi/3)sin(theta) Write everything in terms of x and y Homework Equations cos(a-b)=cos(a)cos(b)+sin(a)sin(b) sina/cosa=tana etc... The Attempt at a Solution I've tried a lot of rearranging, can't figure it out.
  38. S

    Trig identities, what does this mean?

    Tan theta=sin theta/cos theta What does that identity mean? What does it say about the triangle? And what on Earth is an identity? I understand everything taught in the course so far, trig ratios, reciprocal trig ratios, special angles etc etc...but i have no idea what identities mean, some...
  39. C

    Why does sin80°csc80° equal 1?

    I'm on mobile so I can't use regular symbols. Sin80°csc80° = 1 why does this equal one? Is there multiplication involved here? I know csc = 1/sin
  40. Q

    Vector calculus identities - is this right?

    I'm manipulating an equation, and I think I am correct in doing this, but not sure. Could someone tell me if the equality I've written below is true? [\nabla\cdot [\rho\vec{v}\vec{v}] ]\cdot\vec{v} = \nabla\cdot[\frac{1}{2}\rho v^2 \vec{v}] (where \rho is dependent on position) *NOTE* that...
  41. M

    Proof of Trigonometric Identities: Sin(-theta), Cos(-theta), Tan(-theta)

    Homework Statement Can some one please give me the proof or somewhere I can find the proof or just an explanation why Sin(-theta) = -Sin(theta Cos(-theta) = cos(theta) Tan(-theta)= -Tan(theta) Thank you.. Homework Equations The Attempt at a Solution
  42. madmike159

    Proving Trigonometric Identities: cos(3x) & (cos(3x)-cos(7x))/sin(7x)+sin(3x)

    Homework Statement prove that, (cos(3x) - cos (7x)) / (sin(7x) + sin(3x)) = tan(2x) prove that, cos(3x) = 4cos^3(x) - 3cos(x) Homework Equations tan(x) = sin(x)/cos(x) must come into the first one The Attempt at a Solution tried seperating the fraction so there is only one...
  43. S

    Triginometry: Half Angle Identities

    Homework Statement sin(75^\circ) Homework Equations sin(\frac{\alpha}{2}) = \pm \sqrt{\frac{1-cos(\alpha)}{2}} The answer is known to be: \frac{\sqrt{6} + \sqrt{2}}{4} The Attempt at a Solution sin(\frac{150^\circ}{2}) = \pm \sqrt{\frac{1-cos(150^\circ)}{2}} sin(\frac{150^\circ}{2}) = \pm...
  44. M

    Finding Trig Identities: How Do I Solve Trig Equations with Given Values?

    Homework Statement Problem 6. If you know that tan(theta) = -4/5 and sin(theta) > 0, find: (a) sin(theta) (b) cos(theta) (c) tan(theta + pi)Homework Equations cos^2(t)+sin^2(t)=1 tan(t)=sin(t)/cos(t) The Attempt at a Solution My teacher went over this today, but likes to skip over steps that I...
  45. F

    Can You Divide When Formally Proving Trig Identities?

    When formally proving trig identities using algebra are you allowed to divide? Ordinarily I wouldn't think twice about this. Sure, the trig function by which you are dividing might take on 0, but not for all angles. We divide by potential zeros all the time. My advisor was being very coy...
  46. M

    Mastering Trig Identities: Strategies & Tips

    Homework Statement I have been practicing proving trig. ident. for the past couple of days... although I am getting better... I wanted to ask are there any strategies that I should be aware of to make the process easier ? Homework Equations The Attempt at a Solution
  47. T

    General Relativity - Tensor Identities

    I am doing this question on page 75 here (http://www.maths.cam.ac.uk/undergrad/pastpapers/2006/Part_2/list_II.pdf ). Here is my solution so far: \nabla_a\nabla_b\phi=\partial_a\partial_b\phi=\partial_b\partial_a\phi=\nabla_b\nabla_a\phi Then for the energy-momentum tensor I just wrote...
  48. S

    Proving Vector Identities Using the Permutation Tensor and Kroenecker Delta

    Homework Statement Prove using the Levi-Civita Tensor/Kroenecker Delta that: (AxB)x(CxD) = (A.BxD).C-(A.BxC).D Homework Equations εіјkεimn = δjmδkn – δjnδkm (where δij = +1 when i = j and 0 when i ≠ j) The Attempt at a Solution if E = (AxB) then Ei = εіјkAjBk, and if F =...
  49. T

    Different identities in one equation

    1. Homework Statement Sorry for the long intro: An object is propelled up at angle theta 45 deg. < theta < 90 deg. to the horiz. with initial vel. of V0 m/s. from the base of a plane that makes an angle of 45 deg. with the horiz. If air resistance is ingored, the distance, R, traveled by...
  50. B

    How can I expand sin(x + y) + sin(x - y) to get 2sinxcosy?

    Proving trig identities I have 2 more this time, thanks for the time! Homework Statement -sin^2x-sin^2y+1=cos(x+y)cos(x-y) Homework Equations Compound, double, pythagorean, reciprocal, quaotient, etc. The Attempt at a Solution R.H.S. cos(x+y)cos(x-y) =...
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