What is Trig identities: Definition and 126 Discussions

In mathematics, 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

    Using Euler's formula to prove trig identities using "sum to product" technique

    Hello, This is actually not homework. I was google searching for "proving trig identities from geometric point of view), found one of the result which proves trig identities using Euler formula. I really liked it. Easier, quicker & simple. But when the author speak about sum to product formulas...
  2. C

    Another way to find trig identities

    Using the identity's (1) (2) Gives, Why dose this elegant method work? Many thanks!
  3. C

    Proving trig identities -- Is the method related to the unit circle?

    Why when proving trig identities, Do we assume that r = 1 from ## rcis\theta = r[\cos\theta + i\sin\theta]##? This makes me think that this is somehow it is related the unit circle. Note: I am trying to prove the ##cos3\theta## identity and am curious why we assume that the modulus is 1...
  4. 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...
  5. chwala

    Proof of the trig identities for half-angles

    I was just checking this out the sin##\frac {A}{2}## property, in doing so i picked a Right-Angled triangle, say ##ABC##, with ##AB=5cm##, ##BC=4cm## and ##CA= 3cm##. From this i have, ##s=6cm## now substituting this into the formula, ##sin\frac {A}{2}##= ##\frac {1×3}{5×3}##=##\frac...
  6. srfriggen

    B Verifying trig identities.... what about when tan is undefined?

    Hello, If I wanted to verify tan(x)cos(x) = sin(x), what about when x is pi/2? LHS has a restricted domain so it can't equal sin(x). Does this equation only work with a restricted domain? Nothing in my textbook discusses that. Thank you
  7. MichaelRocke

    Trig Identities - Pre-calculus in a Nutshell - Section 4 Question 1

    My latest attempt \begin{align*} \frac{\sin \theta + \tan \theta}{\csc \theta + \cot \theta} = \\ \frac{\sin \theta + \tan \theta}{\csc \theta + \cot \theta} \cdot \frac{\csc \theta - \cot \theta}{\csc \theta - \cot \theta} =\\ \frac{\sin \theta \csc \theta + \tan\theta \csc \theta - \sin...
  8. M

    B Why do we care about trig identities?

    Homework Statement: This is not a homework question. I am trying to understand why we spend so much time studying trig identities. Homework Equations: As far as I understand, the two basic trig functions (sin and cos ) show the relationship between the sides of a right angle triangle in a...
  9. opus

    Trig Identities -- example problem confusion

    Homework Statement Back with more trig identities. Verify that the following is an identity ##-tan\frac{a}{2} = cot\left(a\right)-csc\left(a\right)## Homework Equations All pythagorean identities, double angle formulas, half angle formulas The Attempt at a Solution In the picture that I've...
  10. L

    Integration of even powers of sine and cosine

    Homework Statement Homework Equations cos2x = (1+cos2x)/2 sin2x = (1-cos2x)/2 The Attempt at a Solution I believe you would use the double angle formula repeatedly but that is very tedious; is there a more concise way to solve the problem?
  11. S

    Courses Trig Identities in Calc II?

    I was wondering exactly what parts of trig I need to do to do well in Calc II. I took trig this past spring and aced it and I'm taking Calc I this semester. I'm not worried about this semester because I know my instructor won't use trig outside teaching us how to take the derivatives of the trig...
  12. F

    MHB Solving Trig Identities

    How do you do this one? I can't figure it out! (2 - 5cot x) / (2 + 5cos x) = (2sin x - 5cos x) / (2sin x + 5cos x)
  13. F

    Need help solving this trig equation

    Homework Statement what's the best way to solve this equation: 3cos(θ) + 1.595*sin(θ) = 3.114 Homework Equations (sinθ)^2 + (cosθ)^2 = 1 The Attempt at a Solution I tried using the identity above to solve this equation and ended up with cosθ = +/- 1.0526.
  14. D

    Solve acos²θ+bsinθ+c=0 for all values 0≤θ≤360°

    Homework Statement Solve acos²θ+bsinθ+c=0 for all values 0≤θ≤360° a=16 b=6 c=-12 So 16cos²θ+6sinθ-12=0 Homework Equations Cos²x=1-Sin²x The Attempt at a Solution Identity: Cos²x=1-Sin²x 16(1-Sin²θ)+6Sinθ-12=0 16-16Sin²θ+6Sinθ-12=0 6Sinθ-16Sin²θ=12-16=-4 Divide by 2(?) 3Sinθ-8Sin²θ=-2...
  15. S

    B Factoring quadratic equation (with trig identities used)

    Is it possible to factor a quadratic equation along the lines of asin^2x -bsin2x+c ? If so, how? The sin2x seems to be a problem since when expanded it becomes 2sinxcosx, but I'm wondering if it is possible, and how it would be done? Thanks in advance.
  16. A

    Trig identities (I think?) for precalc review

    Homework Statement My calc class is having me review precalc(which I'm really rusty on...) 21. Find sin θ, sec θ, and cot θ if tan θ = 27 22. Find sin θ, cos θ, and sec θ if cot θ = 4. 23. Find cos 2θ if sin θ = 15 24. Find sin 2θ and cos 2θ if tan θ = √2 25. Find cos θ and tan θ if sin θ =...
  17. Draconifors

    Quick Trigonometric Identity Question

    Hi! I have an integral to solve (that's not the point, though) and the inside of the integral is almost a trig identity: 1. Homework Statement ##sin\frac{(x+y)} {2}*cos\frac{(x-y)} {2} ## Homework Equations I noticed this was very similar to ##sinx+siny = 2sin \frac{(x+y)} {2} *...
  18. N

    I Integrate 1/(y+cos(x))^2 dx

    First part of the question was to work out the integral 1/(y+cos(x)) between x=0 and x=pi/2 by using the substitution t=tan(x/2). Got this to be \frac{2}{\sqrt{y^2-1}}arctan(\sqrt{\frac{y-1}{y+1}}) The next question says HENCE find integral with the same limits of \frac{1}{(y+cos(x))^2} Ive...
  19. Bill_Nye_Fan

    Help with Q7 part a: Trig Identities

    Homework Statement [/B]Q7 part a on one of the attached pictures 2. Homework Equations Trigonometric identities The Attempt at a Solution See attached pages Please help me I've spent onwards of 4 hours trying to figure this out and I can't get anywhere at all
  20. P

    Simplifying a log expression with identities

    I was supposed to simplify the expression ##\ln |\cot {x}|+\ln |\tan {x}\cos {x}|## and apparently it’s wrong. Where’s the mistake? Is it not simplified enough or . . . ? ##\ln |\cot {x}|+\ln |\tan {x}\cos {x}|## ##=\ln |\frac {\cos {x}}{\sin {x}}|+\ln |\frac {\sin {x}}{\cos {x}}\cdot \cos...
  21. A

    Finding the Domain of a Trigonometric Function

    Homework Statement Find the domain of this function and check with your graphing calculator: f(x)=(1+cosx)/(1-cos2x) Homework EquationsThe Attempt at a Solution i get to (1+cosx)/(1+cosx)(1-cosx) which is factored. so then setting each one to zero one at a time i figure out that cosx = -1 and...
  22. C

    Trig Identities: Simplifying Expressions with Cotangent, Secant, and Cosecant

    I have re-post this forum as I should have paid closer attention to rules. I apologized for that. Homework Statement 1) The expression tan^3 θ + sinθ/cosθ is equal to: (a) cot θ (b) tan θ sec^2 θ (c) tan θ (d) sin θ tan θ (e) tan θ csc^2 θ 2) Simplify (cos θ/1+ sin θ - cosθ/sinθ-1)^-1 (a)...
  23. A

    Solving y = sec^2x - 2sin x: Tips & Strategies

    I need to solve y = sec^2 x - 2sin x, I'm fine with the 2sin x, but for the sec^2 x... I can't seem to understand how to do it... I can get it to... sin^2 x + cos^2 x / cos^2 x or 1 / cos^2x but I don't understand how to solve it when the square comes right after the cos/sin etc.
  24. C

    MHB Trig Identities Problem

    If \cos(\pi/3)= \frac{1}{2}, find \sec(\pi-\pi/3) Someone really give me step-by-step explanation. I really don't know what identity to use, and no idea how to get cosine to secant. Please, it would help. I do have more questions if you help me dissect this problem. XD Thanks so much in advance!
  25. C

    Integration via Trigonometric Substitution

    Homework Statement Evaluate \int{\frac{x^2}{(1-x^2)^\frac{5}{2}}}dx via trigonometric substitution. You can do this via normal u-substitution but I'm unsure of how to evaluate via trigonometric substitution. Homework EquationsThe Attempt at a Solution Letting x=sinθ...
  26. T

    Three phase phasor proof or simplification

    So I'm really rusty on phasors, I was reading that a space vector current was i(t) = I(cos(wt)<0 + cos(wt - 120)<120 + cos(wt - 240)<240 ) = 3/2 * I < wt and I couldn't figure out how that could be so (still can't, please help) So I tried to go back to basics and I went back and read: A =...
  27. binbagsss

    Tricky Quadratic formula / trig identities

    Im to solve ##(k+l)^{2}e^{-ila}-(k-l)^{2}e^{ila}=0##, for ## k##, The solution is ##k=l(e^{ial}-1)/(e^{ial}+1)=il tan(al/2)## FIRST QUESTION So it's a quadratic in k, should be simple enough, my working so far using the quad. formula is ##k= (4l^{2}(e^{-ila}+e^{ila})\pm...
  28. earthloop

    Simplifying with trig identities

    Homework Statement [/B] Hi, I am currently working through a textbook, and the following simplification is given for an example question: I can't seem to work out how they have moved from cos(pi+n*pi) to cos(pi)cos(n*pi) so easily? Is there a simple trick I have missed? I understand the...
  29. M

    Help with trig identities

    Homework Statement Show that (sin^4 x + (sin^2 x * cos^2 x)) / (cos^2 x - 1) == -1 Homework Equations Sin^2 x + cos^2 x == 1 The Attempt at a Solution (sin ^4 x + (sin^2 x * cos^2 x)) / (cos^2 x - 1) = ((sin^2 x)(sin^2 x) + (sin^2 x * cos^2 x)) / (cos^2 x - 1) =((sin^2 x)(1 - cos^2 x) +...
  30. D

    MHB Finding Exact Value using Trig Identities and Complementary Angle Theorem

    Hey guys, I've been trying to wrap my mind around this problem but I've really come up short. Any help would be amazing. If tanX=10 Find the exact value of cot(pi/2 - x)
  31. D

    MHB What is the equivalent identity for $\sin^2\omega t$?

    is there an equivalent identity for $\sin^2\omega t?$ please tell me.REGARDS!
  32. C

    Need help proving some trig identities

    Proving identities is a pain! Thanks in advance, guys! Homework Statement 1. 1 + sec^(2)xsin^(2)x = sec^(2)x 2. sinx/1-cosx + sinx/1+cosx = 2cscxHomework Equations The Attempt at a Solution For the first problem, this is the best I got: 1 + sec^(2)x(1-cos(2)x) For the second problem, I...
  33. Dethrone

    Understanding integration with trig identities, and absolute value

    Homework Statement In integration, we are allowed to use identities such as sinx = \sqrt{1-cos^2x}. Why does that work, and why doesn't make a difference in integration? Graphing \sqrt{1-cos^2x} is only equal to sinx on certain intervals such as(0, \pi) and (2\pi, 3\pi). More correctly...
  34. Dethrone

    MHB Integration with trig identities and absolute value

    In integration, we are allowed to use identities such as sinx = \sqrt{1-cos^2x}. Why does that work, and why doesn't make a difference in integration? Graphing \sqrt{1-cos^2x} is only equal to sinx on certain intervals such as (0, \pi) and (2\pi, 3\pi). More correctly, shouldn't we use the...
  35. L

    Trig identities and complex numbers help.

    Please forgive me as I may have to edit this post to get the equations to show properly. I am doing some work with AC circuits and part of one of my phasor equations has this in it: \frac {2i} {1+cos(θ) + i sin(θ)} - i , where i is the imaginary number \sqrt{-1}. However, knowing the...
  36. M

    Simplifying Trig Identities

    Homework Statement The length of the curve r(t) = cos^3(t)j+sin^3(t)k, 0 =< t <= pi/2 is Homework Equations AL in polar = ∫sqrt(r^2 + [dr/dθ]^2) The Attempt at a Solution I am having trouble simplifying the terms within the square root. What method should I use to deal with...
  37. N

    Trig identities, prove (cot(x)-tan(x)=2tan(2x))

    Homework Statement Prove cot(x) - tan(x) = 2tan(2x) Homework Equations Trig identities http://en.wikipedia.org/wiki/List_of_trigonometric_identities The Attempt at a Solution I have worked it down and don't think they are equal. I think it's supposed to be 2cot(2x) not 2tan(2x)...
  38. I

    Use trig identities to show that

    Homework Statement use trig identities to show that (b) cos(tan^(−1)[x])=1/√(1+x^2) for −1/2π<x<1/2π. Homework Equations i think Pythagoras has to applied but that is geometric reasoning hmm The Attempt at a Solution
  39. S

    MHB Antiderivative involving Trig Identities.

    A little confused on something. Suppose I have the integral 2 \int 4 \sin^2x \, dx So I understand that \sin^2x = \frac{1 - \cos2x}{2} BUT we have a 4 in front of it, so shouldn't we pull the 4 out in front of the integral to get: 8 \int \frac{1 - \cos 2x}{2} \, dx then pull out the...
  40. J

    Confusion with product-to-sum trig identities

    I'm having some confusion with a couple trig identities. On wikipedia (http://en.wikipedia.org/wiki/List_of_trigonometric_identities#Product-to-sum_and_sum-to-product_identities), the following two identities are listed: sinθcosβ = (1/2)[sin(θ+β) + sin(θ-β)] and sinβcosθ =...
  41. M

    Using complex numbers to find trig identities

    I can find for example Tan(2x) by using Euler's formula for example Let the complex number Z be equal to 1 + itan(x) Then if I calculate Z2 which is equal to 1 + itan(2x) I can find the identity for tan(2x) by the following... Z2 =(Z)2 = (1 + itan(x))2 = 1 + (2i)tan(x) -tan(x)2 = 1...
  42. QuantumCurt

    Prove that the two trig identities are equivalent

    Homework Statement Prove that the two trig identities are equivalent. cos \ x \ -\frac{cos \ x}{1-tan \ x} \ = \ \frac{sin \ x \ cos \ x \ }{sin \ x \ - \ cos \ x} The Attempt at a Solution My professor recommended that we only work with one side of the equality when we're trying...
  43. S

    Trig identities (is my method correct?)

    Homework Statement Prove the following identities 31c) sin(\frac{\pi}{2}+x)=cosxHomework Equations sin2x+cos2x=1 The Attempt at a Solution The idea here is to prove the identity by making LS=RS so here is what i have done, but I am not sure if it is the right way, since the book shows it...
  44. B

    Can you prove the trig identities given A + B + C = 180 degrees?

    Given: A + B + C = 180 degrees Prove: TanA + TanB + TanC = TanA x TanB x TanC First time submitting something. Thank you all.
  45. C

    How are these two equations equal? trig identities possibly?

    Homework Statement Proof that (1/6)sin(3x)-(1/18)sin(9x) = (2/9)sin^3(3x) Homework Equations The Attempt at a Solution I am just curious exactly how the power on the sine function is cubic on one side. It obviously has to do with something that increases the power on the...
  46. T

    Proving tan2A = 2tanA/1-tan^2A: A Trig Identity Example

    Homework Statement Prove the following: tan2A=2tanA/1-tan^2A Homework Equations The Attempt at a Solution Took the right hand side: =2(sinA/cosA) / 1-(sin^2A/cos^2A) =2sinA/cosA /cos^2A-sin^2A/cos^2A =2sinA/cos2A /cosA/1 Dont know what to do next?
  47. S

    Solve Math Trig Identities w/ Sin & Cos Only

    Hi I am in need of some help for this question: 1+tanx/1-tanx = tan(x+(∏/4)) It is easy to solve with the tan trig identites on the right side however, my teacher had told me to do it with SIN and COS only. I am not sure if its possible and was looking for some insight Left Side...
  48. D

    MHB Trig identities Fourier Analysis

    Prove the identities $$ \frac{\sin\left(\frac{n + 1}{2}\theta\right)}{\sin\frac{\theta}{2}}\cos\frac{n}{2}\theta = \frac{1}{2} + \frac{\sin\left(n + \frac{1}{2}\right)\theta}{2\sin\frac{\theta}{2}} $$ By using the identity $\sin\alpha + beta$, I was able to obtain the $1/2$ but now I am not to...