identities Definition and 422 Threads

Homework Statement Show, using complex numbers, that sin(x)+cos(x)=(√2)cos(x-∏/4) Homework Equations cos(x)=(e^(ix)+e^(-ix))/2 sin(x)=(e^(ix)-e^(-ix))/2i e^ix=cos(x)+isin(x) The Attempt at a Solution I was given the hint that sin(x)=Re(-ie^(ix)), but have thus far not been...

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...

Help with Double Angle Identities! cos θ = 24/25 The angle lies in quadrant 1; 0<θ<90 Find sin2θ I know you would use either the formula cos^2θ - sin^2θ or 2sinθcosθ And I know that the answer is 339/625, but I do not know how to get that answer?

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 =...

I have another answer to this but I believe this one is correct. I need someone else to check it out since I have been looking at it too long. Is the bottom equality correct? \begin{alignat*}{3} \frac{\partial^2}{\partial t^2}x_1 + x_1 & = & F\cos t - 2[-A'\sin t + B'\cos t] - c[-A\sin t +...

Sorry for the format, I'm on my phone. Lets say the matrix is | 1 1 1 | | a b c | | a^2 b^2 c^| Or {{1,1,1} , {a, b,c} , {a^2, b^2,c^2}} Show that it equals to (b-c)(c-a)(a-b) I did the determinant and my answer was (bc^2) - (ba^2) - (ac^2) + (ab^2) + (ca^2) - (cb^2)...

I was wondering if someone can show me or point me to a worked out example using integration by parts for more than one variable (as used in relation to pde's, for example). While I took pdes and calc 3, its been awhile and I don't know if I ever understood how to work out a concrete example...

Apparently our professor expects us to know these half-angle identities (http://www.purplemath.com/modules/idents.htm) Without going through them in class or us learning them in high school.. Can somebody explain how these were derived? Does the derivation come from the angle-sum and...

∫(cot^4 x) (csc^4 x) dx Wolfram wants to use the reduction formula, but I'm meant to do this just using identities and u substitution. I was thinking something along the lines of: =∫cot^4 x (cot^2 x + 1)^2 dx =∫cot^8 x + 2cot^6 x + cot^4 x dx but I don't know where to go from there.

Homework Statement I need to prove the identity: (a×b)\cdot(c×d)= (a\cdotc)(b\cdotd)-(a\cdotd)(b\cdotc) using the properties of the vector and triple products: Homework Equations a×(b×c)=b(a\cdotc)-c(a\cdotb) a\cdot(b×c)=c\cdot(a×b)=b\cdot(c×a) The Attempt at a Solution I...

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!

1.) ∫[(7 sin (x))/[1+cos^2(x)]] * dx 2.) I'm looking at the trig identity sin^2 x+cos^2 x=1, and am wondering if I could use that in solving the problem. Or should I use u=sin x, then du= cos x, then plug those in? 3.) so I thought maybe it would be easier to separate the two...

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...

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

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...

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.

Homework Statement The two vectors a and b lie in the xy plane and make angles α and β with the x axis. a)By evaluating a • b in two ways (Namely a •b = abcos(θ) and a • b = a1b1+a2b2) prove the well-known trig identity cos(α-β)=cos(α)cos(β)+sin(α)sin(β) b)By similarly evaluating...

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...

Hello, Say I'm working with ∫ sqrt(1-cos(t)) dt I end up with a substitution of u = 1-cos(t) and dt = du/sin(t) sub back in: ∫ sqrt(u) / sin(t) du Still got a t in there ... hrrmmm So I go to wolfram alpha for some inspiration and 'show steps'...

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) +...

Hi, I am working on proofs of the difference identities for sine, cosine, and tangent. I am hoping to solve these using a specific diagram (attached). I was wondering if you could help me with the difference of cosines. Is it possible to derive it using the attached diagram? If so, how...

Homework Statement I need to show that ##\displaystyle\int_\Omega (\nabla G)w dxdy=-\int_\Omega (\nabla w) G dxdy+\int_\Gamma \hat{n} w G ds## given ##\displaystyle \int_\Omega \nabla F dxdy=\oint_\Gamma \hat{n} F ds## where ##\Omega## and ##\Gamma## are the domain and boundary respectively...

Homework Statement This isn't really a problem that was assigned to me, (I'm studying independently) I just have a question about the general concept behind some identities. Homework Equations sin(a+b) = sin(a)*cos(b) + cos(a)*sin(b) sin(theta) = sin(180-theta) The...

Fact: The ring of integers Z is totally ordered: for any distinct elements a and b in Z, either a>b or a<b. Fact: The ring of integers is discrete, in the sense that for any element a in Z, there exists an element b in Z such that there is no element c in Z with a<c<b, and the same argument...

I know you can derive the double angle formulas for sin(2a) and cos(2a) from Euler's identity, but is there any way to derive the tan(2a) in a similar manner from an easier formula? What about the addition/subtraction formulas (i.e. sin(a+b), etc.)

Is it possible to prove identities involving arctan by complex exponentiation? I had in mind something like the following for the arctan angle addition formula, but I feel there is something not quite right in the argument. $$\arctan{(a)}+\arctan{(b)}=...

Homework Statement Find the exact value of: sin (-5∏/12) 2. The attempt at a solution sin (-45° + -30°) = sin -45° cos -30° + cos -45° sin -30° = (sqrt (2) / 2 )(sqrt (3) / 2 ) + (sqrt (2) / 2)(1 / 2) = (sqrt (6) + sqrt (2)) / 4 However, the book has (-sqrt (6) -...

Hello all, I was wondering if someone has ever found a purely algebraic proof for the addition/subtraction theorems of trigonometry, mainly sin(a+b)=sin(a)cos(b)+sin(b)cos(a). Given a right triangle: Let x be one of the perpendicular legs and let the other leg be composed of two parts, y1...

Homework Statement Suppose f is a continously differentiable real valued function on R^3 and F is a continously differentiable vector field Prove 1)##\oint (f \nabla g +g\nabla f) \cdot dr=0## 2) ##\oint(f \nabla f)\cdot dr=0##Homework Equations ##\nabla f = f_z i+ f_y j+f_z k## Real valued...

Are there any useful identities for simplifying an expression of the form: $$((\ldots((x_{1} *_{1} x_{2}) *_{2} x_{3}) \ldots) *_{n - 1} x_{n})$$ Where each $$*_{i}$$ is one of $$\cap, \cup$$ and $$x_1 \ldots x_n$$ are sets? I believe I found two; though I haven't proved them, I think they...

Homework Statement Use fundamental identities to simplify the expression: (sinx)^2 - (cosx)^2 ____________________ (sinx)^2 - (sinx cosx)*note: it's a numerator and denominator. The underscore line is the fraction line. *note: The answer in the back of the book is "1 + cotx" but I would...

Is 2a/sin 2x equivalent to a cot x?

Hello! I've been tackling the question 'Express sin3x+sinx as a product and hence solve 1/2(sin3x+sinx)=sin2x ; x∈R' but I'm stumped - I'm not sure whether I've even approached it correctly. This is what I did: sin(3x+x)=sin3x.cosx+sinx.cos3x inserting this into the second equation...

Homework Statement Show the remainder when 43^43 is divided by 17. Homework Equations $$43 = 16 \times 2 + 11$$ $$a^{p-1}\equiv1\ (mod\ p)$$ The Attempt at a Solution I believe I can state at the outset that as: $$43\equiv9\ (mod\ 17)$$ Then $$43^{43}\equiv9^{43}\ (mod\ 17)$$ and that I...

If X and Y are binomial random variables with respective parameters (n,p) and (n,1-p), verify and explain the following identities: a.) P{X<=i}= P{Y>=n-i}; b.) P{X=k}= P{Y=n-k} Relevant Equations: P{X=i}=nCi *p^(i) *(1-p)^(n-i), where nCi is the combination of "i" picks given "n"...

Homework Statement Suppose c and (1 + ic)^{5} are real, (c ≠ 0) Show that either c = ± tan 36 or c = ± tan 72The Attempt at a Solution So I considered the polar form \left( {{\rm e}^{i\theta}} \right) ^{5} and that \theta=\arctan \left( c \right) , so c = tan θ Using binomial expansion, I...

Homework Statement The question is to find sin 2x, cos 2x, tan 2x from the given information: sin x = -\frac{3}{5}, x in Quadrant III Homework Equations Half Angle Identities cos2x = cos^{2}x - sin^{2}x sin2x = 2sinxcosx tan2x = \frac{2tanx}{2-tan^{2}x} The Attempt at a...

Homework Statement Use standard identities to express sin(x+pi/3) in terms of sin x and cos x Homework Equations sin(a+b)=sinAcosB+sinBcosA The Attempt at a Solution sin(x)cos(pi/3)+sin(pi/3)cos(x) 0.5sinx + 0.8660cosx I'm just not sure if i need to simplify it even further...

I am required to show that (i)in the upper limit of very high energies, the Born and eikonal identities are identical. (ii)that the eikonal amplitude satisfies the optical theorem. Regarding (i) I think it will involve changing from an exponential to a trig(Euler's theorem) but I could be...

Homework Statement \frac{∂x_{i}}{∂x_{j}} = δ_{ij} Homework Equations \vec{r} = x_{i}e_{i} The Attempt at a Solution \frac{∂x_{i}}{∂x_{j}} = 1 iff i=j δ_{ij} = 1 iff i=j therefore \frac{∂x_{i}}{∂x_{j}} = δ_{ij} Homework Statement r^{2} = x_{k}x_{k} Homework...

1. Sin^2(x) = 3 – x Answer: 2.97 Attempts: 1-cos^2(x) = 3 – x cos^2(x) - x + 2 = 0 Factored it and got x = pi = 3.14 It’s a multiple choice question, and other answers were 3.02,3.09 which are few decimal places off so the answer must not be pi since it's not even a choice. Is the...

hi.. i came through a problem in which the expression [1-(k(sin^2) θ)] has to be simplified.. can someone help me to solve it.??

Homework Statement Use Euler's identity to prove that cos(u)cos(v)=(1/2)[cos(u-v)+cos(u+v)] and sin(u)cos(v)=(1/2)[sin(u+v)+sin(u-v)] Homework Equations eui=cos(u) + isin(u) e-ui=cos(u)-isin(u) The Attempt at a Solution I was able to this with other trig identities with no...

Homework Statement ((cos x)/(1+sin x))+((1+sin x)/(cos x)) Homework EquationsThe Attempt at a Solution multiplied the left equation by (cos x)/(cos x) and the right fraction by (1+sin x)/(1+sin x) get ((cosx)^2 +(1+sinx)^2) / (1+sinx) (cos x) and I have no idea where to go from here

Homework Statement I have a problem. I need to prove that the divergence of Einstein tensor is 0 using the bianchi identities. I have looked to several sources and I have derived an answer, but I don't fully understand some steps. Homework Equations I have uploaded a document which shows a...

Homework Statement (3/5)cos2x + (3/5)sin2x The Attempt at a Solution I would think the answer would be 6/5, but it looks like the book is saying 3/5. I had a similar problem to this the other day and I tried finding it in my history but I couldn't.

Homework Statement Tan^3Theta Homework Equations Tan^2Theta=1-cos2Theta/1+cosTheta The Attempt at a Solution Attached

If an operation has two left identities, show that it has no right identity. _{} pf/ Let e_{1} and e_{2} be left identities such that e_{1}≠e_{2}. Assume there exist a right identity and call it r. Then we have that e_{1}x=x e_{2}x=x and xr=x. From here I want to...

Any/All help is appreciated :) Thanks! Homework Statement All that has to be done is proving that these two sides are equal. Basically, you just work through the problem until both sides are the same. (csc(x)-sec(x))/(csc(x)+sec(x)) = (tan(x)-1)/(tan(x)+1) Homework Equations...

Homework Statement I'm reading in a fluid dynamics book and in it the author shortens an equation using identities my rusty vector calculus brain cannot reproduce. Homework Equations \vec{e} \cdot \frac{\partial}{\partial t}(\rho \vec{u}) = -\nabla\cdot (\rho\vec{u})\cdot\vec{e} -...