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.
Homework Statement
Prove that this is an identity.
1 + cos² x / sin² x = 2csc² - 1
Homework Equations
cos²x + sin²x = 1 (manipulative equation)
tan²x = sec² - 1
cot²x = csc²x - 1
etc..
The Attempt at a Solution
I attempted this equation more than 10+ times. Each time, I find a...
Homework Statement
2x^2 + 7x - 5 = A(x-1)^2 + Bx + C
Homework Equations
N/a
The Attempt at a Solution
This is an interesting equation as regardless of the whether I substitute x = 0 or x = 1 i still encounter the problem of not be able to make the calculation any easier. I...
Homework Statement
By substituting y for 9x^2 solve 81x^4 - 63x^2 + 10 = 0
Homework Equations
The Attempt at a Solution
My attempt at a solution is:
y^2 - 7y + 10
(y-2)(y-5)
therefore y = 2 y = 5 Can someone double check this?
Cheers
Homework Statement
Prove that
\frac{sin3x}{sinx}-\frac{cos3x}{cosx} = 2Homework Equations
The Attempt at a Solution
LHS:\frac{sin(2x+x)}{sin}-\frac{cos(2x+x)}{cosx}
=\frac{sin2x.cosx + cos2x.sinx}{sin}-\frac{cos2xcosx - sin2x.sinx}{cosx}
Homework Statement
Show that (sin(2theta) - sin(theta)) / (cos(2theta) - cos(theta) + 1) = tan(theta)
sorry if this setting out is unclear but i am not familiar with how to post math symbols and such.
Homework Equations
above
The Attempt at a Solution
I have tried simplifying...
1. (A) sin(-x) = - sin x (C) cos(x+y) = cosxcosy - sinxsiny
(B) cos(-x) = - cos x (D) sin(x+y) = sinxcosy + cosxsiny
Use these equalities to derive the following trigonometric identities.
a. absolute value of cos x/2 = \sqrt{}1 + cosx/2
b. absolute...
Hello.
How can I prove something like
\nabla\cdot(\mathbf fv)=(\nabla v)\cdot\mathbf f+v(\nabla\cdot \mathbf f)
using only the definition of divergence
\text{div}\mathbf V=\lim_{\Delta v\rightarrow0}\frac{\oint_S\mathbf V\cdot d\mathbf s}{\Delta v},
i.e. without referring to...
Trig substitition--lost in identities
Homework Statement
\int\frac{dx}{x^{2}\sqrt{x^{2}+1}}
Homework Equations
It's pretty obvious that this is a trig substitution problem requiring use of tangent.
The Attempt at a Solution
x=tan\theta
dx=sec^{2}\theta
x^{2}=tan^{2}\theta...
[tan(pi/4+x)-tan(pi/4-x)]/[tan(pi/4+x)+tan(pi/4-x)]=2sinxcosx
I tried to prove this trig identity but I an really stuck. I think tan of pi/4 is '1', and if I do that then my numerator becomes zero, thus zero=2sinxcosx. But that can't be right, so I don't know what to do now.
LS=...
I am learning how to prove conditional identities like
(a^2-c^2+b^2+2ab)/(c^2-a^2+b^2+2bc) = (s-c)(s-a)
if a+b+c = 2s
- Derived from Herons formula
I have understood the proof for the above , but i want more problems to work on.
Can anyone suggest some link where i can find similar...
Homework Statement
I'm not looking for anyone's pity, but I can't even begin to tell you how confused I am with this concept. My teacher spent all of five minutes teaching it to us and then leaves us to fend for ourselves with this ridiculously hard worksheet. Please help.
1. 2cscx =...
Homework Statement
cosx - cosy=-2 sin(x + y/2) sin(x - y/2)
Homework Equations
dont know what identities to use
The Attempt at a Solution
ok so when i figure it out, the RHS always comes out to either...
(cos x sin y - sin x cos y)/2
or just
cos x sin y - sin x cos y...
Homework Statement
Prove using index notation that,
the x denoting a cross-product.
(del x f del g)=del f x del g
Homework Equations
The Attempt at a Solution
dif etc. denote partial derivatives.
RHS=eijkdjfdkg
LHS-I'm not even quite sure how to write it in index...
Homework Statement
1) sin(x+y)sin(x-y)=cos^2y-cos^2x
2) tan(∏⁄4+x)+tan(∏/4-x)=2sec2x
3) cosx-cosy=-2sin(x+y/2)sin(x+y/2)
4) 2cotx-2tanx=4-2sec^2x/tanxHomework Equations
all trig identitiesThe Attempt at a Solution
1) i understand that i should show what i have attempted but there is way too...
Homework Statement
How do I put the sign for pi and the sign for theta? I can not solve this problem.
Please help because I am new and I am having difficulty in my precalculus homework. The problem says identify identities of: sin (pi/2 + theta) = cos theta...
hello,
i am supposed to show that
Sigma of k = 0 to m, (n, k) (n - k, m - k) = 2^m (n, m)
So I have after expanding:
(n, k) = n!/(n-k)!k! and (n-k, m-k) = (n-k)!/(m-k)!(n-m)!
so together the (n-k)! cancels out and I have
n!/k!(m-k)!(n-m)!
and that is
n!/m!(n-m)! which is
(n, m)...
Vector Identities ??
Having heaps of trouble with v.identities any help possible would be greatly appreciated.
Let F = (z,y,-x) and f = |F| <--- (magnitude F)
Use vector identities to calculate;
\nabla \cdot (f \nabla \times (f F))
Homework Statement
Question One: Prove that |u x v|^2 = (u . u)(v . v)-(u . v)^2 where u and v are vectors.
Question Two: Given that u = sv + tw, prove algebraically that u . v x w = 0 where u, v and w are vectors and s and t are integers.
Homework Equations
I don't know :(
The...
How do i solve this trig question? identities etc?
4*[cos(B) + 3*sin(B)]=1+ 2*[3*sin(39) - cos(39)]
i can get it to
cosB + 3sinB=0.08054076 using basic algebra, but how do i find a value for B
Homework Statement
The vectors F and G are arbitrary functions of position. Starting w/ the relations F x (∇ x G) and G x (∇ x F), obtain the identity
∇(F . G) = (F . ∇)G + (G . ∇)F + F x (∇ x G) + G x (∇ x F)
Homework Equations
The Attempt at a Solution
I started off...
Homework Statement
Use the definitions of Helmholtz free energy, F, and Gibbs free energy, H, together with the thermodynamic identity, to show that
S = -(\frac{\partial F}{\partial T})_V and S = -(\frac{\partial F}{\partial T})_P
Then use those definitions again, and the...
Homework Statement
The problem is essentially (I've rephrased it, but this is what it is asking)
show that [ (2n1cosa) / (n2cosa + n1cosb) ]2
is equal to (sin2a sin2b) / (sin2(a+b)cos2(a-b))
where these are for refraction through materials and satisfy n1sina=n2sinb
2. The...
Please help me in these identities!
I tried to solve these identities, but I don't think that these are identities exept the first,
1.) sin2α/1+cos2α=tanα
2.) 1-cosα/sinα=tanα/2
3.) tanα+ctanα=csecα
4.) 3cosα+cos3α/3sinα-sin3α=3/2
5.) sin18°+sin30°=sin54
At first i did this...
I was wondering if anyone could help me out about trig identities. I'm a HS trig teacher and I'm going "by the books" and instructing them to manipulate the left and right sides of the identity independently of each other. They are not to treat it like an equation, e.g. no moving terms from...
Homework Statement
Prove the following identities.
a) cosx/1-sinx = secx + tanx
b) cos^2x+ sinxcosx/tanx = 2cos^2x
The Attempt at a Solution
Well what I tried doing was substituting the appropriate compound angle formulas, double angle formulas, quotient identities, and...
Everyone knows the obvious trig identities like sin^2 + cos^2 =1, cosx=1+ sin^2, and tanx =sin/cos. I ran across an old identity the other day: 3cos^2(3x)+3sin^2(3x)=3. Can anyone here figure out why and how? I tried it and couldn't figure it out.
Hi Helpers:blushing:, the following is my problem:
I have to rearrange Equation(1) to make Equation(2) using trigonometric identities
(1): E(θ)=2Eo + (ΔR)^2 (c(x)cos^2(θ-θo) + c(y)sin^2(θ-θo))
(2): E(θ)=2Eo + ½(ΔR)^2 ((c(x)-c(y))cos(2(θ-θo)) + (c(x) + c(y))
I was able to get...
Is it always possible to prove combinatorial identities in a brute force way, as opposed to the preferred way of seeing that the RHS and LHS are two different ways of counting the same thing? For example, the identity
\left (^{n-1}_{k-1}\right) + \left (^{n-1}_{k}\right) = \left...
Homework Statement
Prove that sin (a+b) does not equal sin a + sin b. Let a = pi/3 and b = pi/6
Homework Equations
The Attempt at a Solution
Where I am lost is how to figure out what pi/3 or pi/6 is. Like for example, how can I know sin pi/3 is sqrt3 over 2?
Is this a...
Hi, I'm confused about using trig identities.
Homework Statement
Match the trigonometric function with one of the following: (a) -1, (b) cos(x), (c) cotx (d) 1, (e) -tan(x), (f) sin(x)
(1-cos^2x)(cscx)
Homework Equations
None that I know of.
The Attempt at a Solution
I multiply it...
Homework Statement
Be x an element in the interval [Pi/4, 3Pi/4] express cos(2x), sin x, sin (x+pi) in terms of x. You must know that, for this question, cos x = z and z will always be < 0.
Homework Equations
cos(2x) = 2 (cos(x))^2 - 1
cos(2x) = cos^2 x - sin ^2 x
sin^2 x + cos ^2 x = 1...
Homework Statement
cos(\pi-x)= -cosX
the formula is cos(A-b) = cosAcosB+sinAsinB
so i sub in the given to get..
cos\picosx + sin\pisinX
then where do i go from there? I am new to math like this, its a much higher level than what I am used to, any help would be very apprieciated...
Homework Statement
Solve for x using half-angle identities
cos (x/2) = -√2 / 2
Homework Equations
cos(x/2) = ± √(1+cosx)/2
The Attempt at a Solution
I am trying to figure out what to do with the identity, but I have no idea how to start. I know that x = 270 degrees or 3pi/2, but I do not...
Homework Statement
I have this integral to solve:
\int \frac{ab}{a^2 cos^2 t + b^2 sin^2 t} dt
The limits are 0 to 2*pi.
Homework Equations
The Attempt at a Solution
I've tried using trigonometric identities, trigonometric substitution... and many kinds of algebraic...
Homework Statement
I have been given these two equations:
x=2acos^2(x) , y = 2a(cos(x))(sin(x)) where a ranges from 0 to 5 and -2π < x < 2π
I need to prove that these equations (when you plug in values for x) create points that when plotted, give you a circle with center (x-a) and radius a...
Homework Statement
prove that cos ((pi/2)-x) = sinx
Homework Equations
The Attempt at a Solution
i extended it to: (cos pi/2) (cos -x) + (sin pi/2) (sin -x)
=1-sinx
exam coming up...need some help with these identities for practise.
prove the following:
A)
\frac{tan^3x}{1+tan^2x}+\frac{cot^3x}{1+cot^2x}=\frac{1-2sin^2xcos^2x}{sinxcos}
B)
sec^6x-tan^6x=1+3tan^2xsec^x
C)
cos^4x=\frac{3}{8}+\frac{1}{2}cos2x+\frac{1}{8}cos4x
D)...
Hi there,
I am struggling to solve for x in the following problem:-
Find all values of x in the interval 0<= x <= 360 for which: tan^2(x) = 5sec(x) - 2
I have used the identity tan^2(x) + 1 = sec^2(x) to get:
sec^2(x) - 1 = 5sec(x) - 2
and rearranged to get
sec^2(x) - 5sec(x) +...
Homework Statement
a) If c is a number and \sum a_{n} from n=1 to infinity is convergent to L, show that \sum ca_{n} from n=1 to infinity is convergent to cL, using the precise definition of a sequence.
b)If \sum a_{n} from n=1 to infinity and \sum b_{n} from n=1 to infinity are convergent...
[SOLVED] Convergent Series Identities
Homework Statement
a) If c is a number and \sum a_{n} from n=1 to infinity is convergent to L, show that \sum ca_{n} from n=1 to infinity is convergent to cL, using the precise definition of a sequence.
b)If \sum a_{n} from n=1 to infinity and \sum...
Hi all
Teacher gave me homework that i am having trouble doing it ..
i need help in these two problems which i can't seem to solve.
how can i prove the following :
@= theta
sec@ + tan@ = tan@sec@
cot@ + cos@
and
sin^2@ - tan@= tan^2@
COS^2@ - cot@
my try is...
Theres a few...
Write each expression as a single trigonometric ratio or as the number 1.
1) sint+(cott)(cost)
2) (sec x)(sin^2x)(csc x)
For number one I went like this:
sin t + ((1/cot)(cos/1))
sin t + (cos t/cot t)
sin t + (cos t/1)( sin t/cos x)
(sin t cos t)/1 + (sin t cot...
Prove the following identities:
(cos A = cos B)^2 + (sin A + sin B)^2 = 2[1+cos(A-B)]
I'm really a mess at this stuff. I missed a few important days and fell behind, so I don't reeeally know what to do when things start getting squared and whatnot., but I tried! :bugeye:
Left Side...
Homework Statement
I'm on the last section of identities entitled half angle identities. This one seems to give me some trouble because I have never encountered one with a horizontal shift in it. Tips?
tan 1/2( ß + π/2 ) = ( 1 + sin ß ) / cos ß
The question states, "Prove the identity."
(1) cos(x + y) cos(x-y) = cos(^2)x – sin(^2)y.
Should i start off using the addition and subtraction formulas for the LHS, and breaking down the perfect square for the RHS? If not or if so, how would I go about solving this problem? Step by step...
I just don't get this stuff. I've been trying on my own with the book. Also, is there a better way to post this?
Homework Statement
tanx 1 + secx
_________ + _________ = 2csc x
1 + secx tanx
I need to prove that this side equals the other.
Homework...
I'm solving a pretty descent trig identity question, but I'm stuck. I'm not going to type out the original question, but here the section that I'm stuck on: sin^4x + cos^4x and here is what I have to prove: 1-2sin^2xcos^2x
I know that I'm really close, I just can't get this section. Any help...