What is Trigonometric identity: Definition and 85 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.
I'm trying to use the following trigonometric identity:
$$ a \cos ( \omega t ) + b \sin ( \omega t ) = \sqrt{a^2+b^2} \cos ( \omega t - \phi )$$ Where ##\phi = \tan^{-1} \left( \frac{b}{a} \right)## for the following equation:
$$ x(t) = -\frac{g}{ \omega^2} \cos ( \omega t) + \frac{v_o}{...
Attempt : I could not progress far, but the following is what I could do.
$$\begin{align*}
\mathbf{\text{LHS}} & = (\tan A+\tan B+\tan C)(\cot A+\cot B+\cot C) \\
& = 3+\tan A \cot B+\tan B \cot A+\tan A \cot C+\tan C \cot A+\tan B \cot C+\tan C \cot B\\
& = 3+\frac{\tan^2A+\tan^2B}{\tan A \tan...
(Sinx-2cosx)/ (cotx - sinx)
Substitute tan instead of cot
(Tanx(sinx-2cosx)/(1-sinx)
What do I do from here
I don't think what I did there is correct
That's why I didn't expand the tan to sin/cos
Hi,
K₁cos(θt+φ)=K₁cos(θt)cos(φ)-K₁sin(θt)sin(φ)=K₁K₂cos(θt)-K₁K₃sin(θt)
Let's assume φ=30° , K₁=5
5cos(θt+30°) = 5cos(θt)cos(30°)-5sin(θt)sin(30°) = (5)0.866cos(θt)-(5)0.5sin(θt) = 4.33cos(θt)-2.5sin(θt)
If only the final result, 4.33cos(θt)-2.5sin(θt), is given, how do I find the original...
Homework Statement
Show that ##\arcsin 2x \sqrt{1-x^2} = 2 \arccos{x}## when 1/√2 < x < 1
Homework Equations
All trigonometric and inverse trigonometric identities, special usage of double angle identities here
The Attempt at a Solution
I can get the answer by puting x=cosy, the term inside...
I'm trying to make an approximation to a series I'm generating; the series is constructed as follows:
Term 1:
\left[\frac{cos(x/2)}{cos(y/2)}\right]
Term 2:
\left[\frac{cos(x/2)}{cos(y/2)}-\frac{sin(x/2)}{sin(y/2)}\right]
I'm not sure yet if the series repeats itself or forms a pattern...
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...
\tan\left({^2}\right)-\sin\left({^2}\right)=\tan\left({^2}\right) \sin\left({^2}\right)
i keep on getting \sin\left({^2}\right)-\sin\left({^2}\right) \cos\left({^2}\right)=\sin\left({^2}\right) \sin\left({^2}\right)
\cos\left({^2}\right)...
I have encountered this equation:
##\cos^2 \gamma = \cos^2 \alpha \cdot \cos^2 \beta##
According to the paper, this is a trigonometric identity, but this is the first time I have encountered this. The angles ##\alpha## and ##\beta## are somewhat similar to the components of the distance...
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} *...
Hi,
This is also a sort of geometry question.
My textbook gives a proof of the relation: sin(θ + Φ) = cosθsinΦ + sinθcosΦ.
It uses a diagram to do so:
http://imgur.com/gLnE2Fn
sin (θ + Φ) = PQ/(OP)
= (PT + RS)/(OP)
= PT/(OP) + RS/(OP)
= PT/(PR) * PR/(OP) + RS/(OR) * OR/(OP)
= cosθsinΦ +...
The problem
Show that the left side is equal to right side
## tan (\frac{x}{2}) = \frac{1-cos(x)}{sin(x)} ##
The attempt
##\tan(\frac{x}{2}) = \frac{ sin(\frac{x}{2}) }{ cos (\frac{x}{2}) } = \frac{ sin^2(\frac{x}{2}) }{ cos ^2 (\frac{x}{2}) } = \frac{\frac{1-cos(x)}{2}}{\frac{1+cos(x)}{2}} =...
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...
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
EDIT: I figured out my mistake...no option to delete silly post. Oh well.
1. Homework Statement
The problem is: use iterated integrals in polar form to find the area of one leaf of the rose-shaped curve r = cos(3*theta).
My setup agrees exactly with the solutions manual...but then something...
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...
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...
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...
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 =...
Double integral of (52-x^2-y^2)^.5
2<_ x <_ 4
2<_ y <_ 6
I get up to this simplicity that results in a zero!
1-cos^2(@) - sin^2(@) = 0
This identity seems to be useless.
HELP PLEASE.
Can it be proved?
\left(\frac{-2\sin A}{1-\cos A}\right)\cos\left(\frac{A}{2}\right)\tan^{-1}\left[\cos \left(\frac{A}{2}\right)\right]=\frac{\pi^2-4A^2}{8}
Your help will be greatly appreciated!
Thanks!1. The expression \(\sin\pi\) is equal to \(0\), while the expression $\frac{1}{\csc\pi}$ is undefined. Why is $\sin\theta=\frac{1}{\csc\theta}$ still an identity?
2. Prove $\cos(\theta + \frac{\pi}{2})= -\sin\theta$
So I'm trying to get through euler's introduction to the analysis of the infinite so I could eventually read his books on calculus but I'm stuck somewhere and can't seem to figure out how he equates this identity
so by expanding I get sin(2y) * cos(z) + cos(2y) * sin(z).
I get that the...
"Sum to Product" Trigonometric identity does not work
Hi,
The identity
sin(u) + sin(v) = 2 * sin (\frac{u+v}{2}) * cos(\frac{u-v}{2})
http://en.wikipedia.org/wiki/List_of_trigonometric_identities#Product-to-sum_and_sum-to-product_identities
Does not always work. I put the equation :
(sin(u)...
Here is the question:
Here is a link to the question:
Prove the identity, pre calc!? - Yahoo! Answers
I have posted a link there to this topic so the OP can find my response.
Homework Statement
Given the following two triangles:
Show that v \cos{\delta} = V(1-\cos{\beta})+u\cos(\alpha - \beta)
The Attempt at a Solution
Using the cosine law I've got:
v^{2}=x^{2}+V^{2}-2xV\cos{(\theta + \beta)}
and u^{2}=x^{2}+V^{2}-2xV\cos{(\theta)}
I figured maybe using the...
Homework Statement
(question attached)
Homework Equations
The Attempt at a Solution
Checking solution.. pretty sure I did this wrong.
(solution attached)
Here is the question:
Here is a link to the question:
http://answers.yahoo.com/question/index?qid=20130130130636AAOqgvz
I have posted a link there so the OP can find my response.
Homework Statement
Homework Equations
Any trig formulas
The Attempt at a Solution
The yellow paper is me switching everything to sin and cos to see if that helps but it doesn't. I'm completely stuck here.
Homework Statement
To prove that \sum over m=1 to 15 of sin(4m-2) = 1/4sin2, where all angles are in degress
Homework Equations
The Attempt at a Solution
Tried to solve it using identity sinx+siny=2sin((x+y)/2)cos((x-y)/2)..but all attempts failed..help
Homework Statement
Show that :
1 + cos(2∏/5)= 2 cos(∏/5)
Homework Equations
cos(2x) = cos^2(x)-sin^2(x)
cos^2(x)+sin^2(x) = 1
The Attempt at a Solution
I have tried using the two formulas above but i couldn't show the required result.
Homework Statement
Here is the question given:
A blade for a lawnmower consists of two parts made of the same material and joined together as shown:
The length OP is one unit in length and MPQN is square in shape.
Develop an equation for the cross-sectional area of the blade and...
Homework Statement
Prove that:
tan^2∅/tan∅ - 1 + cot^2∅/cot∅ - 1 = 1 + sec∅cosec∅
Homework Equations
The Attempt at a Solution
I have solved the question taking tan∅ = sin∅/cos∅.
But I want to solve it some other way.
it's bothering my brain..i thought about it many times...i can't make intuition of it
can anyone prove it?
oh by the way... C = Sqrt[A^2 + B^2] and theta is equal to arctan(B/A)