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.
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...
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...
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...
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...
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
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...
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...
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?
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...
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.
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...
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.
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 =...
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} *...
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...
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
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...
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...
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)...
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.
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!
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...
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 =...
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...
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...
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) +...
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)
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...
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...
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...
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...
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...
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)...
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
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...
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 =...
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...
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...
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...
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...
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?
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...
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...