Trigonometric Definition and 1000 Threads

Evaluate \left\lfloor{\tan^4 \frac{3\pi}{7}+\tan^4 \frac{2\pi}{7}+2\left(\tan^2 \frac{3\pi}{7}+\tan^2 \frac{2\pi}{7}\right)}\right\rfloor. Hi MHB, I don't know how to solve the above problem, as I have exhausted all possible methods that I could think of, and I firmly believe there got to be...

Find all reals x,\,y,\,z \in \left[0,\,\frac{\pi}{2}\right] that satisfying the system below: $\sin x \cos y=\sin z\\\cos x \sin y=\cos z$

I have this integral to solve. $$\int_{}^{} (sinx + cos x)^2 \,dx$$ I first start by simplifying the expression: $$\int_{}^{} sin^2x + 2sinxcosx + cos^2x \,dx$$ $$2sinxcosx$$ is $$sin2x$$ (a trigonometric identity) and $$ sin^2x + cos^2x = 1 $$ a trigonometric identity. So, after...

Homework Statement is the circled part wrong ? Homework EquationsThe Attempt at a Solution how could 0.5 sin2theta = ( 1 + cos2theta ) / 2 ?

Given that \frac{\sin 3x}{\sin x}=\frac{6}{5}, what is the ratio of \frac{\sin 5x}{\sin x}?

I noticed the graphs of ##y=\cos^{-1}x## and ##y=\cosh^{-1}x## are similar in the sense that the real part of one is the imaginary part of the other. This is true except when ##x<-1## where the imaginary part of ##y=\cos^{-1}x## is negative but the real part of ##y=\cosh^{-1}x## is positive. I...

Homework Statement a) Differentiate the following equation with respect to: 1) θ 2) Φ 3) ψ (Ua - Ub)' * C * r where: C is a 3 x 3 rotation matrix: [ cos θ cos ψ, -cos Φ sin ψ + sin Φ sin θ cos ψ, sin Φ sin ψ + cos Φ sin θ cos ψ] [ cos θ sin ψ, cos Φ cos ψ + sin Φ sin θ sin...

Say I have the integral of [ 1 / ( sqrt( 1 - x^2) ] * dx . Now I was told by many people in videos that I substitute x = sin theta, and this has me confused. Wouldn't I need to substitute x = cos theta instead? as x = cos theta on the unit circle instead of sin theta? Thanks in advance for...

Let the real $x\in \left(0,\,\dfrac{\pi}{2}\right)$, prove that $\dfrac{\sin^3 x}{5}+\dfrac{\cos^3 x}{12}≥ \dfrac{1}{13}$.

Explain to me: Why the 2πf came in front? I lost touch and sort of forgot.

How to extract l and L from the following system of equations:

Homework Statement ##\lim_{a\rightarrow b} \frac{tan\ a - tan\ b}{1+(1-\frac{a}{b})\ tan\ a\ tan\ b - \frac{a}{b}}## = ...Homework Equations tan (a - b) = (tan a - tan b)/(1+tan a tan b) The Attempt at a Solution [/B] I don't know how to convert it to the form of tan (a-b) since there are...

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} *...

Homework Statement ##tanx=\frac{(1+tan1)(1+tan2)-2}{(1-tan1)(1-tan2)-2}## find x Homework Equations 3. The Attempt at a Solution [/B] I tried multiplying through the paranthesis and arrived at ##tanx=\frac{(tan1tan2-1)+(tan2+tan1)}{(tan1tan2-1)-(tan2+tan1)}## and i don't know if this is any...

To make it useful for any angles. I need a good explanation for this.

Homework Statement 2sin2θ - 1 = sin2θ - cos2θ Homework EquationsThe Attempt at a Solution I am unsure of how to prove these. So far all I have is Left side= 2sin2θ - 1 =sin2sin2-1 And I know that right side is equal to 1. But otherwise not sure where to go from there.

Homework Statement Question: Sum of all the solutions of the equation: ##tan^2 (33x) = cos(2x)-1## which lie in the interval ## [0, 314] ## is: (a) 5050 π (b) 4950 π (c) 5151 π (d) none of these The correct answer is: (b) 4950 π Homework Equations ## cos(2x) = 2cos^2(x) -1 ## The Attempt...

Prove that for all real numbers $x$, we have \left(2^{\sin x}+2^{\cos x}\right)^2\ge2^{2-\sqrt{2}}.

For real numbers 0\lt x\lt \frac{\pi}{2}, prove that $\cos^2 x \cot x+\sin^2 x \tan x\ge 1$.

Homework Statement sin x = C*sin y Find y as a function of x for a given C>0. Homework Equations sin x = C*sin y The Attempt at a Solution This is not actually a problem from a book, but a problem I myself thought about. I was studying elastic collisions in SCM and I obtained 2 equations...

Prove that \frac{\sin^3 x}{(1+\sin^2 x)^2}+\frac{\cos^3 x}{(1+\cos^2 x)^2}\lt \frac{3\sqrt{3}}{16} holds for all real $x$.

Maximize $\sin x \cos y+\sin y \cos z+\sin z \cos x$ for all real $x,\,y$ and $z$.

Homework Statement I want to prove that: Homework EquationsThe Attempt at a Solution I tried using the trigonometric identity: sen2x = senx cosx / 2, so, I got: 1/2m∫(sen2x)mdx, x from 0 to pi/2, but now I don't know how to proceed. Can you help me please?

Homework Statement verify that theta in L = piD + (d-D)theta + 2Csin(theta) is equal to arc-cosine [(D-d)/2C] 2. The attempt at a solution you can see my attempt in the second picture uploaded. i don't think i even got it right

Factorize $\cos^2 x+\cos^2 2x+\cos^2 3x+\cos 2x+\cos 4x+\cos 6x$.

Prove $$\sum_{n=0}^N\cos(nx)=\csc\left(\dfrac x2\right)\sin\left(\dfrac{(N+1)x}{2}\right)\cos\left(\dfrac{Nx}{2}\right)$$ I've tried working from the RHS with various identities but haven't managed to come up with anything that works. I suspect this problem involves some trigonometry that I...

Homework Statement Arbitrary derivative of inverse trigonometric function: (sin-1x) = 1/(√1 - x2) Homework Equations f-1(f(x)) = 1/f`(x) The Attempt at a Solution So basically I learned about derivatives of trigonometric functions in class, and I thought maybe this would work: deriving the...

What's $1. ~ \displaystyle \arccos(\cos\frac{4\pi}{3})?$ Is this correct? The range is $[0, \pi]$ so I need to write $\cos\frac{4\pi}{3}$ as $\cos{t}$ where $t$ is in $[0, \pi]$ $\cos(\frac{4\pi}{3}) = \cos(2\pi-\frac{3\pi}{3}) = \cos(\frac{2\pi}{3}) $ so the answer is $\frac{2\pi}{3}$

Homework Statement I have included the LaTex version of the problem. \int \frac{sin^2 x}{1+cos^2 x} dx Homework Equations Simplifying fraction Partial fractions The Attempt at a Solution I have uploaded my attempt at the solution.

Prove that $\sin 1+\sin 2+\sin 3+\cdots+\sin n<2$.

Prove that $(\sin \theta+ i \cos \theta)^8 = \cos 8\theta - i \sin 8\theta$

Show that this series diverges: $$\sum_{n = 0}^\infty \cos \left ( n^2 \right )$$ (in the sense that it takes arbitrarily large values as $n \to \infty$)

Homework Statement ok so my professor gave me this problem to solve, it goes like this :(I will also have a picture below) In the square (ABCD) is a point P which divides the side BC into 2 halves and point R which divides the side CD into 2 halves The angles at APB and ARB are the same...

When using trigonometric substitution in calculus you're supposed to always keep in mind the domain of the angle. In the case of √(x2-a2) (where "a" is a number >0) you use x=a⋅arcsec Θ for the substitution. For trigonometric substitution, textbooks state that the domain of Θ must be...

This problem probably should be easy, but I don't remember learning the basic way to do these problems: Write the trigonometric expression as an algebraic expression: cos(arccos x + arcsin x) The answer is zero, but I don't know how to get there...

Let $\dfrac{\cos^4 a}{x}+\dfrac{\sin^4 a}{y}=\dfrac{1}{x+y}$ for all real $a,\,b,\,x,\,y$. Prove that $\dfrac{\cos^8 a}{x^3}+\dfrac{\sin^8 a}{y^3}=\dfrac{1}{(x+y)^3}$

Homework Statement There's no reason to give you the problem from scratch. I just want to show that 5 trigonometric functions are linearly independent to prove what the problem wants. These 5 functions are sin2xcos2x. sin2x, cos2x, sin2x and cos2x. Homework Equations...

Homework Statement cos(x - 3p/2) = - 4/5 p <x< p/2 sin(x/2)*cos(5p/4)= ? Homework Equations The Attempt at a Solution I made it as far as to determine that sinx= 4/5 and cosx = - 3/5 but can't seems to progress any further. I am looking for an easier way to find the solution without having to...

http://www5a.wolframalpha.com/Calculate/MSP/MSP238521i5b83i951f19c3000010ca05be63f0bfc0?MSPStoreType=image/gif&s=10&w=219.&h=85. How do I solve this? I know the answers, as Wolphram Alpha has given me only the answers without any steps to how they derived those answers. I know that sin(x)=√3/2...

Homework Statement Evaluate the integral: integral of dx / (4+x^2)^2 Homework Equations x = a tan x theta a^2 + x^2 = a^2 sec^2 theta The Attempt at a Solution x = 2 tan theta dx = 2sec^2 theta tan theta = x/2 integral of dx / (4+x^2)^2 = 1/8 integral (sec^2 theta / sec^4 theta) d theta =...

Homework Statement Show that the real solution ##x## of $$tanhx=cosechx$$ can be written in the form ##x=ln(a \pm \sqrt{a})## and find an explicit value for ##a##. Homework Equations $$cosh^{2}x-sinh^{2}x=1$$ $$coshx=\frac{e^{x}+e^{-x}}{2}$$ The Attempt at a Solution I reduced the original...

Homework Statement For $$\lim _{ x\rightarrow \infty }{ \frac { { x }^{ 2 }+{ e }^{ -{ x }^{ 2 }\sin ^{ 2 }{ x } } }{ \sqrt { { x }^{ 4 }+1 } } } $$, determine whether it exists. If it does, find its value. if it doesn't, explain. Homework Equations Sand witch theorem and arithmetic rule...

I'm struggling to solve the following integral ∫ x/(√27-6x-x2) my attempt is as follows: ∫x/(√36 - (x+3)2) = ∫1/ √(36 - (x+3)2) + ∫x+1/ √(36 - (x+3)2) = arcsin (x + 3)/6 + this is where I got stuck.

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...

Solve for $x$ such that $2\sin(x+30^\circ)\sin 16^\circ \sin 76^\circ=\sin 2028^\circ \sin 210^\circ$ for $0\lt x \lt 180^\circ$.

1) If \tan(\pi/4)=1, find \cot(\pi-\pi/4). 2) If \cot(17^{\circ}) = 3.2709, find \tan(73^{\circ}) 3) If \cot(\theta) = \frac{-9}{2} with \theta in Quadrant II, find \sin (\theta) --------------------------------------------- I really have no idea how to solve any of these problems. I have...

Homework Statement \frac{d}{dx}7.5sin(\frac{pi}{10}x) The Attempt at a Solution 7.5(\frac{pi}{10})cos(\frac{pi}{10}x) Maximum: f'(x) = 0 7.5(\frac{pi}{10})cos(\frac{pi}{10}x) = 0 7.5(\frac{pi}{10})cos^{-1}(0)= \frac{pi}{10}x ** (\frac{pi}{10}\frac{10}{pi})7.5(90) = x (1)(7.5)(90) = x...

Homework Statement ∫(t/√(1-t4))dt Homework Equations ∫(du/√(a2 - u2)) = arcsin (u/a) + C ∫(du/(a2 + u2) = (1/a) arctan (u/a) ∫(du/(u√(u2 - a2))) = (1/a) arcsec (|u|/a) The Attempt at a Solution Edit: I meant to write u where t2 is[/B]

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}} =...

http://tutorial.math.lamar.edu/Classes/CalcII/TrigSubstitutions.aspx In example one, the author drops the absolute value bars and makes the following statement: "Without limits we won’t be able to determine if ##\tan{\theta}## is positive or negative, however, we will need to eliminate them in...