Trigonometric Definition and 1000 Threads

Evaluate: \[\lim_{x\rightarrow 0}\frac{\sin (\arctan x)-\tan (\arcsin x)}{\arcsin (\tan x)-\arctan (\sin x)}\]

Show, that $$\tan^2 18^{\circ} \cdot \tan^254^{\circ} \in \Bbb{Q}.$$

Given cot(gamma)=4/3 find all possible values of the five remaining trigonometric functions of γ. Could somebody help me here? Thanks

Homework Statement The number of solutions of the equation sin2x –2cosx + 4sinx = 4 in the interval [0, 5π] is what? Homework Equations sin2x=2sinxcosx The Attempt at a Solution sin2x-2cosx+4sinx=4 ⇒2sinxcosx-2cosx+4sinx=4 ⇒sinxcosx-cosx+2sinx=2 ⇒cosx (sinx-1)=2-2sinx ⇒cosx (sinx-1)=2...

Prove that for $m=2,3,...$ $$\sin\frac{\pi}{m}\sin\frac{2\pi}{m}\sin\frac{3\pi}{m}\cdots\,\sin\frac{(m-1)\pi}{m}=\frac{m}{2^{m-1}}$$

Prove $Cos^6A+Sin^6A=1-3 \hspace{0.2cm}Sin^2 A\hspace{0.02cm}Cos^2A$ So far, $Cos^6A+Sin^6A=1-3 \hspace{0.2cm}Sin^2 A\hspace{0.02cm}Cos^2A$ $L.H.S=(Cos^2A)^3+(Sin^2A)^3$ $=(Cos^2A+Sin^2A)(Cos^4A-Cos^2ASin^2A+Sin^4A)$...

Homework Statement Find the solution of the inequality ## \sqrt{5-2sin(x)}\geq6sin(x)-1 ## Answer: ## [\frac{\pi(12n-7)}{6} ,\frac{\pi(12n+1)}{6}]~~; n \in Z##Homework Equations None. The Attempt at a Solution There are two cases possible; Case-1: ##6sin(x)-1\geq0## or...

Good day :)! Please advise how to start with the following trigonometric equation: 6*Sin^2(x) - 3*Sin^2(2x) + Cos^2(x) = 0 To be honest, I do not know what is the first steps to start with. I have tried to start with: 5*Sin^2(x) + Sin^2(x) + Cos^2(x) - 3*Sin^2(2x) = 0 1 + 5*Sin^2(x) -...

I got this problem on my term test and it's the first problem I couldn't solve on a test ever since I'm in High School. I've tried to solve it at home even, but I still couldn't manage. The thing is that it doesn't even look difficult, maybe there's something I'm not seeing, so I hope someone...

Homework Statement :[/B] Find the general solution of the Trigonometric equation: $$3\sin ^2 {\theta} + 7\cos ^2 {\theta} =6$$ Given andwer: ##n\pi \pm \frac {\pi}{6}## Homework Equations :[/B] These equations may help: The Attempt at a Solution :[/B] Please see the pic below: It...

Homework Statement :[/B] Find the general solution of the equation: $$\tan {x}+\tan {2x}+\tan {3x}=0$$ Answer given: ##x=## ##\frac {n\pi}{3}##, ##n\pi \pm \alpha## where ##\tan {\alpha} = \frac {1}{\sqrt {2}}##. Homework Equations :[/B] These equations may be used: The Attempt at a...

Homework Statement :[/B] Find the general solution of the Trigonometric equation $$\sin {3x}+\sin {x}=\cos {6x}+\cos {4x} $$ Answers given are: ##(2n+1)\frac {\pi}{2}##, ##(4n+1)\frac {\pi}{14}## and ##(4n-1)\frac {\pi}{6}##. Homework Equations :[/B] Equations that may be used: The...

I am facing a lot of problem in proving trigonometric results (advanced ones). There are a lot of formulas (compound angle related ones, transformation of sum into product ones and vice versa, multiple angle and sub-multiple angle ones). I am unable to figure out how to proceed forward in...

Homework Statement Calculate the following limit: Homework EquationsThe Attempt at a Solution I don't know how to proceed with this. I've tried to multiply by the conjugate, and to simplify the expression (x+\pi) to u, but I wasn't very sucessful. To what kind of algebric device I could...

Homework Statement [/B] In the equation between (3) and (2), why does the author says that ? Isn't the trigonometric identity actually ? 2. Homework Equations The Attempt at a Solution

Hello all, I need some guidance in solving these limits: \[\lim_{x\rightarrow \infty }x\cdot sin(x)\] \[\lim_{x\rightarrow 0 }\frac{sin(x)}{\sqrt{x}}\] \[\lim_{x\rightarrow \infty }\frac{sin(x)}{x}\] I guess that the second and third ones are somehow related to \[\lim_{x\rightarrow 0...

In a proof, I encountered the following expressions: \[\sum_{cyc}\frac{\cos^2 A}{\sin B \sin C}\geq \sum_{cyc}\frac{\cos B \cos C}{\sin B \sin C}=\sum_{cyc}\cot B \cot C =1\] My question is concerned with the validity of the inequality. The inequality is based on the use of the Rearrangement...

Hi all, I have a trigonometric function series $$f(x)={1 \over 2}{\Lambda _0} + \sum\limits_{l = 1}^\infty {{\Lambda _l}\cos \left( {lx} \right)} $$ with the normalization condition $$\Lambda_0 + 2\sum\limits_{l = 1}^\infty {{\Lambda _l} = 1} $$ and ##\Lambda_l## being monotonic decrescent...

Hello, Are there any particular properties, indentities or usages of composite trigonometric functions, say sin(sinx) or cos(sin(x))?

Homework Statement Transform the following equation: X2sin(3x) 1. Stretch vertically by a factor 9 2. Stretch horizontally by a factor 3 3. Shift to the left by a value of 1.2. The attempt at a solution 1. Stretching vertically by a factor 9 gives: 9x2sin(3x) 2. Stretching vertically by...

The number of real roots of the equation $$2cos \left( \frac {x^2 + x} {6} \right)=2^x + 2^{-x}$$ Answer options are : 0,1,2,∞ My approach : range of cos function is [-1,1] thus the RHS of the equation belongs to [-2,2] So, we have -2 ≤ 2x + 2-x ≤ 2 solving the right inequality, i got 2x...

Hi community, I get the concept of trig parallax and the apparent shift of nearby stars when viewed against a distant background, by viewing the star in say summer and then winter and it appears to move against the much further away distant background. I get what the angle p represents...

Homework Statement P(-5,-2) is a terminal point of angle theta in standard position. State the exact values of all the trigonometric rations for theta. Homework Equations x^2+y^2=r^2 csc=1/sin sec=1/cos cot=1/tan The Attempt at a Solution

Acute triangle ABC Prove :Sin A +Sin B +Sin C>Cos A + Cos B + Cos C

Hi! Long ago (more than 5 years now, actually) I got stuck with a trigonometrig formula and I haven't been able to got the point. I had an equation (with respect to f): (r - a)L^2 + r\tan \alpha \cos f \cdot L + a\tan^2 \alpha = 0, where L = \sin f - \tan \alpha \cos f. According to my...

Evaluate \tan^4 10^\circ+\tan^4 50^\circ+\tan^4 70^\circ without the help of a calculator.

Hello all I am struggling with these two limits: \[\lim_{x\rightarrow 1}\frac{sin(x^{2}-1)}{x-1}\] \[\lim_{x\rightarrow -1}\frac{sin(x^{2}-1)}{x-1}\] I know that \[\lim_{x\rightarrow 0}\frac{sin(x)}{x}=1\] but can't see how it helps me here. I tried multiplying by x+1 both the nominator...

Homework Statement I'm searching for the integral that gives arcosu Homework Equations as we know : ∫u'/[1-u^2]^0.5 dx = arcsinu derivative of arccosu = -u'/[1-u^2]^0.5 + C derivative of arcsinu= u'/[1-u^2]^0.5 The Attempt at a Solution when I type the -u'/[1-u^2]^0.5 on the online integral...

Homework Statement Find all solutions of the equation in the interval [0, 2\pi]. sin6x+sin2x=0 Homework Equations Double Angle Formulas sin2x=2sinxcosx cos2x=cos^{2}x-sin^{2}x =2cos^{2}x-1 =1-2sin^{2}x (3 formulas for cos2x) tan2x=\dfrac{2tanx}{1-tan^{2}x} Sum to Product Formula...

Homework Statement Problem 1: csc(tan^{-1}\dfrac{x}{2})=\sqrt{\dfrac{x^{2}+4}{x}} Problem 2: \sqrt{\dfrac{1-sinx}{1+sinx}}=\dfrac{|cosx|}{1+sinx} Homework Equations Quotient Identities tan\theta=\dfrac{sin\theta}{cos\theta} cos\theta=\dfrac{cos\theta}{sin\theta} Pythagorean Identites...

Homework Statement ∫(√(64 - x^2)) / x dx I must solve this using a sin substitution. Homework Equations x = 8sinΘ dx = 8cosΘ dΘ Θ = arcsin(x/8) Pythagorean Identities The Attempt at a Solution (After substitution) = ∫8cosΘ * (√(64 - 64sin^2Θ)) / 8sinΘ dΘ = ∫(cosΘ * (√(64(1 - sin^2Θ))) /...

Hi, I've tried to solve this equation: $(2-\sqrt{2})(1+\cos x)+\tan x=0$ and I've tried everything but nothing works...Does anybody have an idea?

sin\theta 3/5

Can anybody please help me solve either of these equations Solve the following equation for angles between 0 and 360 degrees 4cos²θ + 5sinθ = 3 4cot² - 6 cosec x = -6

I have absolutely no idea how to tackle either of these questions Solve the following equation for angles between 0˚ and 360˚ to 2 decimal places 4cos²θ + 5sinθ = 3 4 cot² - 6 Cosec x = -6

I just want your opinion on my attempt at a solution of this task: \tan{\dfrac{x}{2}}>\dfrac{\tan{x}-2}{\tan{x}-2} My attempt: We know that: \tan{x}=\dfrac{2\tan{\dfrac{x}{2}}}{1-\tan^2{\dfrac{x}{2}}} But, at the beginning we should set limits to tangent function: \dfrac{x}{2} \neq...

\sin{(\pi x)}>\cos{(\pi \sqrt{x})} I don't know how to solve this. I would really appreciate some help. I tried to do something, but didn't get anything. If I move cos to the left side, I can't apply formulas for sum. Since arguments of sin and cos have \pi , I think there is no way I can...

Homework Statement I have a series of 12 values that I need to calculate the Theoretical Intensity, I, using the formula below. I have found values for all variables and their uncertainties, and have calculated the I value for each set using the formula. Now I need to calculate the...

\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)...

1 ___________ =csc2\theta-csc\thetacot\theta 1+cos\theta

Homework Statement \sin (x) = \frac{2}{3} and \sec (y) = \frac{5}{4}, where x and y lie between 0 and \frac{\pi}{2} evaluate \sin (x + y) Homework Equations Looked over some trig laws, don't think I saw anything that's too relevant. There \sec (x) = \frac{1}{\sin (x)} The Attempt at a...

Homework Statement Show that sin 600° . cos 330° + cos 120° . sin 150° = - 1 Homework Equations I know that sinΘ = opposite/hypotenuse and cosΘ = adjacent/hypotenuse. The Attempt at a Solution I am equipped with knowledge about what sinΘ and cosΘ is from right angled triangle. I stand in...

This is not a homework question but a general doubt. Suppose we have a function y = pcosx, where 'p' is an arbitrary constant. So my question is how will the graph of this function change with different values of 'p'? This doubt can also be extended for other functions like y = pex, y = p...

Homework Statement The following equation is to be solved for all x: ## \cos(x) + \cos(3x) = \sin(x) + \sin(3x)## Homework Equations The tripple angle formulas: ## \cos(3x) = 4\cos^3(x) - 3\cos(x) ## ##\sin(3x) = 3\sin(x) - 4\sin^3(x) ## The Pythagorean trig identity: ## \sin^2(x) + \cos^2(x)...

i need help with question 4a. Thank to anyone who helped.

A=3sinx+4cosx and B=3cosx-4sinx if B = 4 find A. What i tried is to use 4=3cosx-4sinx and solve for cosx now cosx = (4+4sinx)/3 plug this into A I end up getting A = (25sinx+16)/3 am I correct?

Homework Statement Mod note: Edited the following to fix the LaTeX[/B] compute ##\lim_{n \rightarrow +0} \frac {8-9cos x+cos 3x} {sin^4(2x)}####\lim_{n \rightarrow +\infty} \frac {\sin(x)} x## ##\lim_{n \rightarrow +\infty} \frac {\sin(x)} x##ok find limit as x→0 for the function ##[ 8-9cos x...

Prove \tan x+\tan y+\tan z\ge \sin x \sec y+\sin y\sec z+\sin z \sec x for $x,\,y,\,z\in \left(0,\,\dfrac{\pi}{2}\right)$.

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...

I have this integral: $$\int_{}^{} \frac {x^2}{{(4 - x^2)}^{3/2}}\,dx$$ I can see that we can substitute $x = 2sin\theta$, and $dx = 2cos\theta d\theta$, but I am unable to see how $\sqrt{4 - x^2} = 2cos\theta$. How can I get this substitution?