What is Trigonometric: Definition and 1000 Discussions

Trigonometry (from Greek trigōnon, "triangle" and metron, "measure") is a branch of mathematics that studies relationships between side lengths and angles of triangles. The field emerged in the Hellenistic world during the 3rd century BC from applications of geometry to astronomical studies. The Greeks focused on the calculation of chords, while mathematicians in India created the earliest-known tables of values for trigonometric ratios (also called trigonometric functions) such as sine.Throughout history, trigonometry has been applied in areas such as geodesy, surveying, celestial mechanics, and navigation.Trigonometry is known for its many identities. These
trigonometric identities are commonly used for rewriting trigonometrical expressions with the aim to simplify an expression, to find a more useful form of an expression, or to solve an equation.

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  1. lfdahl

    MHB Rational trigonometric expression show tan^218°⋅tan^254°∈Q

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

    MHB Find All Trig Functions of Gamma Given cot(gamma)=4/3

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

    How Many Solutions Does the Trigonometric Equation Have in the Given Interval?

    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...
  4. Greg

    MHB Trigonometric Product Challenge sin(π/m)sin(2π/m)sin(3π/m)⋯sin(m−1)π/m=m/2^(m−1)

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

    MHB How Can You Prove the Trigonometric Identity Cos^6A+Sin^6A=1-3Sin^2ACos^2A?

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

    Trigonometric inequality problem.

    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...
  7. I

    MHB How to Solve the Trigonometric Equation 6*Sin^2(x) - 3*Sin^2(2x) + Cos^2(x) = 0?

    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) -...
  8. M

    MHB Prove Trigonometric Identities

    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...
  9. Wrichik Basu

    Problem in finding the General Solution of a Trigonometric Equation v3

    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...
  10. Wrichik Basu

    Problem in finding the General Solution of a Trigonometric Equation v2

    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...
  11. Wrichik Basu

    A problem in finding the General Solution of a Trigonometric Equation

    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...
  12. donaldparida

    B What are some strategies to prove trigonometric results

    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...
  13. Bunny-chan

    Determining a trigonometric limit

    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...
  14. Bunny-chan

    Book demonstration about trigonometric relations

    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
  15. Y

    MHB Limit of a trigonometric function

    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...
  16. lfdahl

    MHB Usage of the Rearrangement Inequality in a trigonometric expression

    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...
  17. G

    I Trigonometric series with normalised coefficients

    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...
  18. D

    I Characteristics of trigonometric function compositions like sin(sin(x))

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

    Trigonometric graph transformation

    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...
  20. M

    B Trigonometric equation -- real roots

    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...
  21. G

    Stargazing Trig Parallax: Measurement & Apparent Shift of Nearby Stars

    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...
  22. Giu1iano

    State the exact values of all the trigonometric rations for theta.

    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
  23. Albert1

    MHB Trigonometric inequality challenge

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

    MHB Trigonometric Simplification for Projectile Motion Equation

    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...
  25. anemone

    MHB Find the sum of three trigonometric terms

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

    MHB Solving Trigonometric Limits: \[\lim_{x\rightarrow 1},\lim_{x\rightarrow -1}\]

    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...
  27. A

    Trigonometric functions and integrals

    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...
  28. FritoTaco

    Solving a Trigonometric Equation

    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...
  29. FritoTaco

    Verify the Trigonometric Identity

    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...
  30. B

    Trigonometric Substitution Problem w/ Sin Substitution

    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))) /...
  31. L

    MHB How can I solve the trigonometric equation $(2-\sqrt{2})(1+\cos x)+\tan x=0$?

    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?
  32. A

    MHB Find the exact value of each of the remaining trigonometric functions of theta

    sin\theta 3/5
  33. S

    MHB How can you solve equations involving trigonometric identities?

    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
  34. S

    MHB Equations that result in quadratics in some trigonometric function

    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
  35. K

    MHB Trigonometric inequation with tangent function

    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...
  36. K

    MHB Trigonometric inequality with pi

    \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...
  37. Ryan Hardt

    Calculating Uncertainty for a Chain of Trig Functions

    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...
  38. 8

    MHB Trigonometric Identity: Tan^2-Sin^2 = Sin^2 Cos^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)...
  39. 8

    MHB Home work help: proving a trigonometric identity

    1 ___________ =csc2\theta-csc\thetacot\theta 1+cos\theta
  40. teetar

    Solving trigonometric equation of a sum of unknowns

    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...
  41. pairofstrings

    Visualize this type of Combined Trigonometric Functions

    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...
  42. ItsAnshumaan

    Graph of trigonometric functions

    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...
  43. Alettix

    Trigonometric Equation Solving: Cosine and Sine Identities for Homework

    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)...
  44. I

    MHB Solve the trigonometric equation sin(2theta-pi/6) = cos(2theta)

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

    MHB Trigonometric equation

    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?
  46. chwala

    Limit of a trigonometric function

    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...
  47. anemone

    MHB Can Trigonometric Inequalities Be Proven with Simple Equations?

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

    I Is this a Trigonometric Identity?

    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...
  49. T

    MHB Solving an integral with trigonometric substitution

    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?
  50. anemone

    MHB Evaluate a floor function involving trigonometric functions

    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...
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