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. chwala

    Solve the given trigonometry problem

    My question is on the highlighted part (circled in red); Why is it wrong to pre-multiply each term by ##e^x##? to realize , ##5e^{2x} -2-9e^x=0## as opposed to factorising by ##e^{-x} ## ? The other steps to required solution ##x=\ln 2## is quite clear and straightforward to me.
  2. T

    I Integrating a product of exponential and trigonometric functions

    I am looking for a closed form solution to an integral of the form: $$ \int_0^\infty \frac{e^{-Du^2t}u \sin{ux}}{u^2+h^2} du $$ D, t, and h are positive and x is unrestricted. I have tried everything, integration by parts, substitution, even complex integration with residue analysis. I've...
  3. P

    Equation involving inverse trigonometric function

    I came across the mentioned equation aftet doing a integral for an area related problem.Doing the maclaurin series expansion for the inverse sine function,I considered the first two terms(as the latter terms involved higher power of the argument divided by factorial of higher numbers),doing so...
  4. Fred1230

    How to Simplify This Trigonometric Equation Using Substitutions?

    Returning if I have to show the effort, I came to this: \frac{\sin4\alpha}{1+\cos4\alpha}\cdot\frac{\cos2\alpha}{1+\cos2\alpha}\cdot\frac{\cos\alpha}{1+\cos\alpha}=\tan\frac{\alpha}{2}. =...
  5. chwala

    Prove the hyperbolic function corresponding to the given trigonometric function

    ##8 \sin^4u = 3-4\cos 2u+\cos 4u## ##8 \sinh^4u = 3-4(1+2\sinh^2 u)+ \cosh ( 2u+2u)## ##8 \sin^4u = 3-4-8\sinh^2 u+ \cosh 2u \cosh 2u + \sinh 2u \sinh 2u## ##8 \sinh^4u = 3-4+1-8\sinh^2 u+ 4\sinh^2u +4\sinh^4 u + 4\sinh^2 u + 4\sinh^4 u## ##8 \sinh^4u = -8\sinh^2 u+ 8\sinh^2u +8\sinh^4 u##...
  6. R

    B Are Both Answers Correct for Trigonometric Substitution Integral?

    Last night I tried to calculate from an automatically generated Wolfram Alpha problem set: $$\int{\frac{1}{\sqrt{x^2+4}}}dx$$ I answered $$\ln({\frac{\sqrt{x^2+4}}{2}+\frac{x}{2}})+C$$ The answer sheet gave: $$\ln({\sqrt{x^2+4}+x})+C$$ I couldn't see where I had gone wrong, so I tried...
  7. E

    B Trigonometric Identity involving sin()+cos()

    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}{...
  8. E

    Apply trigonometric methods in solving problems

    Summary: Hey, I'm getting confused with this question and don't think I'm doing it right, I was wondering if anyone could help me Tides vary so the high tide and low tide height of the water is different every day. At certain times of the year, such as a Spring tide, the water can be very deep...
  9. WMDhamnekar

    Evaluation of integral having trigonometric functions

    R is the triangle which area is enclosed by the line x=2, y=0 and y=x. Let us try the substitution ##u = \frac{x+y}{2}, v=\frac{x-y}{2}, \rightarrow x=2u-y , y= x-2v \rightarrow x= 2u-x + 2v \therefore x= u +v## ## y=x-2v \rightarrow y=2u-y-2v, \therefore y=u- v## The sketch of triangle is as...
  10. Eobardrush

    Trigonometric problem: Sin7x= Sin24 * 6.4

    So I came across a question regarding sine rule. This is to find a missing angle. The question goes as follows Sin7x= Sin24 * 6.4 I tried two methods to solve this. Method 1: x= (Sin24 * 6.4)/ Sin 7 =21.35 I basically divided both sides by Sin 7 Method 2: Sin x= (Sin 24 * 6.4)/ 7 x=...
  11. M

    Simultaneous Trigonometric Equations - solving for angles

    Summary:: I have a series of three equations that transform three angles of a system (J1, J2, J3), into three spatial x, y, z coordinates. I want to invert them to find the angles from the coordinates. Reference: https://www.physicsforums.com/forums/general-math.73/post-thread I have a series...
  12. D

    MHB Can you prove the following two difficult trigonometric identities?

    Can you prove the following? [sec(x)]^6 - [tan(x)]^6 = 1 + 3*[tan(x)]^2*[sec(x)]^2 [sin(x)]^2*tan(x) + [cos(x)]^2*cot(x) + 2*sin(x)*cos(x) = tan(x) + cot(x) If not, the following free math tutoring video shows you the method:
  13. vibha_ganji

    Help Solving a Trigonometric Equation

    I’m stuck on how to begin. I’ve tried to factor out sin theta from both of the terms on the left hand side but that led to nowhere. Could I have a hint on how to continue? Than you!
  14. B

    MHB Problems involving Trigonometric Identities

    What are the step-by-step in solving these problems?
  15. rudransh verma

    Trigonometric ratios of angles above 90 degrees

    I have been doing the resolutions of vectors on x and y-axis with making triangles and reference angles in all quadrants. But I want to calculate now how to find something like ##\sin 235## without the help of reference angles. I know we don’t need to. Calculator and Taylor theorem is handy here...
  16. chwala

    Solve the trigonometric equation below

    Solve the equation, $$cos ∅+ \sqrt 3⋅ sin ∅=1$$ in the interval, $$0≤∅≤2π$$
  17. K

    MHB Proving trigonometric functions

    How can i prove that 6cos(x+45) cos(x-45) is equal to 3cosx?
  18. P

    I Solving for ##\theta## in a Trigonometric Equation

    Can a situation at angle ##\theta## happen, where: ##d sin \theta = 3 \lambda## ##b sin \theta = 2 \lambda## ##d/b = 3/2##
  19. Leo Liu

    How to simplify an iterated trigonometric expression

    eg ##\cos (\sin x)## Asking this question out of curiosity.
  20. brotherbobby

    Trigonometric equation of two sines

    Given : The equation ##\sin m\theta + \sin n\theta = 0##. Attempt : Using the formula for ##\text{sin C + sin D}## (see Relevant Equation 3 above), the given equation simplifies to \begin{equation*} 2 \sin \frac{(m+n)\theta}{2} \cos \frac{(m-n)\theta}{2} = 0 \end{equation*} This implies the...
  21. N

    MHB Limit of Trigonometric Function....2

    Find the limit of csc(2x) as x tends to pi/2 from the right side. I decided to graph the function. Based on the graph, I stated the answer to be positive infinity. According to the textbook, the answer is negative infinity. Why is negative infinity the right answer? Thanks
  22. N

    MHB Limit of Trigonometric Function....1

    Find the limit of cot (x) as x tends to pi from the left side. Seeking a hint or two. Does the graph of the given function help in terms of finding the limit?
  23. anemone

    MHB Is this Trigonometric Expression a Constant Function of x?

    Prove $\sin^2(x+a)+\sin^2(x+b)-2\cos (a-b)\sin (x+a)\sin (x+b)$ is a constant function of $x$.
  24. anemone

    MHB What is the remainder when m+n is divided by 1000 in a trigonometric challenge?

    Let $x$ be a real number such that $\dfrac{\sin^4 x}{20}+\dfrac{\cos^4 x}{21}=\dfrac{1}{41}$. If the value of $\dfrac{\sin^6 x}{20^3}+\dfrac{\cos^6 x}{21^3}$ can be expressed as $\dfrac{m}{n}$ where $m$ and $n$ are relatively prime positive integers, find the remainder when $m+n$ is divided by 1000.
  25. U

    MHB Solve Situational Problems Involving Trigonometric Identities

    Hi! I am so confused about the given and what is being asked, I don't know how to solve it. This topic is solving situational problems involving trigonometric identities. Your help would be a big one for me :) Thank you so much in advance!
  26. anemone

    MHB Can the Sum of Two Trigonometric Functions Be Less than pi/2?

    Prove that $\cos(\sin x))+\cos(\cos x))<\dfrac{\pi}{2}$.
  27. anemone

    MHB Trigonometric of tangent and sine functions

    Simplify $\left(\tan \dfrac{2\pi}{7}-4\sin \dfrac{\pi}{7}\right)\left(\tan \dfrac{3\pi}{7}-4\sin \dfrac{2\pi}{7}\right)\left(\tan \dfrac{6\pi}{7}-4\sin \dfrac{3\pi}{7}\right)$.
  28. brotherbobby

    To prove a trigonometric identity with tan() and cot()

    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...
  29. anemone

    MHB Can You Prove this Trigonometric Inequality?

    If $x\in \left(0,\,\dfrac{\pi}{2}\right)$, $0\le a \le b$ and $0\le c \le 1$, prove that $\left(\dfrac{c+\cos x}{c+1}\right)^b<\left(\dfrac{\sin x}{x}\right)^a$.
  30. John Greger

    I Units of trigonometric functions?

    What are the units of the trigonometric functions sinus, cosinus etc? If I take say Sin(0.5), what would the units of the output be?
  31. mcastillo356

    Calculators How can I prove my calculator calculates a trigonometric function?

    Considering the measure of angles in radians, that are real numbers, the concept of of trigonometric function spreads to all real numbers. Any real number can be considered as an angle of the first circumference and a ##\mathbb{K}## number of circumferences. We can consider the function...
  32. M

    Inverse trigonometric functions

    Create one equation of a reciprocal trigonometric function that has the following: Domain: ##x\neq \frac{5\pi}{6}+\frac{\pi}{3}n## Range: ##y\le1## or ##y\ge9## I think the solution has to be in the form of ##y=4sec( )+5## OR ##y=4csc( )+5##, but I am not sure on what to include...
  33. AN630078

    Solving Reciprocal Trigonometric Equation cot^2θ+5cosecθ=4

    cot^2θ+5cosecθ=4 cot^2θ+5cosecθ-4=0 cosec^2θ+5cosec-4-1=0 cosec^2θ+5cosec-5=0 Let u=cosecθ u^2+5u-5=0 Solve using the quadratic formula; u=(-5± 3√5)/2 u=(-5+ 3√5)/2=0.8541... Substitute cosecθ=u Therefore, cosecθ=0.8541 1/sinθ=0.8541 sinθ=1/0.8541=1.170... which is not true since sin x cannot be...
  34. AN630078

    Factor Theorem and Trigonometric Equations Help

    1. The factor theorem states that (x-a) is a factor of f(x) if f(a)=0 Therefore, suppose (x+1) is a factor: f(-1)=3(-1)^3-4(-1)^2-5(-1)+2 f(-1)=0 So, (x+1) is a factor. 3x^3-4x^2-5x+2=(x+1)(3x^2+...) Expand the RHS = 3x^3+3x^2 Leaving a remainder of -7x^2-5x+2 3x^3-4x^2-5x+2=(x+1)(3x^2-7x+...)...
  35. AN630078

    Trigonometric equation solving 2cos x=tan x

    a. I have just plotted the graph using desmos and attached an image here. Clearly, there are two values of x that satisfy the equation in the range. Do I need to add anything to this statement, I feel the response is a little brief for the question? b. Using the trigonometric identities; tan...
  36. AN630078

    Solving trigonometric equations as fractions of π

    Question 1; a. sin θ=√3/2 θ=arcsin √3/2 θ=π/3 rad sin √3/2=60 degrees 60 degrees *π/180=π/3 rad. To find the other solutions in the range, sin θ=sin(π-θ) π-π/3=2π/3 The solutions are π/3 and 2π/3 in the range 0 ≤θ ≤2 π b. cos2θ=0.5 2θ=arccos 0.5 2θ=π/3 rad Divide both sides by 2; θ=π/6 rad...
  37. AN630078

    Trigonometric Equations Problems - Rather Confused

    Question 1; a. sin θ=√3/2 θ=arcsin √3/2 θ=π/3 rad sin √3/2=60 degrees 60 degrees *π/180=π/3 rad. To find the other solutions in the range, sin θ=sin(π-θ) π-π/3=2π/3 The solutions are π/3 and 2π/3 in the range 0 ≤θ ≤2 π b. cos2θ=0.5 2θ=arccos 0.5 2θ=π/3 rad Divide both sides by 2; θ=π/6 rad...
  38. anemone

    MHB What is the Trigonometric Inequality for $0<x<\dfrac{\pi}{2}$?

    Show that for all $0<x<\dfrac{\pi}{2}$, the following inequality holds: $\left(1+\dfrac{1}{\sin x}\right)\left(1+\dfrac{1}{\cos x}\right)\ge 5\left[1+x^4\left(\dfrac{\pi}{2}-x\right)^4\right]$
  39. anemone

    MHB Prove Trig Identity: $\sin^7 x=\dfrac{35\sin x-21\sin 3x+7\sin 5x-\sin 7x}{64}$

    Prove that $\sin^7 x=\dfrac{35\sin x-21\sin 3x+7\sin 5x-\sin 7x}{64}$.
  40. xyz_1965

    MHB Is tan(x)^2 proper notation for the trig function tangent squared?

    Is tan^2 (x) the same as tan(x)^2? Note: I could have used any trig function. I know that tan^2 (x) means (tan x)^2. What does tan (x)^2 mean? Is it proper notation?
  41. anemone

    MHB What is the sum of these trigonometric fractions?

    Evaluate $\dfrac{1}{1-\cos \dfrac{\pi}{9}}+\dfrac{1}{1-\cos \dfrac{5\pi}{9}}+\dfrac{1}{1-\cos \dfrac{7\pi}{9}}$.
  42. anemone

    MHB Integral of trigonometric function

    Prove that if $[a,\,b]\subset \left(0,\,\dfrac{\pi}{2}\right)$, $\displaystyle \int_a^b \sin x\,dx>\sqrt{b^2+1}-\sqrt{1^2+1}$.
  43. agnimusayoti

    Fourier series for trigonometric absolute value function

    First, I try to define the function in the figure above: ##V(t)=100\left[sin(120\{pi}\right]##. Then, I use the fact that absolute value function is an even function, so only Fourier series only contain cosine terms. In other words, ##b_n = 0## Next, I want to determine Fourier coefficient...
  44. anemone

    MHB What is the solution to this trigonometric challenge?

    Evaluate $\dfrac{\sin^2 \dfrac{\pi}{7}}{\sin^4 \dfrac{2\pi}{7}}+\dfrac{\sin^2 \dfrac{2\pi}{7}}{\sin^4 \dfrac{3\pi}{7}}+\dfrac{\sin^2 \dfrac{3\pi}{7}}{\sin^4 \dfrac{\pi}{7}}$ without the help of a calculator.
  45. L

    Evaluate this trigonometric identity

    (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
  46. S

    Value of this trigonometric expression

    Let: equation 1 : sin A + sin B = 1 equation 2 : cos A + cos B = 0 Squaring both sides of equation 1 and 2 then add the result gives me: cos (A - B) = -1/2 Then how to proceed? Thanks
  47. anemone

    MHB Unsolved Challenge: Trigonometric Identity

    Prove $\tan 3x=\tan \left(\dfrac{\pi}{3}-x\right) \tan x \tan \left(\dfrac{\pi}{3}+x\right)$ geometrically.
  48. agnimusayoti

    Absolute value of trigonometric functions of a complex number

    So far I've got the real part and imaginary part of this complex number. Assume: ##z=\sin (x+iy)##, then 1. Real part: ##\sin x \cosh y## 2. Imaginary part: ##\cos x \sinh y## If I use the absolute value formula, I got ##|z|=\sqrt{\sin^2 {x}.\cosh^2 {y}+\cos^2 {x}.\sinh^2 {y} }## How to...
  49. PainterGuy

    B How Can I Reverse a Trigonometric Identity to Find Original Constants?

    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...
  50. D

    Solve the trigonometric equation involve sin(x), cos(x) and sin(x)cos(x)

    I can’t get the angle, answer given is x=56.33 , x=9.545. (All steps before the equation are correct.)