# 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.

View More On Wikipedia.org
1. ### 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. ### 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. ### 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. ### 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. ### 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. ### 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...

29. ### 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. ### 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. ### 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. ### 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. ### 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. ### 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. ### 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. ### 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. ### 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. ### 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. ### 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. ### 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. ### 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. ### 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. ### 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. ### 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. ### 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. ### 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. ### 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. ### 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. ### 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. ### 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.)