Normal And Inverse Trig Functions Cancel Each Other Out

Home » How To Cancel Out Cosine » Normal And Inverse Trig Functions Cancel Each Other Out

Maybe your like

Physics Forums Physics Forums
  • Insights Blog -- Browse All Articles -- Physics Articles Math Articles Education Articles Bio/Chem/Tech Articles
  • Forums General Math Calculus Differential Equations Topology and Analysis Linear and Abstract Algebra Differential Geometry Set Theory, Logic, Probability, Statistics MATLAB, Maple, Mathematica, LaTeX
  • Trending
Log in Register What's new
  • General Math
  • Calculus
  • Differential Equations
  • Topology and Analysis
  • Linear and Abstract Algebra
  • Differential Geometry
  • Set Theory, Logic, Probability, Statistics
  • MATLAB, Maple, Mathematica, LaTeX
Menu Log in Register Navigation More options Style variation System Light Dark Contact us Close Menu You are using an out of date browser. It may not display this or other websites correctly.You should upgrade or use an alternative browser.
  • Forums
  • Mathematics
  • General Math
Normal and inverse trig functions cancel each other out
  • Context: High School
  • Thread starter Thread starter TSN79
  • Start date Start date Nov 9, 2004
  • Tags Tags Functions Inverse Normal Trig Trig functions
Click For Summary

Discussion Overview

The discussion centers around the properties of normal and inverse trigonometric functions, particularly focusing on the equations involving sine and arcsine, and tangent and arctangent. Participants explore the conditions under which these functions can be considered inverses, including the implications of function injectivity and the effects of domain restrictions.

Discussion Character

  • Exploratory
  • Technical explanation
  • Debate/contested
  • Mathematical reasoning

Main Points Raised

  • Some participants note that while \(\sin(\arcsin(x)) = x\) holds true, \(\arcsin(\sin(x)) = x\) is not universally valid due to the non-injectivity of the sine function outside specific intervals.
  • One participant points out that \(\arcsin(x)\) is undefined for \(|x| > 1\), which affects the validity of the equations.
  • Another participant discusses the necessity of restricting the domain of sine to \([- \frac{\pi}{2}, \frac{\pi}{2}]\) to ensure that \(\arcsin(\sin(x)) = x\) holds true.
  • There is a mention of the tangent function's similar issues with injectivity, and that \(\arctan(\tan(x)) = x\) is valid only when \(x\) is restricted to \((- \frac{\pi}{2}, \frac{\pi}{2})\).
  • One participant raises a question about finding solutions for \(\tan(x/2) = 0\) and whether using \(\arctan(\tan(x/2)) = x/2\) would work under certain conditions.
  • Another participant clarifies that the period of \(\tan(x/2)\) is \(2\pi\) and discusses the implications of this on the range of the inverse function.

Areas of Agreement / Disagreement

Participants express differing views on the validity of the inverse relationships for trigonometric functions, particularly regarding domain restrictions and injectivity. There is no consensus on the conditions under which these relationships hold true, and multiple competing views remain throughout the discussion.

Contextual Notes

Limitations include the need for specific domain restrictions for the sine and tangent functions to ensure their inverses are valid. The discussion also highlights the undefined nature of arcsine for inputs outside the range of \([-1, 1]\) and the implications for the validity of certain equations.

TSN79 Messages 422 Reaction score 0 I'm a bit confused at the moment. All my books say that normal and inverse trig.functions cancel each other out like this \sin(\arcsin(x))=x and \arcsin(sin(x))=x But when I try this out on my calculator - TI-89 - it only wants to recognize the first equation as being equal to x. Is that true or...? :confused: Last edited: Nov 9, 2004 Mathematics news on Phys.org
  • Student serves up fresh solutions to the pancake problem
  • How long does it take to get last liquid drops from kitchen containers? These physicists know the answer
  • Putting sports stats to the test: Unpredictable play helps pick a winner in soccer
Galileo Science Advisor Homework Helper Messages 1,980 Reaction score 7 For a function to have an inverse, it needs to be injective and surjective. The problem is that sin(x) is generally not injective, however it is injective on (eg) the interval [-\pi/2,\pi/2]. This interval can be chosen as the domain of sin(x) and becomes the range of arcsin(x), so for x in this interval the second equation will hold as well. Last edited: Nov 10, 2004 TSN79 Messages 422 Reaction score 0 But since my calculator always accepts sin(arcsin(x)) as being equal to x, does that mean that this equation is always true no matter what? :confused: Galileo Science Advisor Homework Helper Messages 1,980 Reaction score 7 That's funky. You cannot take the arcsin of x if |x|&gt;1, since such an x wouldn't be in the range of sin(x). My calculator gives an error, as it should. (Unless you use complex numbers). krab Science Advisor Messages 892 Reaction score 3 Think it through. Try x=360 degrees. sin(360 deg.)=0, but arcsin(0)=0. So arcsin(sin(x)) is not equal to x. OTOH, sin(arcsin(x))=x HallsofIvy Science Advisor Homework Helper Messages 42,895 Reaction score 983 Another function that does not have a "true" inverse is f(x)= x2 and you might find it easier to look at that function (you can do the calculations yourself rather than use a calculator). If we restrict the domain to non-negative numbers, THEN we can use f-1(x)= &radic;(x) as the inverse. &radic;(x) only makes sense when x is positive. For any positive number, if you first take the square root and then square, you get x back again: f(f-1(x))= x. In particular, if x= 4, f-1(4)= &radic;(4)= 2 and f(2)= 22= 4 again. However, f(x) is defined for x positive or negative. If x= 2, f(2)= 22= 4 and then f-1(4)= &radic;(4)= 2. In that case, f-1[\sup](f(x))= x. But if x= -2, f(-2)= (-2)2= 4 and then f-1(4)= &radic;(4)= 2. In that case, f-1(f(x)) is NOT equal to x. tmwong Messages 24 Reaction score 0 sin(arcsin(x))=x true while arcsin(sin(x))=x is not true. Galileo Science Advisor Homework Helper Messages 1,980 Reaction score 7 sin(arcsin(x))=x does not make sense if |x|>1, since it is undefined. It's only true if |x|<=1. TSN79 Messages 422 Reaction score 0 So how would I go about it to find the solutions for let's say tan(x/2)=0 ? I thought I could make it easier by using arctan(tan(x/2))=x/2, but you're telling me that this wouldn't work? What if I decided to look only at a part of tan(x/2) that is increasing or decreasing? Let's say from -pi to pi. Would this work? Because that's the requirement for a trig.function to have an inverse isn't it? Would arctan(tan(x/2))=x/2 be true then? Galileo Science Advisor Homework Helper Messages 1,980 Reaction score 7 tan(x) has the same problem: It isn't injective. However, it IS injective if it is restricted to the interval (-\frac{\pi}{2},\frac{\pi}{2}) Then for x in this interval \arctan(\tan(x))=x holds. (Notice that range of arctan(x) is (-\frac{\pi}{2},\frac{\pi}{2})) If you wish to solve \tan(x/2)=0, then using arctan will give you an answer that lies in the interval (-\frac{\pi}{2},\frac{\pi}{2}). (In this case x=0). For a complete answer, you need additional info. In this case you can use that the tangent is periodic (period \pi) so that x/2=k\pi where k is an integer will give the general solution). Last edited: Nov 10, 2004 TSN79 Messages 422 Reaction score 0 Wouldn't period of tan(x/2) be 2pi? When the interval of tan(x) is restricted to <-pi/2 , pi/2>, the range of arctan(x) is the same. But when the interval of tan(x/2) is <-pi , pi>, but now the range of arctan(x/2) is not the same. Why is this? Shouldn't the range of arctan(x/2) the same as the interval of tan(x/2)? Galileo Science Advisor Homework Helper Messages 1,980 Reaction score 7 Yes, the period of tan(x/2) is 2\pi (I meant the period of tan(x) in my post). If y=\tan(x/2). Then for x \in (-\pi,\pi): \arctan(y)=x/2 x=2\arctan(y) So the inverse function of tan(x/2) is 2arctan(x) and has range (-\pi,\pi)

Similar threads

MHB How do trigonometric functions and their inverses relate to each other?
  • Jul 27, 2020 · Replies 16 · Jul 29, 2020
Replies 16 Views 2K High School Brushing Up Math Knowledge: Basic Trig Questions
  • Apr 24, 2018 · Replies 14 · Apr 25, 2018
Replies 14 Views 2K How to Determine Correct Inverse Trig Angle?
  • Apr 10, 2024 · Replies 2 · Apr 10, 2024
Replies 2 Views 1K High School Inverse Functions vs Inverse Relations
  • Jun 23, 2018 · Replies 4 · Jun 25, 2018
Replies 4 Views 4K High School Are Inverse Problems More Complex Than Direct Problems in Mathematics?
  • Jul 30, 2022 · Replies 13 · Jul 30, 2022
Replies 13 Views 3K High School How Can Trigonometry Calculate Distances in Surveying?
  • Jan 18, 2018 · Replies 5 · Jan 19, 2018
Replies 5 Views 3K High School How would I write this trig solution?
  • Jan 14, 2019 · Replies 19 · Jan 22, 2019
Replies 19 Views 3K High School Can Six Pencils Be Arranged to Each Touch the Other Five?
  • Jul 23, 2025 · Replies 30 · Oct 21, 2025
2 Replies 30 Views 3K High School Can you solve this geometry problem for nine year olds?
  • Oct 26, 2016 · Replies 30 · Sep 12, 2017
2 Replies 30 Views 11K MHB Domain and range of inverse functions (circular and hyperbolic)
  • Aug 31, 2014 · Replies 1 · Aug 31, 2014
Replies 1 Views 2K
  • Forums
  • Mathematics
  • General Math

Hot Threads

  • Charles Link

    Undergrad Trigonometry problem of interest

    • Started by Charles Link
    • Oct 18, 2025
    • Replies: 81
    • General Math
  • bob012345

    High School Area of Overlapping Squares

    • Started by bob012345
    • Dec 6, 2025
    • Replies: 10
    • General Math
  • bob012345

    Undergrad Find the Number of Triangles

    • Started by bob012345
    • Dec 29, 2025
    • Replies: 33
    • General Math
  • G

    High School Potato paradox

    • Started by Gavran
    • Feb 26, 2026
    • Replies: 28
    • General Math
  • bob012345

    Undergrad Three Circle Problem

    • Started by bob012345
    • Feb 3, 2026
    • Replies: 14
    • General Math

Recent Insights

  • Greg Bernhardt

    Insights Remote Operated Gate Control System

    • Started by Greg Bernhardt
    • Feb 28, 2026
    • Replies: 3
    • General Engineering
  • Greg Bernhardt

    Insights AI Enriched Problem Solving

    • Started by Greg Bernhardt
    • Feb 28, 2026
    • Replies: 1
    • General Math
  • Greg Bernhardt

    Insights Thinking Outside The Box Versus Knowing What’s In The Box

    • Started by Greg Bernhardt
    • Oct 13, 2025
    • Replies: 27
    • Other Physics Topics
  • Greg Bernhardt

    Insights Why Entangled Photon-Polarization Qubits Violate Bell’s Inequality

    • Started by Greg Bernhardt
    • Sep 29, 2025
    • Replies: 28
    • Quantum Interpretations and Foundations
  • Greg Bernhardt

    Insights Quantum Entanglement is a Kinematic Fact, not a Dynamical Effect

    • Started by Greg Bernhardt
    • Sep 2, 2025
    • Replies: 22
    • Quantum Physics
  • Greg Bernhardt

    Insights What Exactly is Dirac’s Delta Function? - Insight

    • Started by Greg Bernhardt
    • Sep 2, 2025
    • Replies: 33
    • General Math
Back Top

Tag » How To Cancel Out Cosine