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Reading and commenting on What are some common ways students get confused about finding an inverse of a function? I was kindly set straight that the use of $\sin^{^{-1}}(x)$ to mean $\arcsin(x)$ has been rendered obsolete. International Standard ISO 80000-2 offers the following:

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This notation is preferable to me, with one point of confusion. The text we use, Algebra and Trigonometry by Paul Foerster, uses $\textrm{Arcsin}(x)$ to indicate the restricted domain, and the lower case as above for all real domain. This notation leaves me with my question -

How do we ask for the arcfunction with no domain restriction, i.e we actually want the multiple answer as in "solve for the first 6 values when $y=2$ for the following sinusoidal function" ?

Disclosure - This question is in reaction to my discovering that a notation that I never cared for has been eliminated by an international standard for nearly 10 years, but still used at my high school. I am trying to understand where to go with this as people in general aren't so open to change.

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  • $\begingroup$ You mean $\sin x= 2$? $\endgroup$ – Paracosmiste May 3 at 19:37
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    $\begingroup$ No. I mean a function that's sinusoidal, shifted up, left, a larger amplitude, etc.... A ferris wheel rotating and asking what 6 times your car is 2 feet off the ground... I'd have spelled it out, but it seemed TMI. I am delighted to get rid of the bad notation, but for word problems we need to have a way of asking for the multiple solutions for some domain. To be real clear - "When is sin(x) = .5 ?" And I want 30, 150, 390, 510 degrees if I ask for first 4 solutions. $\endgroup$ – JoeTaxpayer May 3 at 19:46
  • $\begingroup$ I usually ask "Solve $\sin x=0.5$ where $x\in [0,550]$" but if you want a word problem: "A mass attached to a spring is oscillating with amplitude $2$ and frequency $0.5/\pi$ Hz starting from its equilibrium point...." but it seems difficult. Your example of the ferris wheel is better. $\endgroup$ – Paracosmiste May 3 at 19:56
  • $\begingroup$ Fair enough, the request is within the domain when stating the word problem, not in the notation of arcsin(x). That can work for me. Thanks. $\endgroup$ – JoeTaxpayer May 3 at 19:58
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  • Solve the equation $\sin x=0.5$ in $[0,7\pi]$.
  • The equation $y=-10\cos(3t)$ represent the motion of a weight hanging on a spring after it has been pulled $10$ cm below its equilibrium point and released. The output $y$ gives the position of the weight in cm above (positive $y$) or below (negative $y$) the equilibrium point after $t$ seconds. Find the first four times when the weight is $5$ cm above its equilibrium point.
  • A ferris wheel with a diameter of $36$ m rotates counterclockwise at $5$ revolutions per minute. At its lowest point, each seat is $0.9$ m from the ground. For a rider starting at the lowest point, his height (in meters) above the ground after $t$ minutes is given by $y=18.9-18\cos(10\pi t)$. Find all the times in the first half minute when the rider is $24$ m above the ground. Round to the nearest tenth of a second.
  • An alternating current generator produces the current (in amperes) $i=30\sin(120\pi t)$ where $t$ is the time in seconds. Find the least positive $t$ for which $i=25$ A.
  • A polarizing filter for a camera contains two parallel plates of polarizing glass, one fixed and the other able to rotate. If $\theta$ is the angle of rotation from the position of maximum light transmission then the intensity of light leaving the filter is $\cos^2 \theta$ times the intensity entering the filter. Find the least positive $\theta$ so that the intensity of light leaving is $70$ % of that entering.
  • The planet Mercury travels around the sun in an elliptical orbit given by $\displaystyle r=\frac{3.44\times 10^7}{1-0.206\cos\theta}$. Find the least positive $\theta$ for which mercury is $3.78\times 10^7$ miles from the sun.
  • Given the equation $xy=-2$ replace $x$ and $y$ with $x=u\cos\alpha-v\sin\alpha$ and $y=u\sin\alpha+v\cos\alpha$. Find the least positive $\alpha$ so that coefficient of the $uv$ term will be $0$.
  • What are the $x$-intercepts of the graph of the function $f$ defined by $f(x)=4\sin^2x-3$ on the interval $[0,2\pi]$?
  • Blood pressure is a way of measuring the amount of force exerted on the walls of blood vessels. It is measured using two numbers: systolic (as the heart beats) blood pressure and diastolic (as the heart rests) blood pressure. Blood pressures vary substantially from person to person, but a typical blood pressure is 120 80, which means the systolic blood pressure is 120 mmHg and the diastolic blood pressure is 80 mmHg. Assuming that a person’s heart beats 70 times per minute, the blood pressure P of an individual after t seconds can be modeled by the function $P(t)=100+20\sin\left(\frac{7\pi t}{3}\right)$. In the interval $[0,1]$, determine the times at which the blood pressure is between 100 and 105 mmHg.
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    $\begingroup$ You’ve basically told me that I seek a notation that’s not needed. That what I want to instruct the students to do is requested via context. That’s great. I now have an answer to that issue when I present this at the next staff meeting. $\endgroup$ – JoeTaxpayer May 3 at 20:28
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    $\begingroup$ For me $\arcsin$ is always limited to the restricted domain $[-\pi/2,\pi/2]$ otherwise it won't be a function. $\endgroup$ – Paracosmiste May 3 at 20:30
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    $\begingroup$ I know you are stating a fact to me. In the book I referenced, arcsin(x) is not a function, but Arcsin(x) is. This is what I need to address. The book revision we use is 1994, and it’s an affluent community for what that’s worth. $\endgroup$ – JoeTaxpayer May 3 at 20:38

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