# How to formulate this type of arcsin problem?

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:

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.

• You mean $\sin x= 2$? – llllllllllllllllllllllllllllll May 3 '19 at 19:37
• 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. – JTP - Apologise to Monica May 3 '19 at 19:46
• 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. – llllllllllllllllllllllllllllll May 3 '19 at 19:56
• 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. – JTP - Apologise to Monica May 3 '19 at 19:58

• Solve the equation $$\sin x=0.5$$ in $$[0,7\pi]$$.
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• 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.
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• 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.
• For me $\arcsin$ is always limited to the restricted domain $[-\pi/2,\pi/2]$ otherwise it won't be a function. – llllllllllllllllllllllllllllll May 3 '19 at 20:30