Solve sin(y/2)=y/4: Step-by-Step Guide

  • Thread starter garyng2001hk
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In summary, there are three distinct solutions to the given equation and one way to find them is by graphing the two given functions and finding their points of intersection. Another method is using the bisection method, where you continuously divide the interval between two values until you reach the desired accuracy. It is also possible to use the half-angle formula and solve algebraically, although it may not always work.
  • #1
garyng2001hk
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Please show the steps
 
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  • #2
There is no "algebraic" way to solve that equation. (There are, in fact, three distinct solutions. Here's one for free: x= 0!)

Simplest way: graph y= sin(x/2) and graph y= x/4 on the same coordinates. The solution is the x coordinate of the points where the two graphs cross. On my TI85 calculator, that's quick and simple. It even let's me "zoom" in on a point of intersection.

Almost as simple but tedious is 'bisection": going by steps of 1, I see that sin(3/2)= .997 and 3/4= .75 so sin(x/2) is larger. But sin(4/2)= .909 while 4/4= 1 so now x/4 is larger. Since sin(x/2) and x/4 are continuous functions, that means that they must be equal somewhere between 3 and 4. We don't know where so 1/2 way between is as good as any: sin(3.5/2)= .984 and 3.5/4= .875. sin(x/2) is larger there so they must be equal somewhere between 3.5 and 4. Half way between again is 3.75. sin(3.75/2)= .984 and 3.75/4= .934. sin(x/2) is still larger so we try half way between 3.75 and 4, 3.875. sin(3.875/2)= .9335 (I added an addtional decimal place because these are all the same to 3 places) and 3.875/4= .9687. Now x/4 is larger so there must be a solution between 3.75 (the last value at which sin(x/2) was larger) and 3.875. Half way between them is 3.8125. Continue this until you have the desired accuracy. Once you get that root, it should be obvious what the third solution is.
 
  • #3
I'm sure you could try the half-angle formula and then solve algebraically from there. I'm not sure if it would work out though.
 

What does "sin(y/2)=y/4" mean?

The equation "sin(y/2)=y/4" means that the sine of half of y (y/2) is equal to one fourth of y (y/4).

How do I solve this equation?

To solve this equation, you can use the trigonometric identity "sin(2x)=2sin(x)cos(x)." First, use this identity to rewrite the equation as "2sin(y/2)cos(y/2)=y/4." Then, divide both sides by 2 to get "sin(y/2)cos(y/2)=y/8." Next, use the double angle formula "sin(2x)=2sin(x)cos(x)" again to rewrite the equation as "sin(y)=y/8." Finally, use a calculator or a table of values to approximate the solution for y.

What if I don't know the value of y?

If you do not know the value of y, you can use a calculator or a table of values to approximate the solution. Alternatively, you can use a graphing calculator to graph both sides of the equation and find the point(s) of intersection.

What is the domain and range of this equation?

The domain of this equation is all real numbers, as there are no restrictions on the values of y. The range is limited to values between -1 and 1, as the sine function has a range of -1 to 1.

Can I solve this equation without using a calculator?

Yes, you can solve this equation without a calculator by using trigonometric identities and approximations. However, using a calculator or a graphing calculator can make the process easier and more accurate.

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