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Fed [463]
3 years ago
14

For what values of θ, −π ≤ θ ≤ π, is y = tan θ undefined?

Mathematics
1 answer:
pogonyaev3 years ago
3 0

Answer:

The answer is II and IV ⇒ the 3rd answer

Step-by-step explanation:

* lets check the domain of angle Ф

∵ -π ≤ Ф ≤ π

∵ y = tanФ

∵ tanФ = sinФ/cosФ

* The effective term which makes tanФ undefined is cosФ

- If cosФ = 0, than tanФ will be undefined

* Lets check the angles that have cosФ = 0

∵ The unit circle intersect x-axis at point (1 , 0) and (-1 , 0)

∵ The unit circle intersect y-axis at point (0 , 1) and (0 , -1)

∵ cosФ = x-coordinates of the points

∵ The points of intersection with the y-axis have x- coordinates = 0

∴ The angles on the y-axis have cosФ = 0

* The angles on the +ve part of y-axis are π/2 and -3π/2

 The angles on the -ve part of y-axis are -π/2 and 3π/2

∴ The tan of π/2 , 3π/2 , -π/2 , -3π/2 undefined

* In the problem

I. -π ⇒ defined ⇒ on the -ve part of x-axis

II. -π/2 ⇒undefined

III. 0 ⇒ defined ⇒ on the +ve part of the x-axis

IV. π/2 ⇒ undefined

V. π ⇒ defined ⇒ on the -ve part of x-axis

∴ The answer is II and IV ⇒ the 3rd answer

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There are a variety of methods of performing this subtraction, and corresponding different methods of regrouping.

If you're taught (as I was) to do the subtraction right-to-left, then the first regrouping you need to do is when you try to subtract 500 from 100. You must regroup the 8 thousands and 1 hundred to 7 thousands and 11 hundreds. Then, when you subtract 5 hundreds, you end with 7 thousands and 6 hundreds.

The next regrouping you need to do is when you try to subtract 6 ten-thousands from 5 ten-thousands. You must regroup the 2 hundred-thousands and 5 ten-thousands to 1 hundred-thousand and 15 ten-thousands. Then, when you subtract 6 ten-thousands, you end with 1 hundred-thousand and 9 ten-thousands.

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If you use an abacus or soroban or similar tool to help you keep track of the numbers, you were likely taught to do the subtraction left-to-right. In this case, the first regrouping  comes when you want to subtract 6 ten-thousands. Practitioners of this method know that -6 = -10 +4, so the number represented on the tool becomes (2-1) hundred-thousands and (5+4) ten-thousands plus the rest of the initial number, or 198,197 after subtracting the 6 ten-thousands.

The subtraction proceeds until you find you need to subtract 500 from 100. At this point, the tool is representing the partial result as 194,197. Again, if you practice this method, you know that -5 = -10 +5, so you reduce the thousands digit by 1 (to 3) and add 5 to the hundreds digit to get 193,697.

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An attempt is made to show the regroupings in the attachment. In each case there are two of them. However, working left-to-right, the result of the first subtraction of 6 ten-thousands is 19 ten-thousands, so you never actually write down anything else. Of course, if you're using an abacus or soroban, you don't write down anything—you simply change the position of the beads on the tool.

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