Wikipedia:Reference desk/Archives/Mathematics/2007 July 5

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July 5

angles

What is "sin²θ", θ being "theta"? Is it the same as "sin(θ²)" ? --Confused Student 03:42, 5 July 2007 (UTC)

sin²θ = (sin(θ))² = sin(θ) * sin(θ). This is different from, and generally more important than, sin(θ²). J Elliot 04:06, 5 July 2007 (UTC)

See also sine#Identities. PrimeHunter 04:10, 5 July 2007 (UTC)

Excuse me, one more question. I'm supposed to solve the equation

sinθ = -1/√2

in interval 0<θ<2π without using a calculator. I asked around and searched, but so far nothing's been able to help. --Confused Student 04:25, 5 July 2007 (UTC)

I think the most important thing might be to have a picture of the unit circle in your head. Using the Pythagorean theorem, you can come up with a relation between sin²(θ) and cos²(θ). In this case, finding cos²(θ) and thinking about (cos(θ),sin(θ)) as a point on the unit circle should help. Check back if you can't get that to work for you. J Elliot 04:34, 5 July 2007 (UTC)

But that's the thing. All I have is sinθ as a y value, and the CAST Rule says that it can be in either quadrant II or III! Besides, there's no "point of special interest" with the value of -1/√2 --Confused Student 04:45, 5 July 2007 (UTC)

Try taking $-{1 \over {\sqrt {2}}}$ and multiplying both the top and the bottom by ${\sqrt {2}}$. —Bkell (talk) 04:51, 5 July 2007 (UTC)

Also, many equations in trigonometry have more than one solution. That's perfectly acceptable; just be sure you list all of the solutions. —Bkell (talk) 04:52, 5 July 2007 (UTC)

Try not to just read it off the circle. What does the trig identity in the article say about sin²(θ) and cos²(θ)? How does that help? J Elliot 04:55, 5 July 2007 (UTC)

Oh, YEAH! I forgot to rationalize it. Thanks! --Confused Student 17:30, 6 July 2007 (UTC)

Another small point: When you use the CAST rule, you label the quadrants like this:

S A

T C

But that's not the order the quadrants are numbered. Quadrant I is the one with the A in it; quadrant II has the S; quadrant III has the T; and quadrant IV has the C. —Bkell (talk) 04:59, 5 July 2007 (UTC)

Have the days gone when students considered triangles with angles of 45, 45, 90 degrees and of 30, 60, 90 degrees, to get the sine, cosine and tangent of 30, 45 and 60 degrees as a fraction or a surd? The latter triangle comes by halving an equilateral one, then Pythagoras will give the less-obvious side. Also, it's worth knowing that the squares of the sines of angles 0, 30, 45, 60 and 90 degrees are in even steps from 0 to 1. …81.153.220.5 11:25, 5 July 2007 (UTC)

0_o Please explain. --Confused Student 17:30, 6 July 2007 (UTC)

Just fill in the dots and it will become clear:

• 4 × sin²  0° = ...
• 4 × sin² 30° = ...
• 4 × sin² 45° = ...
• 4 × sin² 60° = ...
• 4 × sin² 90° = ...

 --Lambiam^Talk 20:20, 6 July 2007 (UTC)

(Clever, how you used underlining to get the appearance of "≤". For future reference, just click on the handy inserters below the edit window, or type "&le;".) So, you've realized that a negative value for a sine means an angle in the left half-plane, quadrants II or III. Good. Now consider the implications of the squares in the fundamental identity

${\Big (}\sin(\theta ){\Big )}^{2}+{\Big (}\cos(\theta ){\Big )}^{2}=1.\,\!$

First, notice that for any given value of the sine, if c = cos(θ) will work, then so will −c. So from this alone you should expect there may be at least two solutions. Second, notice that your given sine value is (the negative of) an exact square root, and since you're asked to work without a calculator there must be something special about the angle. Try solving for c, and ask yourself what the solutions suggest. --KSmrq^T 13:51, 5 July 2007 (UTC)

"So, you've realized that a negative value for a sine means an angle in the left half-plane, quadrants II or III." Don't you mean, the lower half-plane, or quadrants III and IV? -GTBacchus^(talk) 13:43, 6 July 2007 (UTC)

The CAST rule above doesn't say so. --Confused Student 17:30, 6 July 2007 (UTC)

KSmrq made a mistake, as also noticed above by GTBacchus.  --Lambiam^Talk 20:23, 6 July 2007 (UTC)

Additional Mathematics Project Work 2007 (Form 5)

Question: Ang, Bakar and Chandran are friends and they have just graduated from a local university. Ang works in a company with a starting pay of RM2000 per month. Bakar is a sales executive whose income depends solely on the commission he receives. He earns a commission of RM1000 for the 1st month increases by RM100 for each subsequent month. On the other hand, Chandran decides to go into business. He opens a cafe and makes a profit of RM100 in the first month. For the first year, his profit in each subsequent month is 50% more than that of the previous month.

In the second year, Ang receives a 10% increment in his monthly pay. On the other hand, the commission received by Bakar is reduced by RM50 for each subsequent month. In addition, the profit made by Chandran is reduced by 10% for each subsequent month.

1. a) How much does each of them receive at the end of the 1st year? (2 or more method are required for this ques.)

b) What is the percentage change in their total income for the 2nd year compared to the 1st year? Comment on the answers.

c) Ang, Bakar and Chandran, each decided to open a fixed deposit account of RM10000 for 3years without any withdrawal. - Ang keeps the amount at an interest rate of 2.5% per annum for a duration of 1month renewable at the end of each month. - Bakar keeps the amount at an interest rate of 3% per annum for a duration of 3months renewable at the end of every 3months. - Chandran keeps the amount at an interest rate of 3.5% per annum for a duration of 6months renewable at the end of every 6months.

(i) Find the total amount each of them will receive after 3years. (ii) Compare & comment on the difference in the interests received. If you were to invest RM10000 for the same period of time, which fixed deposit account would you prefer? Give your reasons.

Further Exploration

2. a) When Chandran’s 1st child, Johan is born, Chandran invested RM300 for him at 8% compound interest per annum. He continues to invest RM300 on each of Johan’s birthday, up to and including his 18th birthday. What will be the total value of the investment on Johan’s 18th birthday?

b) If Chandran starts his investment with RM500 instead of RM300 at the same interest rate, calculate on which birthday will the total investment be more than RM25000 for the 1st time.

• I have used arithmetic progression to calculate how much bakar earns in the second year and came up with RM 8700...Is this right? If not, what did i do wrong and how do i calculate the answer?

I used geometric progression to calculate how much chandran earns in the second year and came up with RM 55861.31..Is this right?

How do i start question 1c)?

And lastly can you please tell me what and how to count compound interest?

Thanks alot..

Compound_interest#Mathematics_of_interest_rates will hopefully help you with your final question. Lanfear's Bane

Bakar's initial income is RM 1000. What is it in the 12th month? In the second year it decreases, but not as much as it increased in the first year. So at the end of the second year his monthly income is still higher than what he started with. Each month he earns more than RM 1000, for 12 months. So is it possible then that his total earnings are RM 8700? If a is the initial term of an arithmetic progression, b is the final term, and n is the number of terms, then the sum is equal to n×(a+b)/2.

I don't get quite the same amount for Chandran's earnings in year 2. A problem with these kind of word problems is that they are often ambiguously phrased. This one is no exception. What does "in the second year" mean here? Does the change of income from month 12 to month 13 take place, so to speak, still in the first year (so that we have 11 increases followed by 12 decreases) or does the change of income only take effect at the end of month 13? In either case, I get a slightly larger result.  --Lambiam^Talk 16:59, 5 July 2007 (UTC)

Compound Interest

Please tell me what compound interest is in simple english...What does it mean?

Thank You

Compound interest. The mathematics part should be the easiest explanation. x42bn6 Talk Mess 12:10, 5 July 2007 (UTC)

Here is an example. John comes to you and asks: can you loan me 400 ringgit for one month? You say: OK, but you have to pay 20% interest on the money you owe me, so at the end of the month you pay me back not only the 400 ringgit, but also 20%, which amounts to 80 ringgit; all together you will have to fork over 480 ringgit. The month is over and John comes to you and says: I'm awfully sorry, but I can't pay you back right now – I've invested the money in a very good deal which will pay off any moment, and at the end of this next month I'll surely give you the 480 ringgit I owe you. You reply: wait a second, brother; you now owe me 480 ringgit, and one month from now you will not only need to repay me the money you owe me now, but also interest on that money: 20% of 480 ringgit, or 96 ringgit, making a total of 480 plus 96 is 576 ringgit.

In this example, in the second month 20% interest is paid on the original "principal" amount of 400 ringgit, and additionally on the accrued unpaid interest to the amount of 80 ringgit. Because John now also has to pay interest on interest, it is called compound interest. If he defaults for a third time, he will have to pay 20% interest on 576 ringgit.  --Lambiam^Talk 14:35, 5 July 2007 (UTC)

What the hell is a ringgit? 202.168.50.40 23:51, 5 July 2007 (UTC)

In less time than it took you to ask that, you can type ringgit into the search box at the top left of this page. 169.230.94.28 23:58, 5 July 2007 (UTC)

Look it up. Ringgit. —Bkell (talk) 23:55, 5 July 2007 (UTC)

Let me add my 2 cents in here. No doubt that Lambian's reply was accurate and well-meaning. However, if the individual is having trouble understanding the concept behind compound interest, it's probably safe to say that he is unfamiliar with foreign currency terms as obscure as "ringgit". Lambian's reply would have worked just as well (nay, better) if the word "dollar" was substitued for "ringgit". Why convulate and obfuscate the issue? The person doesn't understand compound interest, and I am sure that the use of the term "ringgits" only served to confuse him more. It's (almost) like referring a kindergartener to a calculus textbook, when he asks why does 1+1=2. That's just my 2 cents. But, Lambian, your explanation (minus my above concerns) was good. Thanks. (JosephASpadaro 01:10, 6 July 2007 (UTC))

Furthermore, the original questioner specifically asked, "please use simple English" to help me understand the concept. (JosephASpadaro 01:14, 6 July 2007 (UTC))

Since the person who asked this question (219.95.47.214) was the same person who asked the previous question, and the previous question dealt with ringgit, I'd say Lambiam's choice of currency was well motivated. —Bkell (talk) 01:15, 6 July 2007 (UTC)

Well, yes, then that makes perfect sense. Re: the last question ... I assume that RM is the notation for ringgits? (JosephASpadaro 07:17, 6 July 2007 (UTC))

It is the official abbreviation used in Malaysia for Ringgit Malaysia, which is Malay for "Malaysian ringgit".  --Lambiam^Talk 11:21, 6 July 2007 (UTC)

And ringgit is the singular as well as plural form. x42bn6 Talk Mess 14:56, 11 July 2007 (UTC)

Trig functions without a calculator

If all you had was a pencil and paper and wanted to find sin(5), how would you do it? 68.231.151.161 15:54, 5 July 2007 (UTC)

Either you'd look it up in a table (the usual way back before calculators), or you'd estimate it with the Taylor expansion: $\sin x=x-{\frac {1}{3!}}x^{3}+{\frac {1}{5!}}x^{5}-\ldots$ where x is in radians. — Laura Scudder ☎ 16:12, 5 July 2007 (UTC)

This is where it pays off if you have memorized the decimal expansion of π to a sufficient number of digits. Then you can use the identity sin(5) = − sin (2π − 5). The Taylor series converges much faster for the latter argument. To compute an approximation of sin(x) to 5 decimals, you'd only need 5 terms, instead of 10 for the series with x = 5.  --Lambiam^Talk 19:18, 5 July 2007 (UTC)

Right, and it is even better to use the tailor series around $3\pi /2$, that is, $\sin 5=-\cos(5-3\pi /2)\approx$$1-(5-3\pi /2)^{2}/2+(5-3\pi /2)^{4}/24$. – b_jonas 16:33, 8 July 2007 (UTC)

Caution: Mathematicians will answer this question assuming that "5" is not degrees, but natural measure where a full turn is 2π. For a good time and a good education, do some research into how people have done this over the millennia. (We have examples from thousands of years ago!) One interesting approach is mentioned in HAKMEM, Item #158. The idea is to exploit two facts:

1. $\sin x=4\left(\sin {\tfrac {-x}{3}}\right)^{3}-3\left(\sin {\tfrac {-x}{3}}\right)\,\!$
2. $\sin x\approx x\qquad {\text{for very small }}x\,\!$

This also can be handy in using a table, because we can use a small table and just a little recursion. Calculator software may use a version of the CORDIC approach, which can also be done by hand. --KSmrq^T 00:00, 6 July 2007 (UTC)