Math and Arithmetic

Trigonometry

To solve this, you need to find values of x where cos(x)

=

xsin(x).

First of all, 0 is not a solution because cos(0) =

1, and sin(0) =

0. Since 0 is not a solution, divide both sides of the equation by sin(x)

to get cot(x)

=

x (remember that cos divided by sin is the same as cot). The new question to answer is, when is cot(x)

=

x? Using Wolfram Alpha, the results are

x Â±9.52933440536196...

x Â±6.43729817917195...

x Â±3.42561845948173...

x Â±0.860333589019380... there will be an infinite number of solutions.

If you'd like to do the calculation yourself (not asking WolframAlpha)

then there's a trick which almost always works, even for equations which cant be done analytically.

Starting with the basic equation, cos(x)

=

x*sin(x),

transpose it to a form starting with "x =".

In this case you could get: x =

1/tan(x), x =

cot(x)

or from tan(x)

=

1/x you get x =

Arctan(1/x).

Because I like to do my calcs

on an old calculator which only has Arctan

and not Arccotan

(Inverse cotangent(x))

I use the last above - x =

Arctan(1/x)

Starting with a value like 0.5, hit the 1/x key then shift tan keys. Just keep repeating those two operations and the display will converge on 0.860333. Too easy. This example of the method is not a good one as it takes about 25 iterations to converge to within 0.0000001 of the right answer. It is unusually slow.

And finally, this method has only 50% chance of working first try. We were lucky picking x =

Arctan(1/x). x =

1/tan(x) diverges ind the iterations do not converge on the answer.

So if you try this method on another problem and it diverges, just transpose the equation again and have another go.

Starting with x^2 + x - 3 =

0,

and iterating x =

3-x^2, you find it diverges, so

try x =

sqr(3-x) which (with care and about 25 iterations) converges on 1.302775638.

499500501

Math and Arithmetic

Trigonometry

first divide each side by 2 so you get...

sine^2(X)=1/2

Then make sine ^2(X)=sine(x^2)

SO you get... sine(X^2)=1/2

Then take the sine^-1 of each side it will look like this X^2=sine^-1(1/2)

type the right side into a calc which will give you a gross decimal but it works (0.5235987756)

so now you have

X^2=0.5235987756

then take the square root of each side to make it linear and you will get X=.7236012546

and that is your answer!!!! make sure to check it on your calculator...I did and it worked

* * * * *

Not quite correct, I fear. Try this:

Let s = sin θ.

Then,

2s2 = 1;

s2 = ½; and

s = ±½√2.

Therefore,

θ = 45°, 135°, 225°, or 315°;

or, if you prefer,

θ = ¼π, ¾π, 1¼π, or 1¾π.

412413414

Math and Arithmetic

Trigonometry

2cos2x - cosx -1 = 0

Factor:

(2cosx + 1)(cosx - 1) = 0

cosx = {-.5, 1}

x = {...0, 120, 240, 360,...} degrees

426427428

Calculus

Trigonometry

The problem x = 2 sin x cannot be solved by using algebraic methods.

One solution is to draw the graphs of y = x and y = 2 sin x.

The two lines will intersect. The values of x where the intersection takes place are the solutions to this problem.

You can tell from the graph that one solution is x=0 and verify this contention by noting that 2 sin(0) = 0.

You can find the other solution through numerical methods and there are many that will give you the correct solution. Perhaps the simplest is to start with a value of X like pi/2 and then take the average of 2*sin(X) and X. Using that as your new value, again take the average of 2*sin(X) and X. As you continue to do this, the value will get closer and closer to the desired value. After 20 steps or so, the precision of your calculator will probably be reached and you will have a pretty good answer of about 1.89549426703398. (A spreadsheet can be used to make these calculations pretty easily.)

381382383

Math and Arithmetic

Algebra

Trigonometry

Let 'theta' = A [as 'A' is easier to type]

sec A - 1/(sec A)

= 1/(cos A) - cos A

= (1 - cos^2 A)/(cos A)

= (sin^2 A)/(cos A)

= (tan A)*(sin A)

Then you can swap back the 'A' with theta

325326327

Algebra

Trigonometry

It is not clearly mentioned that whether it is the "THIRD" root we have to calculate.. or the "SQUARE ROOT THREE TIMES.. i.e the EIGHTth ROOT".. The answer posted here before was not correct in a sense that the if the question demands "third square root" which can also mean the "EIGHTth" root..

Re-edit:

It seems rather obvious that this answerer is new to wiki answers and is not in the habit of question interpretation. The question is most likely thus; " What is the cubic root of 0.125. The answer then would be...,

cubic root(0.125)

= 0.5

====( calculate the square root three times?!? sounds like a stretched interpretation )

Do you know what square root means? -_- if u know.. then u clearly know what "third square root means" .. huh -_-

297298299

Math and Arithmetic

Geometry

Trigonometry

The formula for converting radians to degrees is the given angle in radians multiplied by (180 degrees / pi), so in this case, 7 pi radians would be equal to 1260 degrees. (By the way, the greek letter pi isn't spelled with an e).

290291292

Electronics Engineering

Physics

Trigonometry

Waves Vibrations and Oscillations

Suppose a sine wave of the form y = A*sin(k) with

- A = amplitude or maximum value of the function y (namely when k = pi/2 or 90°)
- k = the value on the x-axis of the function

It's typical of a sine wave that it's periodic, which means the function y repeats itself after a certain period. This period is equal to 2*pi or 360°, for example:

for k = pi/2, 5*pi/2, 9*pi/2, ... the value of y will be the same and equal to A (notice that 5*pi/2 = pi/2 + 2*pi and 9*pi/2 = 5*pi/2 + 2*pi)

In physics it's a more common practice to write a sine wave as y = A*sin(omega*t) with omega the angular frequency specified in radians/s (omega refers to the Greek letter) and t the time specified in seconds.

Now, when you want to calculate the frequency f of a sine wave (which is not equal to the angular frequency) or in other words the number of complete cycles that occur per second (specified in cycle/s or s-1 or Hz), you need to know the time T required to complete one full cycle (specified in s/cycle or just s or Hz-1). The frequency f is then equal to 1/T.

Knowing omega you can calculate the frequency in a different and more common way:

since the sine wave is periodic and after a time T one cycle has been completed (thus one period), it follows that omega*T = 2*pi for the function y to have the same value after one period (the function y having the same value is equal to completing one cycle).

Let's rearrange this formula by bringing 2*pi to the left and T to the right, so we get:

omega/(2*pi) = 1/T and since 1/T = f we finally get:

f = omega / (2*pi)291292293

Math and Arithmetic

Trigonometry

There are 6 basic trig functions.

sin(x) = 1/csc(x)

cos(x) = 1/sec(x)

tan(x) = sin(x)/cos(x) or 1/cot(x)

csc(x) = 1/sin(x)

sec(x) = 1/cos(x)

cot(x) = cos(x)/sin(x) or 1/tan(x)

---- In your problem csc(x)*cot(x) we can simplify csc(x).

csc(x) = 1/sin(x)

Similarly, cot(x) = cos(x)/sin(x).

csc(x)*cot(x) = (1/sin[x])*(cos[x]/sin[x])

= cos(x)/sin2(x) = cos(x) * 1/sin2(x)

Either of the above answers should work.

In general, try converting your trig functions into sine and cosine to make things simpler.

283284285

Math and Arithmetic

Algebra

Trigonometry

This is so much work that it is not worthwhile to do in practice, although the formulae themselves are actually quite simple. The basic method is to use a so-called "infinite series". The angle must be expressed in radians. If the angle is in degrees, multiply it by (pi/180), to get the equivalent angle in radians. Then, use the formula:

sin(x) = x - x3/3! + x5/5! - x7/7! + x9/9!...

The individual terms become smaller and smaller, quite quickly, so the idea is to continue adding more terms until you see that the terms become so small that you can ignore them (depending on the desired degree of accuracy). An expression like 5!, read "five factorial" or "the factorial of five", means to multiply all natural numbers up to five: 5! = 1 x 2 x 3 x 4 x 5.

Similarly,

cos(x) = 1 - x2/2! + x4/4! - x6/6! + x8/8!...

There is a more complicated formula for tan(x), or simply calculate as follows:

tan(x) = sin(x) / cos(x)

The formulae for sin(x) and cos(x) are derived from the Taylor expansion, explained in basic calculus books.

277278279

Statistics

Trigonometry

Mathematical Analysis

distance and angle

277278279

Math and Arithmetic

Algebra

Trigonometry

It is a trigonometric function, equivalent to the sine of an angle divided by the cosine of the same angle.

261262263

Science

Geometry

Trigonometry

A five sided shape is called a pentagon.

848586

Math and Arithmetic

Algebra

Trigonometry

(/) = theta

sin 2(/) = 2sin(/)cos(/)

257258259

Trigonometry

The answer is cos A .

cos A = 1/ (sec A)

253254255

Math and Arithmetic

Algebra

Trigonometry

cos(A + B) = cosAcosB - sinAsinB, with A=B, cos(2A) = cos2A - sin2A, then you can use cos2A + sin2A = 1, to produce more, like: [2cos2A - 1] or [1 - 2sin2A], and others.

251252253

Math and Arithmetic

Trigonometry

Without an "equals" sign somewhere, no question has been asked,

so there's nothing there that needs an answer.

Is it the sum that you're looking for ?

csc(x) + cot(x) = 1/sin(x) + cos(x)/sin(x) = [1 + cos(x)] / sin(x)

251252253

Calculus

Trigonometry

1/sin(x) is also known as cosec(x).

It looks a bit like a U, starting at "infinity" when x = 0, bottoming out at 1 when x = pi/2 radians and then returning to "infinity" at x = pi. Next, it is an upside down U, below the axis and peaking at -1: between x = pi and 2*pi. These U shapes alternate.

227228229

Math and Arithmetic

Trigonometry

Yes they are. Both have a a period of 2 pi

219220221

Geometry

Trigonometry

triangular prism

757677

Algebra

Trigonometry

Assume the angle u takes place in Quadrant IV.

Let u = arctan(-12). Then, tan(u) = -12.

By the Pythagorean identity, we obtain:

sec(u) = √(1 + tan²(u))

= √(1 + (-12)²)

= √145

Since secant is the inverse of cosine, we have:

cos(u) = 1/√145

Therefore:

sin(u) = -√(1 - cos²(u))

= -√(1 - 1/145)

= -12/√145

Otherwise, if the angle takes place in Quadrant II, then sin(u) = 12/√145

195196197

Math and Arithmetic

Algebra

Trigonometry

A sphere is not usually divided into 4 quadrants. Dividing by 2 along each of the 3 orthogonal axes partitions the sphere into 8.

185186187

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