# Trigonometry

## Trigonometry is a field of mathematics. It is the study of triangles. Trigonometry includes planar trigonometry, spherical trigonometry, finding unknown values in triangles, trigonometric functions, and trigonometric function graphs.

###### Asked in Math and Arithmetic, Trigonometry

### How do you solve cosx - xsinx equals 0?

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 ±9.52933440536196...
x ±6.43729817917195...
x ±3.42561845948173...
x ±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.

###### Asked in Math and Arithmetic, Trigonometry

### How do you solve 2 sin squared theta equals 1?

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.

###### Asked in Calculus, Trigonometry

### How do you solve x equals 2 sin x?

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.)

###### Asked in Algebra, Trigonometry

### What is the third square root of 0.125?

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 -_-

###### Asked in Electronics Engineering, Physics, Trigonometry, Waves Vibrations and Oscillations

### How do you calculate the freq of a sine wave?

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)

###### Asked in Math and Arithmetic, Trigonometry

### How do you simplify csc theta cot theta?

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.

###### Asked in Math and Arithmetic, Algebra, Trigonometry

### How do you find sin cos and tan values manually?

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.

###### Asked in Algebra, Trigonometry

### Find the exact value of the expression sinarctan-12?

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

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