Questions on Algebra: Trigonometry answered by real tutors!

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multiplying by sinx __ (1-cosx) + (sinx^2)/(1-cosx) = 2

substituting __ (1-cosx) + (1-cosx^2)/(1-cosx) = 2

factoring __ (1-cosx) + [(1+cos)(1-cosx)]/(1-cosx) = 2

cancelling __ 1-cosx + 1+cosx = 2 __ 2=2

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Something is missing from our posting.
Cheers,
Stan H.

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Combine similar terms:




, Answer
Let's check,




Thank you,
Jojo

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-5/2x+1/2=-18
Multiply thru by 2 to get rid of the denominators:
-5/2x+1/2=-18
-5x + 1 = -36
-5x = -37
x = 37/5
==========
Cheers,
Stan H.

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): I need help verifying these identities.

 
Work with the right 
side.  Substitute

for  and
substitute 
for 
 
                      = 
Distribute out the
numerator:
                      = 
 
Put a 1 under the 
 so everything
will be a fraction 
                      = 


Invert the fraction
in the denominator 
and multiply:
 
                      = 
 
Swap the fraction factors:

                      =                    
 
 
Distribute:

                      =  

                      = 

                      = 

AND 


Work with the left side. 
Square out the two 
binomials:

 =




 


Use the identity  and
replace  by 



Distribute:



Combine like terms:



Edwin

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Solve the following equation: cos(x/3 - pi/4) = 1/2 , x is btw 0 and 2pi
---------------------------
cos(x/3 - pi/4) = 1/2
(x/3 - pi/4) = pi/6, 5pi/6 (or -pi/6)
x/3 = 5pi/12, pi/12 (5pi/6 + pi/4 times 3 exceeds 2pi)
x = 5pi/4, pi/4
x = 225 degs, 45 degs

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In this question you need to find what angle has a tangent of 0.5. (Tan A = 0.5)
You need to use your calculator to find the size of angle A to the nearest 1/10 of a degree.
The actual buttons to press on your calculator depend on the type of calculator which you are using.
Step 1. Find the button with [tan-1] or arctan written above it. This button will probably have tan written on the button itself.
Step 2. On your calculator press the button which is used to access the second meaning for a button (this button may be [INV] or [2nd]).
Step 3. Now press the button which you identified in Step 1.
Step 4. Press the buttons for 0.5 and then press [=] or [ans] or what ever button you use to get an answer from your calculator.
Step 5. You will now get an answer of approximately 26. Take that answer and round it to one decimal place.
If this is too difficult to follow let me know what brand and model of calculator you have and I will try to find what the buttons to press on your calculator are.

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Hi here i answered questions 2) and 3) contact me if you need more help
2. Calculate the points of the tangent of the curve : y = ex(x2 – 8) where and in which the tangent line is horizontal
,
y'=0 if only if x=0
then the point is (0,0) (where the tangent of the curve is horizontal)
3. Write one of the two tangents to the curve : y = 1 + x³ which are parallel to this equation 12x - y = 1.

then

then
then 12 (first term of 12x-1 and y'(x)= 3x^2 are the same}
then then or
taking x=2 then equation of the line is where
then line is
Answer:

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If that's a right triangle, then right?
Since in that right triangle , then



Let's see the triangle below:
---> RED Line =unknown?; Blue Line =13"; GREEN Line =5"
Now, if , by trigo functions ---->


Check: Remember in a right triangle:
--->


, close enough, round off.
Thank you,
Jojo

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cos (theta) = 3/5 = .6
theta = 53.130... degrees.
sin (4*theta) = sin (212.520 degrees) = -.5376
-----
i used the calculator to get the angle whose cosine was .6.
since that angle was between 0 and 90, i used it directly.
i then multiplied that angle * 4.
then i used the calculator to get the sine of that angle.
-----
i assumed you meant sin (4*theta) when you showed "sin 4 (theta)"
-----

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Find the solutions of the equation in the interval [0,2∏)
cos(X/2)-sin(X)= 0
------------------
cos(x/2) - 2sin(x/2)cos(x/2) = 0
cos(x/2) = 0
x/2 = 90, 270
x = 180º = pi (540 is 180, and exceeds 2pi)
-------------
1 - 2sin(x/2) = 0
sin(x/2) = 1/2
x/2 = 30º, 150º
x = 60º, 300º = pi/3, 5pi/3

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Let u=x/2. So this means that 2u=2(x/2)=x (ie 2u=x)


cos(x/2)-sin(x)= 0 ... Start with the given equation


cos(u)-= 0 ... Replace x/2 with u and x with 2u


cos(u)-2sin(u)cos(u)= 0 ... Use the identity sin(2A) = 2sin(A)cos(A)


cos(u)(1-2sin(u))= 0 ... Factor out the GCF cos(u)


cos(u)=0 ... or ... 1-2sin(u)=0 ... Set each factor equal to zero


Let's solve the first equation cos(u)=0

By taking the arccosine of both sides, we get u=pi/2 or u=3pi/2


But remember, we let u=x/2. So this means that x/2=pi/2 or x/2=3pi/2


Solving for x, we get x=pi or x=3pi (we must discard this solution as it is NOT in the given interval)

So the first solution is x=pi
-----------------

Now let's solve the second equation 1-2sin(u)=0

-2sin(u)=-1 ... Subtract 1 from both sides


sin(u)=1/2 ... Divide both sides by -2

Taking the arcsine of both sides gives us u = pi/6 or u = 5pi/6


Once again, recall that we said that u = x/2. So x/2=pi/6 or x/2=5pi/6


So this means that the values of x are x=pi/3 or x=5pi/3 (both of which are within the given interval)


==================================================


Answer:

So the three solutions in the interval [0,2∏) are


x=pi, x=pi/3 or x=5pi/3

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A which is 29 degrees,
side CA of 18 meters,
and a right angle which is angle C.
---------------------------------------
Here's what I found out so far:
that angle b = 61 degrees
but I do not know how to figure out CB and BA
--------------------
Draw the picture.
CB is opposite angle A=29 degrees.
BA is opposite angle C so BA is the hypotenuse
CA is opposite angle B
----------------------
So, cos(A) = adjacent/hypotenuse
cos(29) = 18/hypotenuse
hypotenuse = 18/cos(29) = 20.58
-----------------------------------
sin(A) = CB/hypotenuse
CB = hypotenuse*sin(A)
CB = 20.58*0.4848..
CB = 9.97738....
==========================
Cheers,
Stan H.

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The largest of the pyramids of Egypt has a square base with sides 755 feet long. Angles PQR has a measure of 52 degrees. The top of the pyramid is no longer there. What was the pyramid's oringinal height RP to the nearest foot?
----------------------------------
The diagonal of the pyramid = 755sqrt(2)
Assuming PQR is an angle at one of the corners of the pyramid, you have
a right triangle with base = (755sqrt(2))/2, a base angle of 52 degrees,
and height = RP.
-----------------------
EQUATION:
tangent(52) = h/[755sqrt(2)/2]
h = [755sqrt(2)/2]*tangent(52)
h = 533.8656..* 1.2799...
h = 683.32 feet
======================
Cheers,
Stan H.

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Suppose the blimp was directly over the 50-yard line. A photograper on the ground estimated the angle of elevation to be 85 degrees when she was 65 yards away from the point on the ground directly under the blimp. How high is the blimp? Round your answer to the nearest foot.
--------------------------
Draw the picture: a right triangle with a base of 65 yds.,
base angle of 85 degrees and height of "h".
---------------------------
EQUATION:
tan(85) = h/65
h = 65*tan(85)
h = 65*11.43005..
height = 742.95.. yrds. = 2228.86 ft.
========================================
Cheers,
Stan H.

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On September 30, 1995, the Ohio State Buckeyes defeated the Notre Dame Fighting Irish in football as a blimp relayed aerial viewes from above. Suppose the blimp was directly over the 50-yard line. A photograper on the ground estimated the angle of elevation to be 85 degrees when she was 65 yards away from the point on the ground directly under the blimp. How high is the blimp? Round your answer to the nearest foot.
.
You must use trigonometry. Draw a diagram to see where the right triangle is.
.
tan = opposite/adjacent
.
tan(85) = x/65
x = 65 * tan(85)
x = 65 * 11.4301
x = 742.9534 yards
.
Converting the above into feet:
x = 742.9534 * 3 = 2228.86
Rounding to nearest foot:
x = 2229 feet

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... Start with the given equation


... Factor out (good so far)


Rearrange the terms.


Factor 4 into 2*2


Use the identity


Use the identity


Let


Substitute "z" in for 2x


Use the identity (similar to the previous used identity just with a different variable)



Plug in


Multiply


Since both sides of the equation are equal, this means that we've verified the identity

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Note: I'm only manipulating the left side


... Start with the given equation



... Use the identity to simplify the denominator


... Now use the identity to get the denominator in terms of cosine


... Multiply the first fraction by the reciprocal of the second fraction


... Get tangent in terms of sine and cosine


... Distribute


... Multiply and simplify


... Factor out a negative 1


... Rearrange the terms.



... Use the identity



... Use the identity


... Distribute


... Rearrange the terms.


Since the two sides are equal, this confirms the identity

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Prove that the maximum height of the projectile launched upward is: .
Starting with the function for the height (as a function of time, t) of a projectile launched upward from an initial height of with an initial velocity of :

In the given problem:
Projectile is launched from the ground, so you have:

When graphed, this equation describes a parabola that opens downward (negative coefficient of so the vertex is a maximum.
The abscissa (corresponds to the x-coordinate) of the vertex is given by:
where the a and b come from the general form of a quadratic equation:, but, in this problem, the independent variable is t rather than x, and and , so you have:
This is the time at which the projectile will be at its maximum height. Let's call this . Simplifying this, you get:
Now substitute this value of t into the original equation to find the maximum height of the projectile:
Simplifying:


QED
The maximum height attained by the projectile in this problem is:

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It might be hard to see, but the equation can be rearranged to and fits the form where , , and


So to find the greatest height, we need to find the vertex.


To do that, we use this formula





Plug in ,


Multiply and reduce


Reduce


So the max height will occur when the time is


---------------------------------


Go back to the original equation


Plug in


Square to get


Cancel like terms. (one pair of "g" terms cancel)


Multiply


Multiply the first fraction by


Combine the fractions


Subtract


So at the time the max height will be

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I'll leave you the sketch, hopefully we can follow together.
In the triangle ABC, given:
; ;

Pardon me for the illustration below is just for reference --> NOT TO SCALE. So it will be easy for us to go along.
---> BLUE circle>>>; GREEN Circle>>> ; RED Circle>>> . Also,BLUE LINE =; GREEN LINE=; RED LINE=
a)the lengths of AC and BC
In order to find and with given data,w e draw a line from to line and mark LINE . See below:
---> BLACK Line=
We know -------------> Eqn 1 >>>> as shown on the graph
We find forms right triangle. By Trigo Functions:
Find first Line ---> --->---------> Eqn 2:
But, --> --------------> Eqn 3
Subst. Eqn 3 in Eqn 2:
------------------> Eqn 4
.
Next, Line , On Angle :

-----------> Eqn 5
Therefore, Via Eqn 1:
------------------> Eqn 6
*Note: as per given
.
Next, Line =?



---------------> Eqn 7
.
b)sin 75degrees, cos 75degrees, tan75degrees
First =?,
On Right Triangle with :




Next, =?,




.
Next, =?




.
c)sin 15degrees, cos 15degrees, tan 15degrees
We draw a line from to as shown:
Mark Line .See below:
----> We call it angle from A --> E.
.
We solve for :

------------> Eqn 8
Next, =?

-------------------> Eqn 9
Next, =?

Try to finish it off with what we've computed on the above EQUATIONS 1-9:
.
Thank you,
Jojo

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If in radians,



If in degrees,

°

Edwin

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Edwin

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Different authors and different teachers use 
different domains for the domain of  
and . See this site for a discussion
of this:

http://sci.tech-archive.net/Archive/sci.math/2006-03/msg02129.html

Edwin

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Different authors and different teachers use 
different domains for the domain of  
and . See this site for a discussion
of this:

http://sci.tech-archive.net/Archive/sci.math/2006-03/msg02129.html

Edwin

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1.Derivatives:
i)(Sin4x) /( x² +1)
Let y = (sin4x)/x^2 +1
dy/dx use u/v method
d(u/v) = (vdu-udv)/v^2
Here , u = sin4x
v= x^2 +1
du = 4 cos4x
dv = 2x
dy/dx = ((x^2 +1)4cos4x- sin4x(2x))/(x^2 +1)^2
= (4x^2 cos4x + 4 cos4x - 2x sin4x)/(x^2+1)^2
ii)10t^2 +1
y = 10 t^2 +1
dy/dt = 20 t
iii)y= sqrt((1+u)/(1-u))
y^2 = (1+u)/1-u)
2y dy = ((1-u) du + (1+u) du)/(1-u)^2
2y dy = 2du/(1-u)^2
y dy = du/(1-u)^2
dy/du = y/(1-u)^2
= sqrt((1-u)(1+u))/(1-u)^2
= (sqrt(1+u))/(1+u)^3/2
iv)question not clear

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eq 1
eq 2
subtract 2nd equation values from 1st equation to eliminate
divide by -3
plug a's value into either equation to find .
--->

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here's what i think.
the volume of the cube = s^3 (side * side * side)
-----
if you plane off .5 centimeters from each face, then each side will have 1 centimeter taken off of it (.5 centimeters from each end).
i will explain this better later.
-----
the volume of the planed cube will therefore be (s-1)^3.
-----
the volume of the planed cube is 169 cubic centimeters less than the volume of the unplaned cube.
-----
what this says is that s^3 - (s-1)^3 = 169 cm^3.
-----
i worked the equation which i will detail later.
for now, i will tell you that the side of the unplaned cube is 8 centimeters.
this makes the side of the planed cube 7 centimeters.
-----
(8cm)^3 = 512 cubic centimeters
(7cm)^3 = 343 cubic centimeters
-----
512 cm^3 - 343 cm^3 = 169 cm^3.
-----
this satisfies the original statements of the problem so i assume that the values for each side of the planed cube and the unplaned cube are correct.
-----
now some details.
-----
first the equation:
s^3 - (s-1)^3 = 169 is where i started from.
multiplying out, this equation becomes:
s^3 - (s^3 - 3s^2 + 3s - 1) = 169
remove the parentheses:
s^3 - s^3 + 3s^2 + 3s - 1 = 169
combine like terms:
3s^2 + 3s - 1 = 169
subtract 169 from both sides of the equation and combine like terms:
3s^2 + 3s - 168 = 0
divide both sides of equation by 3:
s^2 + s - 56 = 0
factor:
(s-8)*(s+7) = 0
roots are:
s = 8
or
s = -7
-----
since a negative measurement is not possible, the answer has to be s = 8.
-----
each side measures 8 centimeters in length.
this is equivalent to:
length = 8, width = 8, height = 8 since a cube is a rectangular solid where length = width = height.
-----
the assumption of 1 centimeter off of each dimension was made by visualizing what happens to the cube when you remove .5 centimeters off of each face.
look at it from one side and you can see that removing .5 centimeters from each end reduces that dimension by 1 centimeter (.5 from each end).
look at it from all 3 sides and you can then determine that each dimension is reduced by that same amount.
if you have trouble visualizing this, then take some soap or styro-foam or anything easy to cut and remove a measured amount from each face. you will see that each dimension has been reduced by double that number.
-----
hope this helps.

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When 0.5 cm was planed off of each of the 6 faces of a wooden cube its voume decreased by 169 cm^2. Find its new volume.
.
A "cube" implies that all edges are the same length.
Volume of a cube is "length of an edge" cubed.
.
Let x = original length of one edge
then
x-1 = edge of new cube
.
x^3 - 169 = (x-1)^3
x^3 - 169 = (x-1)(x-1)(x-1)
x^3 - 169 = (x^2-2x+1)(x-1)
x^3 - 169 = (x)(x^2-2x+1)-(x^2-2x+1)
x^3 - 169 = (x^3-2x^2+x)-(x^2-2x+1)
x^3 - 169 = x^3-2x^2+x-x^2+2x-1
x^3 - 169 = x^3-3x^2+x+2x-1
x^3 - 169 = x^3-3x^2+3x-1
-169 = -3x^2+3x-1
0 = -3x^2+3x+168
dividing both sides by 3:
0 = -x^2+x+56
0 = x^2-x-56
0 = (x-8)(x+7)
.
x = {-7, 8}
We can toss out the negative solution so we end up with:
x = 8 cm (length of an edge of the original cube)
.
Length of new cube edge:
x-1 = 8-1 = 7 cm
.
Finally, volume of new cube is:
7^3 = 343 cubic cm

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tan(a)=tan(a+pi)=tan(a+2pi) so
tan(a)+tan(a+pi)+tan(a+2pi)=3+3+3=9

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For the life of me, I cannot solve 4cos^2x - sinx -5 = 0. Any help would be appreciated

Use the Pythagorean identity 
to replace  by 







Multiply thru by 



This is a quadratic in 

But it can have no solution since the discriminant
would be negative  

Edwin

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tan(x)sin(x)-sin(x)=0 ... Start with the given equation

sin(x)(tan(x)-1)=0 ... Factor out the GCF sin(x)

sin(x)=0 or tan(x)-1=0 ... Set each factor equal to zero

Solve the first equation:

sin(x)=0 ---> x = 0 + 2pi*n ... or ... x = pi + 2pi*n


Solve the second equation:

tan(x)-1=0=0 ---> tan(x)=1 ----> x = pi/4 + pi*n

So the solutions are

x = 0 + 2pi*n ... or ... x = pi + 2pi*n ... or ... x = pi/4 + pi*n

where "n" is an integer.

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I am trying to verify the Identity of: cot x - csc x = cos x-1/ sin x can you please help?
----------------------
cot x - csc x = cos x-1/ sin x
cot(x) = cos(x)/sin(x) and csc(x) = 1/sin(x)
(cos/sin) - 1/sin = (cos - 1)/sin

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2 csc2x + 4 = 1
2 csc(2x) = -3
csc(2x) = -3/2
2x = arccsc(-3/2) = -41.81 degrees
x = -20.09 degrees
or
x = 339.09 degrees
============================
2 csc2x + 3 = 0
2csc(2x) = -3
csc(2x) = -3/2
Same answer as abvove
============================
Cheers,
Stan H.

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Use the angle sum or difference identity to find the exact value of
cos195

You have to figure out a way to make 195° using 
the sum or difference of two special angles from this list:

30°, 45°, 60°, 90°, 120°, 135°, 150°, 180°, 210°, 
225°, 240°, 270°, 300°, 315°, 330°.

You can write 195° as any one of these:

150°+45°, 135°+60°, 225°-30°, 240°-45°, 315°-120°, 330°-135°

If you choose to use the first one, 150°+45°, you then use



=







--------------------------------



Let's convert that to degrees:

°

Then we can write 75° as 30°+45°

=

Then we can use the identity: 

=

=

=

=





Rationalize the denominator:



Indicate multiplication of numerators and denominators:



Multiply out the bottom:

=

=

=

=

=
Move the negative sign out front, and cancel the 4's

=







Edwin

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Write a polynomial equation of the least degree with the roots 4,1,i,and -1.
:
Write factors from the roots
(x-4)
(x+1)*(x-1) = (x^2 - 1) = 0
and
x = i
x^2 = i^2
x^2 = -1
(x^2+1) = 0
:
(x^2-1) * (x^2+1) = x^4 -x^2 + x^2 - 1 = x^4 - 1
then
(x-4)*(x^4-1) = x^5 - x - x^4 + 4
:
Arrange to: x^5 - 4x^4 - x + 4

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x^2 - 20 = 0
:
x^2 = +20
:
x = +/-
:
x = +/-
extract the perfect square
x = -2*
and
x = +2*

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The Rational Root Theorem tells you that if the polynomial has a rational zero then it must be a fraction p/q, where p is a factor of the last constant and q is a factor of the first coefficient.
Since both the first and the last coefficients are 2 the factors are 1 & 2
The possible rational roots are +- 1/2, 1, 2
If you test them all you will find that -2 is the only rational root. (Answer) If you use long division (2x^3-7x+2)/(x+2)=2x^2-4x+1. Then if you use the quadratic formula on this result it gives the other 2 irrational roots (2-sqrt(2)/2) and (2+sqrt(2))/2.
.
Ed
.

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Please help me solve this equation. Find the discriminant of
------------------------
6-5x=6x^2
---
6x^2 +5x -6 = 0
-----
D = b^2-4ac
D = 5^2 - 4*6*-6
D = 25 + 144
D = 169
=================
Cheers,
Stan H.

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subtracting 10 __ x^2+6x=-10

adding half of the x coefficient, squared __ x^2+6x+(6/2)^2=-10+(6/2)^2 __ x^2+6x+9=-1

taking square root __ x+3=±i __ subtracting 3 __ x=-3±i

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We have to be careful here!  We might be tempted
to use the zero-factor principle. However that 
will lead us astray because  is not 
defined at ANY of the values of x for which
!

So do NOT do this:



Instead use the identity  to replace .




We can cancel 's as long as we are careful 
not to allow any values of x which cause the 
denominator  to be zero.

 

That leaves:



which has solution 

, where n is any integer.

And this is a valid solution since 
is always either  or , and never .

Edwin

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Lets solve the first equation for either A or B;
B = 100-A
Now plug this into the second equation for b;





Now we have A, lets plug that into the first equation;


check;


:)

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2 * sin^2(x) = sin(x)
dividing both sides of equation by sin(x) gets:
(2* sin^2(x))/sin(x) = 1
dividing both sides of eqution by 2 gets:
sin^2(x)/sin(x) = 1/2
since sin^2(x) = sin(x) * sin(x), equation becomes:
(sin(x)*sin(x))/sin(x) = 1/2
which becomes:
sin(x) = 1/2
-----
if sin(x) = 1/2, then x = 30 degrees.
-----
since sin(x) = hypotenuse / y value on the graph, and since hypotenuse is always positive, and since y value on the graph is positive in quadrants 1 and 2, then sin (180-x) = sin(x).
-----
since 180-30 = 150, then x can be either 30 degrees or 150 degrees.
-----
using the calculator to prove the answer is correct.
sin (30) = 1/2
sin (150) = 1/2
-----
calculator confirms logic and answer is good.

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you're OK so far

multiplying by sin^2 __ cos^2-cos^2*sin^2=cos^4

substituting sin^2=1-cos^2 __ cos^2-cos^2(1-cos^2)=cos^4

cos^2-cos^2+cos^4=cos^4 __ cos^4=cos^4

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.055x+.045(6,000-x)=305
.055x+270-.045x=305
.-1x=305-270
.01x=35
x=35/.01
x=$3,500 is the amount invested @ 5.5%
6,000-3,500=$2,500 is the amount invested @ 4.5%
Proof:
.055*3,500+.045*2,500=305
192.50+112.50=305
305=306