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tan x = \(\frac{6}{g}\) so the side opposite to x is 6 and the adjacent side (Not the hypotenuse) is g

sin x =\(\frac{6}{h}\) so the side opposite to x is 6 (already know) and the hypotenuse is h

so

cos x = \(\frac{g}{h}\) (adjacent/hypotenuse) 

Guest used the right process but forgot that a "4" was placed on the right side of the equation.

                            2x² - 8x + 2y² + 12y + 14  =  0

Divide by 2:         x² - 4x     + y² + 6y     + 7  =  0

Subtract the 7:    x² - 4x     + y² + 6y            =  -7

Complete the squares by adding  4  and  9  to both sides:

                        ( x² - 4x + 4 ) + ( y² + 6y + 9 )  =  -7 + 4 + 9

Rewrite:                 (x - 2)² + (y + 3)²  =  6

Center:  (2, -3)

Radius:  sqrt(6)

What's the parameter of the triangle ABC?

thank u so much!

Just divide the powers of 10 and the numbers separately...

3 * (10^15)/6 * (10^10) = 0.5 * 10^5 

Now, we can simplify that into 5 * 10^4...

Therefore, the answer is 5 * 10^4. 

If you draw the square with vertices:  (a, a)     (-a, a)     (a, -a)     (-a, a)

and the line  y  =  x/2  this line will pass through the points  (a, a/2)   and  (-a, -a/2).

Consider the quadrilateral whose enpoints are (a, a/2)    (a, a)     (-a, a)     (-a, -a/2).

The distance from  (a, a/2)  to  (a, a)  is  a/2.

The distance from  (a, a)  to  (-a, a)  is  2a.

The distance from  (-a, a)  to (-a, -a/2)  is  3/2·a

To find the distance from  (-a, -a/2)  to  (a, a/2)  we can use the distance formula:

   distance  =  sqrt[ (a - a)² + (a/2 - -a/2)² ]  =  sqrt[ (2a)² + (a)² ]  =  sqrt[ 4a² + a² ]  =  sqrt( 5a² )  =  a·sqrt(5).

Adding these togerther:  a/2 + 2a + 3/2·a + a·sqrt(5)  =  4a + a·sqrt(5)

Dividing this by  a  gives  4 + sqrt(5).

we can use difference of squares. 

first, let's divide everything on both sides of the equation by 2... then, we have: 

x^2 - 4x + y^2 + 6y + 7 = 0

Now, we add 4 on both sides in order to have (x^2 - 4x + 4) + y^2 + 6y + 7 = 4, meaning we can rewrite (x^2 - 4x + 4) as (x-2)^2. 

Again, we can do the same... 

we subtract 7 on both sides to get (x-2)^2 + y^2 + 6y = -7...

Seeing the y^2 + 6y, we can also think of y^2 + 6y + 9 which allows us to rewrite this as (y+3)^2...

So, we add 9 to both sides to get (x-2)^2 + (y+3)^2 = 2...

Now, this equation is in the circle form (x-h)^2 + (y-k)^2 = r^2 where r is the radius and (h,k) is the center of the circle. 

2 = r^2 

r = sqrt2 

So, this circle has center (2,-3) and has radius sqrt2. 

hope this helped! :D

I would use the Law of Cosines to find the sizes of the angles. (Or, at least two of them, you can subtract from 180^o to find the third angle.)

Knowing these, you can find the answers.

Find the size of angle(C) first; so, if you want to find the size of angle(A) or angle(B) by using the Law of Sines, you can do so.

You won't get the correct answer for angle(C) if you use the Law of Sines to try to find it.

100! · 99! · 98! ·  ...  · 3! · 2! · 1!

=  [100! · 99!] · [98! · 97!]  · ...  · [4! · 3!] · [2! · 1!]

=  [100 · 99! · 99!] · [98 · 97! · 97!] · ... [4 · 3! · 3!] · [2 · 1! · 1!]

=  [100 · 99!²] · [98 · 97!²] · ... · [4 · 3!²] · [2 · 1!²]

=  [100 · 98 · 96 ·... ·  6 · 4 · 2] · [99!² · 97!² · ... · 3!² · 1!²]

=  2⁵⁰[50 · 49 · 48 · ... · 3 · 2 · 1] · [99!² · 97!² · ... · 3!² · 1!²]

=  2⁵⁰ · [50!] · [99!² · 97!² · ... · 3!² · 1!²]

So, if you let k!  =  50!, the result will be a perfect square (because 2⁵⁰ and [99!² · 97!² · ... · 3!² · 1!²] are perfect squares)

but I can't guarantee that it is unique.

Thanks!!

Here is the question as I understand it.

there are 7 distinct points on a circle. ANY 4 lines are drawn. each one connecting one point to a different point.

What is the probability that these 4 lines form a bowtie?

I think there are 7C4 ways to choose 4 points. = 35 ways

For each of these there are 2 ways they can be joined to make a bow tie. So that is 70 ways.

There are 21 chords so I think there must be 21*20*19*18 ways to choose 4 chords.  6840ways

So maybe if my logic is right the prob of forming a bow tie is

70 / 6840 = 7/684

Can you check if this is correct or not?

2 x 10¹²  /  5 x 10⁶  =    .4 x 10⁶ =   4 x10⁵  =   4 00,000  times larger

draw a triangle to find sin(x) and sin(y)

(little drawing thing)

sin(x) = 4/5, sin(y) = 3/5 from the picture

cos(x-y) = cos(x)cos(y) + sin(x)sin(y)
= (3/5)(4/5) + (4/5)(3/5) = 12/25 + 12/25 = 24/25

theta should be 5·pi/6

50p q²  + 6p-p

50 p q²  + 5p

5p ( 10q² + 1)   

 I am not sure, but I think the theoretical probability of rolling a 2  is  1 out of 6    1/6    (assuming a fair die)

     your EXPERIMENTAL probability is  4 out of 24   ....  also  1/6

From what I can see, there are 6: (-14,4),(-8,6),(-6,12),(-4,-6),(-2,0),(4,2)

I looked at the graph \(b= \frac {3a+6} {a+5}\), since there is a horizontal asymptote at b=3, there are only a finite number of points to check

It is flipped     so it is   -x^2

Now look at the value at x = 1     it changes from    y = 1   to   y =  -4

   so the original f(x) is flipped AND multiplied by 4

     and becomes   g(x) = - 4 x^2

A number cube is rolled 24 times and lands on 2 four times and on 6 three times. 

Find the experimental probability of not landing on a 6. 

aa 22 3 3  43  3 3 33  3

P=100;   R=0.04 /12;  N=4*12;    FV=P*((1 + R)^N - 1)/ R;print"FV = $",FV

FV = $ 5195.96

S₁₀ = S₉ + a₁₀

a₁₀ = S₁₀ - S₉

S₁₀ = 10(11)(12) = 1320

S₉ = 9(10)(11) = 990

a₁₀ = 1320 - 990 = 330

\(1)\frac{7}{4n}-\frac{2n+4}{15n-3}=\frac{7}{4n}-\frac{2n+4}{3(5n-1)}\\ 2)\frac{21(5n-1)}{12n(5n-1)}-\frac{8n^2+16n}{12n(5n-1)}\\ 3)\frac{105n-21-16n-8n^2}{12n(5n-1)}\\ 4)\frac{89n-21-8n^2}{12n(5n-1)} \)
further explanations:
step 1 --> i just factored out the 3 from the right expression's denominator to make it easier to work with

step 2 --> trying to get both expressions to have common denominators, i multiply the left expression's numerator and denominator by 3(5n-1); i also multiply the right expression's numberator and denominator by 4n

step 3 --> i subtract the numerator of the left and right expressions since the denominators are now the same

step 4 --> simplify and get choice #3

Is the answer 432?

(1,432)

(2,216)

(3,144)

(4,108)

(6,72)

(8,54)

(9,48)

(12,36)

(16,27)

(18,24)