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guest,

you had been helpful beyond believe!!...Thank you for your inputs and I will definitely make a copy of this and go study....Thank you very much!

Another thing, yes, there are a lot of identities in trigonometry. 

But do not worry about it. It just takes time. The more questions you solve, the more identities you will use, and so you will remember them after using them a lot.

But, here, I will Include a summary of the most common ones:

\(sin^2x+cos^2x=1 \\ sin(2x)=2sin(x)cos(x) \\ cos(2x)=cos^2x-sin^2x=2cos^2x-1=1-2sin^2(x) \\ tan(2x)=\dfrac{2tan(x)}{1-tan^2x} \\ 1+tan^2(x)=sec^2(x) \\ 1+cot^2(x)=csc^2(x) \\ cos^2(x)=\dfrac{1+cos(2x)}{2} \\ sin^2(x)=\dfrac{1-cos(2x)}{2} \\ \)

\(sin(x+y)=sin(x)cos(y)+sin(y)cos(x) \\ cos(x+y)=cos(x)cos(y)-sin(x)sin(y) \\ tan(x+y)=\dfrac{tan(x)+tan(y)}{1-tan(x)tan(y)}\)

(Notice: In the last three, if you let x=y, you get sin(2x), cos(2x), tan(2x) formula).

So by studying these, you will solve most trigonometric questions.

I hope this helps!

Hi JurieMagic!

Question: \(16sin(x)cos^3(x)-8sin(x)cos(x)\)

When you factored it, you forgot to remove it from the other bracket. 
I.e. the correct factorization is:

\(8sin(x)cos(x)[2cos^2(x)-1]\)

Now, we use this identity: \(cos(2x)=cos^2(x)-sin^2(x)=2cos^2(x)-1\)

So:

\(8sin(x)cos(x)(cos(2x))\)

is the answer. However, we can even simplify further:

\(4(2*sin(x)cos(x))*cos(2x)\)

And another identity: \(sin(2x)=2sin(x)cos(x)\)

So:

\(4(sin(2x))cos(2x)\)

But we use this identity again:

\(4sin(2x)cos(2x)=2(2sin(2x)cos(2x))=2sin(4x)\)

I hope this helps, and do not hesitate to ask further questions!

You need to memorize

\(sin(A+B) = sinAcosB + cosAsinB\\ so\\ sin(2A)=sinAcosA+cosAsinA=2sinAcosA\)

It can be provben of course but you really need to memorize it.

so Sin24cos24 can be written as   0.5(2sin24cos24) = 0.5*sin48

Note: I have not looked at the whole question.

I do not understand what you are asking ??

Where is the diagram?

Since G is the centroid of triangle ABC, we know that it is the intersection point of the medians of the triangle. A median divides a triangle into two smaller triangles of equal area. In this case, we have triangle ABG as one of those smaller triangles, and it's given that ABG is an equilateral triangle with side length 2.

The area of an equilateral triangle can be found using the formula:

Area = (side^2 * √3) / 4

For triangle ABG, we have:

Area(ABG) = (2^2 * √3) / 4 = 4√3 / 4 = √3

Since triangle ABC is composed of 3 smaller triangles with equal area (ABG, ACG, and BCG), we can find the area of triangle ABC by multiplying the area of triangle ABG by 3:

Area(ABC) = 3 * Area(ABG) = 3 * √3 = 3√3

So the area of triangle ABC is 3√3 square units.

a. counting from 1 to 99 would take exactly 99 numbers

b. Do you mean the number 0, or a number containing 0?

c. take all the pairs of numbers that add up to 99 (aka 1 and 98, 2 and 97, 3 and 96 etc.) all the way to the end. the total should be around (49*99)+50

Dividing the numerator and denominator by x^2, we get

y = (2x^2 - 3/x^2)/(1 - 4/x + 1/x^2).

As x approaches infinity, this approaches y = 2x^2.  Thus, the parabolic asymptote is y = 2x^2.

To write the given equation in the form (x + a)(y + b) = c, we need to expand the right-hand side and match the terms with the given equation.

(x + a)(y + b) = c

Expanding the left-hand side, we get:

xy + xb + ay + ab = c

Comparing the coefficients with the given equation, we have:

ab = 3 (coefficient of x in numerator) a + b = -4 (coefficient of x in denominator) c = -21 (constant term)

Solving for a and b, we can use the fact that a + b = -4 to express one variable in terms of the other:

a = -4 - b

Substituting into the equation ab = 3, we have:

(-4 - b) * b = 3

Expanding and solving for b, we get:

b^2 + 4b + 3 = 0

(b + 3)(b + 1) = 0

So, b = -3 or b = -1.

If b = -3, then a = -4 - (-3) = -1. This gives us:

(x - 1)(y - 3) = -21

If b = -1, then a = -4 - (-1) = -3. This gives us:

(x - 3)(y - 1) = -21

Therefore, the equation y = (x - 3)/(x + 7) can be expressed in the form:

(-1, -3, -21) or (-3, -1, -21)

|x − 4| ≥ 7  

Since an absolute value is positive, what you need inside the pipes is 11 < x < –3  

If x is positive, x has to be 11 or greater. 

If x is negative, x has to  be –3 or smaller. 

Are you blind ?? Do you look at your question when you copy and paste it to see if anything is missing? Don't you know that " the product of her rolls is"..........(missing)! Without that product, your question CANNOT be answered. Understood?

40 T-shirts must be sold.

There are two cases:

Case 1: You draw two black balls on the first draw. The probability of this is 3/8*2/7 = 3/28.

Case 2: You draw one black ball and one white ball on the first draw. The probability of this is 3/8*5/7 = 15/56​. You then put the white ball back in the bag and draw two more balls. The probability of drawing two black balls in this case is 3/8*2/7​=3/28​.

The total probability of drawing two black balls is 3/28 + 15/56 = 45/56​​.

how many feet are in a mile?  

Just to add to the first answer, there is a distance they call a nautical mile.

A nautical mile is usually used in the context of velocity, i.e., 1 knot is 1 nautical mile per hour. 

A nautical mile is 6,076.12 feet  

The two trees will have the same height in 10 months.  They will both be 20 inches.

5/8 times 7/8 times 3/5 as a fraction  

    5             7             3 

 ––––   x   ––––   x   –––– 

    8             8             5 

multiply across the numerators       5 x 7 x 3 = 105 

multiply across the denominators    8 x 8 x 5 = 320 

so there's the answer        105                                             21 

                                        –––––      which will reduce to    –––––  

                                          320                                             64 

You could have cancelled the 5's and saved yourself a little arithmetic.  

_.

Acute isosceles triangle.

Sides: a = 15   b = 20   c = 20

Area: T = 139.054
Perimeter: p = 55
Semi-perimeter: s = 27.5

Angle ∠ A =44.049° = 44°2'55″ = 0.769 rad
Angle ∠ B =67.976° = 67°58'32″ = 1.186 rad
Angle ∠ C =67.976° = 67°58'32″ = 1.186 rad

Height: ha = 18.54
Height: hb = 13.905
Height: hc = 13.905  {You have 2 of them because it is an isosceles triangle}

ererer

The answer is 3/8.

Why is it "hard counting question" ?

When you roll 2 dice, how would you get the sum of 2? Well, you can only get the sum of 2 if you roll 2 "ones". Because: 1 + 1=2. Right? How would you get the sum of 3? Again, you can only get the sum of 3 if you roll one "1" and one "2". Because: 1 + 2=3 and 2 +1 =3.......and so on.

Go through all the numbers from 2 to 12, because 12 is a maximum you can get from rolling 2 dice, and that can only happen if you roll 2 "sixes", because: 6 + 6=12

See the answer here:  https://web2.0calc.com/questions/counting_34169