Answer:
See explanation below
Step-by-step explanation:
Recall the identity for the cosine of an addition of angles:
[tex]cos(\theta+\gamma)= cos(\theta)*cos(\gamma)- sin(\theta)*sin(\gamma)[/tex]
and that for a subtraction of angles:
[tex]cos(\theta-\gamma)= cos(\theta)*cos(\gamma)+sin(\theta)*sin(\gamma)[/tex]
Now, use the property of addition of angles above to write :
[tex]cos(\frac{A+B}{2} +\frac{A-B}{2} )=cos(\frac{A+B}{2} )*cos(\frac{A-B}{2} )-sin(\frac{A+B}{2} )*sin(\frac{A-B}{2} )[/tex]
but since : [tex]\frac{A+B}{2} +\frac{A-B}{2}= A[/tex]
Then the cosine of addition of angles above can be written as:
[tex]cos(\frac{A+B}{2} +\frac{A-B}{2} )=cos(\frac{A+B}{2} )*cos(\frac{A-B}{2} )-sin(\frac{A+B}{2} )*sin(\frac{A-B}{2} )\\cos(A)=cos(\frac{A+B}{2} )*cos(\frac{A-B}{2} )-sin(\frac{A+B}{2} )*sin(\frac{A-B}{2} )[/tex]
We do something similar now with the cosine of the subtraction of angles:
[tex]cos(\frac{A+B}{2} -\frac{A-B}{2} )=cos(\frac{A+B}{2} )*cos(\frac{A-B}{2} )+sin(\frac{A+B}{2} )*sin(\frac{A-B}{2} )[/tex]
and we notice that: [tex]\frac{A+B}{2} -\frac{A-B}{2}= B[/tex]
Then the cosine of subtraction of angles above can be written as:
[tex]cos(\frac{A+B}{2} -\frac{A-B}{2} )=cos(\frac{A+B}{2} )*cos(\frac{A-B}{2} )+sin(\frac{A+B}{2} )*sin(\frac{A-B}{2} )\\cos(B )=cos(\frac{A+B}{2} )*cos(\frac{A-B}{2} )+sin(\frac{A+B}{2} )*sin(\frac{A-B}{2} )[/tex]
and finally, we add term by term the expressions we got for cos (A) and for cos(B) and notice that the terms that contain the sine functions cancel each other because they have opposite signs, while the terms in cosine add up to double themselves:
[tex]cos(A )=cos(\frac{A+B}{2} )*cos(\frac{A-B}{2} )+sin(\frac{A+B}{2} )*sin(\frac{A-B}{2} )\\+\\cos(B )=cos(\frac{A+B}{2} )*cos(\frac{A-B}{2} )-sin(\frac{A+B}{2} )*sin(\frac{A-B}{2} )\\=\\cos(A) +cos(B)=2\,cos(\frac{A+B}{2} )*cos(\frac{A-B}{2} )[/tex]
which is what we wanted to prove.
If the simple interest on $3000 for 3 years is $720 , then what is the interest rate?
Answer:
8%
Step-by-step explanation:
Starting Principal: $
3,000.00
Years:
3
Annual Interest Rate:
8
%
Results
Future Value, using...
Simple Interest: $
3,720.00
Annually Compounded Interest: $
3,779.14
Evaluate the function for x = 2,3,4
F(x) =2x-7-3
Answer:
below in bold
Step-by-step explanation:
f(x) = 2x - 7 - 3
~Combine like terms
f(x) = 2x - 10
When x = 2...
f(x) = 2(2) - 10
f(x) = 4 - 10
f(x) = -6
When x = 3...
f(x) = 2(3) - 10
f(x) = 6 - 10
f(x) = -4
When x = 4...
f(x) = 2(4) - 10
f(x) = 8 - 10
f(x) = -2
Best of Luck!
Evaluate the expression 8-3×2+5
Answer:
7
Step-by-step explanation:
First do 3x2 to get 6, and then do 8-6, which is 2, and add that to 5 to get 7
Check the picturee need helppp
how do you find the distance between two negative fractions
Answer:
Use the number line and split the whole numbers into fractions
Step-by-step explanation:
Example: If a number line goes from -10 all the way to +10, and you are trying to find the distance between -2 1/4 and -7 3/4. You would split the number line into fourths and then locate -7 3/4. You would then count up to -2 1/4 to get 4 3/4.
Hopefully this makes sense!
Bella wants to create a tea-mix. She can buy Asian Treasure tea for $7.50 per ounce and Paradise tea for $5.00 per ounce. If she wants to make 60 ounces of tea-mix at a cost of $6.00 per ounce, how many ounces of Asian Treasure tea and how many ounces of Paradise tea should she buy?
9514 1404 393
Answer:
24 oz Asian Treasure36 oz ParadiseStep-by-step explanation:
The proportion of Asian Treasure is the difference in price of the mix and Paradise tea, divided by the difference in price of Asian Treasure and Paradise tea. So the amount of Asian Treasure is ...
(6.00 -5.00)/(7.50 -5.00)(60 ounces) = 24 ounces . . . Asian Treasure tea
The remaining 60-24 = 36 ounces will be Paradise tea.
__
Using an equation
You can get the same result if you define a variable (a) to the amount in ounces of the higher-priced contributor. Then the lower-priced contributor will be in the amount (60-a) ounces. The total cost of the mix is ...
7.50a +5.00(60 -a) = 60(6.00)
(7.50 -5.00)a +60(5.00) = 60(6.00) . . . rearrange the left side
(7.50 -5.00)a = 60(6.00 - 5.00) . . . . . . subtract 60(5.00)
a = 60(6.00 -5.00)/(7.50 -5.00) . . . . . . divide by the coefficient of 'a'
Note that this last expression is identical to the one we used above to find the quantity of Asian Treasure tea.
The GCF of 15 and 27
Karen spent 39 minutes knitting 3 hats. In all, how many hats could Karen knit in 117 minutes
5. Cherries are on sale for $16 for 4 pounds. Jordyn pays $28.00 for
cherries to make a cobbler. How many pounds of cherries does she buy? *
Answer:
7 pounds
Step-by-step explanation:
1 pound is $4 28/4 is 7
Which angles are adjacent angles to 3? Select all that apply
Two Answers: Angle 1, Angle 4
Adjacent angles share a common segment, line, or ray. Think of two adjacent rooms sharing a common wall.
Peter collected 171 marbles but lost 3/4 of them on his way home through of hole in his bag when he arrived at home now many marbles did Peter have left?
The number of marbles Peter had when he arrived home was 43 marbles
What is Percentage?
A percentage is a number or ratio expressed as a fraction of 100. It is often denoted using the percent sign, %
Given data ,
The number of marbles Peter had on his way home = 171
The amount of marbles he lost on his way home = ( 3/4 ) th
Now , the amount of marbles Peter have left = 1 - amount of marbles he lost on his way home
The amount of marbles Peter have left = 1 - 3/4
= ( 1 / 4) th
So , The number of marbles Peter had left when he arrived home is =
Amount of marbles Peter have left x Number of marbles Peter had
The number of marbles Peter had left when he arrived home is = (1/4) x 171
= 171 / 4
= 42.75
≈ 43 marbles
Hence , the number of marbles Peter had left when he reached home is 43
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Find the value of this question
Answer:
the expression is equal to 0
What are all of the possible classifications for a 3 × 3 system? Check all of the boxes that apply.
consistent and dependent
consistent and independent
inconsistent and dependent
inconsistent and independent
Answer:
consistent and dependent
consistent and independent
inconsistent and independent
Step-by-step explanation:
Answer:
1
2
4
Step-by-step explanation:
has nice day
Find the slope using the formula.
(-2,-9) and (-6,-6)
Answer:(3,-4) you just have to follow the equation.
Slope = change in y over change in x.
Slope = (-6 - -9) / (-6 - -2)
Slope = (-6 + 9) / (-6 + 2)
Slope = 3/-4
Slope = -3/4
Can someone help? I don't really understand this question...
If 6 x 7 = 26 +0, find the number that belongs in the box.
211
The cost of an item is $0.30 per ounce. If total cost of the item is $3.60, how many ounces are in the item?
Answer:
Example 1: - If the item's label displays the weight as 3 lb 2 oz, type the shop name and item's brand, enter 3 in the pounds field, leave the 00 (decimal) value unchanged, enter 2 in the ounces field, leave the 00 (decimal) value unchanged, fill out the Price field and Calculate to get the price per ounce
Answer:
12 ounces
Step-by-step explanation:
3.60 = .3x
divide by .3 on both sides
12 = x
2688 divided by 96 with remainder
Answer: OK, the answer is 28.
Step-by-step explanation: Also there is no remainder for this division problem, you could have the remainder if the numbers are not the same as a multiplication problem, so did you get it?
Out of 1,200 students surveyed, 92% of them like country music. How many students like rap music?
Answer:
96
Step-by-step explanation:
1200*92=110400/100=1104
1200-1104=96
Answer:
1152 students like rap just take the numbers n divide
Are they equal or diffrent angles
How many solutions does the absolute value equation have?
I x - 3 I + 2 = -2
9514 1404 393
Answer:
the correct answer is marked
Step-by-step explanation:
When the graph of the left side of the equation does not intersect the graph of the right side of the equation, there are no solutions.
__
Additional comment
Even if you extend the domain to complex numbers, there are no solutions. The f(x) = |x| function always returns a positive real value, even for complex numbers.
PLEASE HELP QUICKLY!!!!! PLZZZZZZ!!!!!!!
What do you call a female bug that floats?
Answer:
A buoyant girl ant
Step-by-step explanation:
You can tell if the value of a function for a certain interval is positive just by 2 points looking at its graph. If the value is positive that part of the graph will:
a: be to the right of the y-axis
b: be above the x-axis
c: intersect the x-axis
d: be below the x-axis
What is the place value of 3 of 652.4397. N Place Value: Select an answer Select an answer Get help: Vi tenths Points possible hundredths This is attemp Post this questhousandths ten thousandths Submit . hundred thousandths
Answer:
100 thousandths
Step-by-step explanation:
Please help geometry find the solution
Answer:
Directions Unclear:
Step-by-step explanation:
I am now drowning on air.
Answer:
ok
Step-by-step explanation:
Which phrase describes how to divide a complex number z by r(cos θ + i sin θ)?
Scale by a factor of StartFraction 1 Over r EndFraction and rotate clockwise by θ.
Scale by a factor of r and rotate clockwise by θ.
Scale by a factor of StartFraction 1 Over r EndFraction and rotate counterclockwise by θ.
Scale by a factor of r and rotate counterclockwise by θ.
Answer:
its a
Step-by-step explanation:
The phrase is Scale by a factor of 1 /r and rotate clockwise by θ.
What is complex number?Complex numbers are numbers that consist of two parts — a real number and an imaginary number.
Given:
z=r(cos θ + i sin θ)
So, to divide a complex number the best phrase to describe is Scale by a factor of 1 /r and rotate clockwise by θ.
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Students will be going to the zoo in a field trip entrance fee is $6.00 and you will be able to purchase drinks or food items for $2.00 each write an equation indicating the total cost of going to the zoo
Answer:
c = 6+2x
Step-by-step explanation:
c = total cost, 6 = entrance fee, 2x = 2 dollars per food/drink item
9x+6+4x-2x+1-15
What is the answer?
Answer: 3x - 8
Hope it helps!
Answer:
9 + 6 + 4 − 2 + 1 − 1 5
9x+{\color{#c92786}{6}}+4x-2x+{\color{#c92786}{1}}{\color{#c92786}{-15}}
9x+6+4x−2x+1−15
9 − 8 + 4 − 2
Step-by-step explanation:
Suppose that Y has density function
f(y) = ky(1 − y) if 0 ≤ y ≤ 1 0, elsewhere
A) Find the value of k that makes f (y) a probability density function.
B) Find P(Y ≤ .4|Y ≤ .8).
C) Find the .95-quantile, i.e., find a point φ.95 such that P(Y ≤ φ.95) = .95.
I'm assuming
[tex]f(y)=\begin{cases}ky(1-y)&\text{for }0\le y\le1\\0&\text{otherwise}\end{cases}[/tex]
(a) f(x) is a valid probability density function if its integral over the support is 1:
[tex]\displaystyle\int_{-\infty}^\infty f(x)\,\mathrm dx=k\int_0^1 y(1-y)\,\mathrm dy=k\int_0^1(y-y^2)\,\mathrm dy=1[/tex]
Compute the integral:
[tex]\displaystyle\int_0^1(y-y^2)\,\mathrm dy=\left(\frac{y^2}2-\frac{y^3}3\right)\bigg|_0^1=\frac12-\frac13=\frac16[/tex]
So we have
k / 6 = 1 → k = 6
(b) By definition of conditional probability,
P(Y ≤ 0.4 | Y ≤ 0.8) = P(Y ≤ 0.4 and Y ≤ 0.8) / P(Y ≤ 0.8)
P(Y ≤ 0.4 | Y ≤ 0.8) = P(Y ≤ 0.4) / P(Y ≤ 0.8)
It makes sense to derive the cumulative distribution function (CDF) for the rest of the problem, since F(y) = P(Y ≤ y).
We have
[tex]\displaystyle F(y)=\int_{-\infty}^y f(t)\,\mathrm dt=\int_0^y6t(1-t)\,\mathrm dt=\begin{cases}0&\text{for }y<0\\3y^2-2y^3&\text{for }0\le y<1\\1&\text{for }y\ge1\end{cases}[/tex]
Then
P(Y ≤ 0.4) = F (0.4) = 0.352
P(Y ≤ 0.8) = F (0.8) = 0.896
and so
P(Y ≤ 0.4 | Y ≤ 0.8) = 0.352 / 0.896 ≈ 0.393
(c) The 0.95 quantile is the value φ such that
P(Y ≤ φ) = 0.95
In terms of the integral definition of the CDF, we have solve for φ such that
[tex]\displaystyle\int_{-\infty}^\varphi f(y)\,\mathrm dy=0.95[/tex]
We have
[tex]\displaystyle\int_{-\infty}^\varphi f(y)\,\mathrm dy=\int_0^\varphi 6y(1-y)\,\mathrm dy=(3y^2-2y^3)\bigg|_0^\varphi = 0.95[/tex]
which reduces to the cubic
3φ² - 2φ³ = 0.95
Use a calculator to solve this and find that φ ≈ 0.865.
Using probability concepts, it is found that
a) The value is k = 6.
b) P(Y ≤ .4|Y ≤ .8) = 0.554.
c) The 95th percentile is x = 0.86465.
The density function is:
[tex]f(y) = ky(1 - y)[/tex]
Item a:
The condition that makes it a probability density function is:
[tex]\int_0^1 f(y) dy = 1[/tex]
Thus:
[tex]\int_0^1 f(y) dy = \int_0^1 ky - ky^2 dy[/tex]
[tex]\int_0^1 f(y) dy = k(\frac{y^2}{2} - \frac{y^3}{3})|_{y = 0}^{y = 1}[/tex]
[tex]\int_0^1 f(y) dy = k(\frac{1}{2} - \frac{1}{3})[/tex]
Then
[tex]k(\frac{1}{2} - \frac{1}{3}) = 1[/tex]
[tex]k(\frac{3-2}{6}) = 1[/tex]
[tex]k = 6[/tex]
The value is k = 6.
Item b:
This probability is:
[tex]\int_0.4^0.8 f(y) dy = 6(\frac{y^2}{2} - \frac{y^3}{3})|_{y = 0.4}^{y = 0.8}[/tex]
Then
[tex]p = 6\left(\frac{0.8^2}{2} - \frac{0.8^3}{3} - \frac{0.4^2}{2} + \frac{0.4^3}{3}\right)[/tex]
[tex]p = 0.554[/tex]
Thus, P(Y ≤ .4|Y ≤ .8) = 0.554.
Item c:
This is x for which:
[tex]\int_0^x f(y) dy = 0.95[/tex]
Thus:
[tex]6(\frac{x^2}{2} - \frac{x^3}{3}) = 0.95[/tex]
[tex]-2x^3 + 3x^2 - 0.95 = 0[/tex]
Solving a cubic equation with the help of a calculator, and considering [tex]0 \leq x \leq 1[/tex], this is x = 0.86465.
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