Answer:
Proof in the explanation
Step-by-step explanation:
Trigonometric Equalities
Those are expressions involving trigonometric functions which must be proven, generally working on only one side of the equality
For this particular equality, we'll use the following equation
[tex]\displaystyle cos(x-y)=cos\ x\ cos\ y+sin\ x\ sin\ y[/tex]
The equality we want to prove is
[tex]\displaystyle arccos\ \sqrt{\frac{2}{3}}-arccos\left(\frac{1+\sqrt{6}}{2\sqrt{3}}\right)=\frac{\pi}{6}[/tex]
Let's set the following variables:
[tex]\displaystyle x=arccos\ \sqrt{\frac{2}{3}},\ y=arccos(\frac{1+\sqrt{6}}{2\sqrt{3}})[/tex]
And modify the first variable:
[tex]\displaystyle x=arccos\ \frac{\sqrt{6}}{3}}=>\ cos\ x= \frac{\sqrt{6}}{3}}[/tex]
Now with the second variable
[tex]\displaystyle y=arccos\ \frac{1+\sqrt{6}}{2\sqrt{3}}=>cos\ y=\frac{1+\sqrt{6}}{2\sqrt{3}}=\frac{\sqrt{3}+3\sqrt{2}}{6}[/tex]
Knowing that
[tex]sin^2x+cos^2x=1[/tex]
We compute the other two trigonometric functions of X and Y
[tex]\displaystyle sin \ x=\sqrt{1-cos^2\ x}=\sqrt{1-(\frac{\sqrt{6}}{3})^2}=\sqrt{1-\frac{6}{9}}=\frac{\sqrt{3}}{3}[/tex]
[tex]\displaystyle sin\ y=\sqrt{1-cos^2y}=\sqrt{1-\frac{(\sqrt{3}+3\sqrt{2})^2}{36}}}[/tex]
[tex]\displaystyle sin\ y=\sqrt{\frac{36-(3+6\sqrt{6}+18)}{36}}=\sqrt{\frac{15-6\sqrt{6}}{36}}[/tex]
Computing
[tex]15-6\sqrt{6}=(3-\sqrt{6})^2[/tex]
Then
[tex]\displaystyle sin\ y=\frac{3-\sqrt{6}}{6}[/tex]
Now we replace all in the first equality:
[tex]\displaystyle cos(x-y)=\frac{\sqrt{6}}{3}.\frac{\sqrt{3}+3\sqrt{2}}{6}+\frac{\sqrt{3}}{3}.\frac{3-\sqrt{6}}{6}[/tex]
[tex]\displaystyle cos(x-y)=\frac{3\sqrt{2}+6\sqrt{3}}{18}+\frac{3\sqrt{3}-3\sqrt{2}}{18}[/tex]
[tex]\displaystyle cos(x-y)=\frac{9\sqrt{3}}{18}=\frac{\sqrt{3}}{2}=cos\ \pi/6[/tex]
Thus, proven
4(a+5) in distributive form
Answer:
4a+20
Step-by-step explanation:
Answer:4a + 20
Step-by-step explanation:
a. 8% of ? = 20
b. 75% of ? = 30
c. 20% of ? = 45
d. 150% of ? = 36
Answer:
A. 250
B.40
C.225
D.24
Step-by-step explanation:
20/0.08 = 250
30/0.75 = 40
45/0.2 = 225
36/1.5 = 24
Answer:
a. 8% of 250 = 20
b. 75% of 40 = 30
c. 20% of 225 = 45
d. 150% of 24 = 36
Step-by-step explanation:
20×100÷8
=250
30×100÷75
=40
45×100÷20
=225
36×100÷150
=24
20 blocks 6 are shaded what percent are shaded?
Amanda is at a store for apples and bananas. if she buys 6 apples and 5 bananas, it will cost $8.00. is she buys 5 apples and 7 bananas, it will cost $9.50. write and solve a system to determine how much the apples cost
Answer:
Therefore the cost of each apple is $ 0.50 and each banana is $1.
Step-by-step explanation:
Given , Amanda buys 6 apples and 5 bananas , it will cost $8.00 and if she buys 5 apples and 7 bananas it will cost $9.50.
Let the cost of one apple be =$ x
and the cost of one banana be =$y
According to the problem,
6x+5y=8.........(1) 5x+7y= 9.50............(2)
Equation (1)×5 - equation (2) × 6
30x +25 y-(30x+42y)= 40-57
⇔30x+25y -30x-42y= -17
⇔-17y =-17
⇔y=1
putting the value of y in equation (1)
6x+5×1=8
⇔6x = 8-5
⇔6x=3
⇔[tex]x =\frac{3}{6}[/tex]
⇔x = 0.50
Therefore the cost of each apple is $ 0.50 and each banana is $1.
What is the answer to 4•4+4(5-2)+7
Answer:
35
Step-by-step explanation:
4 x 4+4(5-2)+7
4 x 4+4(3)+7
16+12+7
28+7
35
Angie wants to purchase mulch for her tulip garden. The dimensions of her garden are 6 feet by 8 feet. If mulch is sold by the square yard, determine how many square yards of mulch she must purchase and how much it costs. The price of the mulch is 1.50 per square yard.
Answer:
72 dollars
Step-by-step explanation:
Doug makes a parallelogram-shape sign for his shop.
What is the area of this parallelogram?
Round to the nearest hundredth.
A. 5.32 ft²
B. 8.96 ft²
C. 14.28 ft²
D. 17.02 ft²
Parallelogram A B C D with side A D parallel to side B C and side A B is parallel to side D C. Point E is on side A B and is connected to point D by a dotted line. Point F is on side DC and connected to B by a dotted line. Angle D E B and angle F B E are right angles. A E and F C are 1.9 feet each. E B is 3.2 feet. F B is 2.8 feet.
Answer:
14.28 ft squared
Step-by-step explanation:
The formula for the area of a parallelogram is base times height.
Add 3.2 to 1.9 to get the base, and 2.8 is the height.
(3.2 + 1.9) x 2.8
5.1 x 2.8 = 14.28
14.28
Your answer should be a polynomial in standard form.(x-5)(x-4)
Answer: x^2-9x+20
Step-by-step explanation:
Answer:
x² - 9x + 20
Step-by-step explanation:
Given
(x - 5)(x - 4)
Each term in the second factor is multiplied by each factor in the first factor, that is
x(x - 4) - 5(x - 4 ) ← distribute both parenthesis
= x² - 4x - 5x + 20 ← collect like terms
= x² - 9x + 20
At ulrich 45% of the seventh graders ride the school bus to school if there are 360 seventh graders how many ride the bus
Basically 45% of 360.
45/100 * 360
45/360 * 100
16,200 / 100
162
Answer:
162
Step-by-step explanation:
There are 360 seventh graders and 45% of this 360 ride the school bus
All we need to do is find 45% of 360
we have;
45 / 100 X 360
= 162
Hence 162 seventh graders ride the school bus
What value of r makes the equation true
2(3r + 11) = (7r - 3r)
Answer:
r=-11
Step-by-step explanation:
Solve for r:
2 (3 r + 11) = 7 r - 3 r
7 r - 3 r = 4 r:
2 (3 r + 11) = 4 r
Expand out terms of the left hand side:
6 r + 22 = 4 r
Subtract 4 r from both sides:
(6 r - 4 r) + 22 = 4 r - 4 r
6 r - 4 r = 2 r:
2 r + 22 = 4 r - 4 r
4 r - 4 r = 0:
2 r + 22 = 0
Subtract 22 from both sides:
2 r + (22 - 22) = -22
22 - 22 = 0:
2 r = -22
Divide both sides of 2 r = -22 by 2:
(2 r)/2 = (-22)/2
2/2 = 1:
r = (-22)/2
2 | 1 | 1
| 2 | 2
- | 2 |
| | 2
| - | 2
| | 0:
Answer: r = -11
What is the algebraic expression for three more cookies than are in the jar
Answer:
x+3
Step-by-step explanation:
Since the number of cookies in the jar could be anything we'll use a variable to represent it, x. so three more than x is what?
Well, what's thee more than 5? 8, so you add 3.
So x+3
The following data represent the number of people aged 25 to 64 years covered by health insurance (private or government) in 2018. Approximate the mean and standard deviation for age.
Age: 25-34. 35-44. 45-54. 55-64
Number 22.1. 31.5. 37.7. 25.3
(Millions)
Answer:
Mean = 45.178
Standard Deviation = 10.33
Step-by-step explanation:
class m frequency, f m [tex]\times[/tex] f f [tex]\times[/tex] [tex]m^{2}[/tex]
25 - 34 29.5 22.1 651.95 19232.525
35 - 44 39.5 31.5 1244.25 49147.875
45 - 54 49.5 37.7 1866.15 92374.425
55 - 64 59.5 25.3 1505.35 89568.352
116.6 5267.7 250323.15
mean = [tex]\frac{5267.7}{116.6} = 45.178[/tex] standard deviation = [tex]\sqrt{\frac{250323.15 - \frac{(5267.7)^2}{116.6} }{(116.1 - 1)} } = 10.33[/tex]
The mean of a distribution is the average of the distribution ; it is the sum of observation divided by the count of the observation.
The mean age is 45.18The standard deviation is 10.29
Given that:
[tex]\begin{array}{ccccc}{Age}&25-34&35-44&45-45&55-64\\{Number}&22.1&31.5&37.7&25.3\end{array}[/tex]
First, we calculate midpoint of each class.
This is the sum of the class interval divided by 2.
So, we have:
[tex]x_1 = \frac{25+34}{2} = 29.5[/tex]
[tex]x_2 = \frac{35+44}{2} = 39.5[/tex]
And so on...
The table becomes:
[tex]\begin{array}{ccccc}{Age}&25-34&35-44&45-45&55-64&{Number(f)}&22.1&31.5&37.7&25.3&{x}&29.5&39.5&49.5&59.5\end{array}[/tex]
The mean of the distribution is:
[tex]\bar x = \frac{\sum fx}{\sum f}[/tex]
[tex]\bar x = \frac{29.5 \times 22.1 + 39.5 \times 31.5 + 49.5 \times 37.7 +59.5 \times 25.3}{22.1 + 31.5 + 37.7 +25.3}[/tex]
[tex]\bar x = \frac{5267.7}{116.6}[/tex]
[tex]\bar x = 45.18[/tex]
The standard deviation is calculated as follows:
[tex]\sigma = \sqrt{\frac{\sum f(x - \bar x)^2}{\sum f}}[/tex]
[tex]\sigma= \sqrt{\frac{(29.5 - 45.1)^2\times 22.1 + (39.5 - 45.1)^2 \times 31.5 + (49.5 - 45.1)^2 \times 37.7 + (59.5 - 45.1)^2 \times 25.3}{22.1 + 31.5 + 37.7 +25.3}}[/tex]
[tex]\sigma= \sqrt{\frac{12342.176}{116.6}}[/tex]
[tex]\sigma= \sqrt{105.85}[/tex]
[tex]\sigma= 10.29[/tex]
Hence:
The mean age is 45.18The standard deviation is 10.29Read more about mean and standard deviation at:
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What is 3/2+2/3 in fraction form
Answer:
[tex]\frac{13}{6}[/tex] or [tex]2\frac{1}{6}[/tex]
Step-by-step explanation:
Answer:
Step-by-step explanation:3/2 x 3= 9/6
2/3 x 2= 4/6
9/6 + 4/6 = 13/6
13/6=2 1/6
Name the property the equation illustrates.
-4x-1/4=1
Answer:
Distrbutive Property
Step-by-step explanation:
The equation -4x - 1/4 = 1 illustrates the addition property of equality.
Explanation:The property illustrated by the equation -4x - 1/4 = 1 is the addition property of equality. According to this property, if you add the same number to both sides of an equation, the equality of the equation is maintained. To isolate the variable, we need to get rid of the constant term (-1/4) on the left side of the equation by adding 1/4 to both sides. This simplifies the equation to -4x = 1 + 1/4. Then, divide both sides by -4 to solve for x.
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Question is in the picture, please help
Answer:
18.9 units to 1 decimal Place
Step-by-step explanation:
You need the diameter so do 3 X 2 = 6
Circumference is pi X diameter so
Do 6 X pi = 6pi or 18.9 to 1 dp
Michael is organizing a game of chance for his class. He asks each student to pick a ribbon from a bag and then put it back into the bag. The bag has 12 yellow ribbons, 7 red ribbons, and 3 blue ribbons. If Jane picks a ribbon, what is the probability that the ribbon will be yellow and what is the likelihood of this event?
A.
The probability is 1, and the event is certain.
B.
The probability is 3/11 , and the event is likely.
C.
The probability is 6/11, and the event is unlikely.
D.
The probability is 6/11 , and the event is neither likely nor unlikely.
Answer:
D. The probability is 6/11, and the event is neither likely nor unlikely
Step-by-step explanation:
54% = 6/11
Answer:
D
Step-by-step explanation:
Identify the Pythagorean triple. A) 3, 6, 12 B) 8, 16, 32 C) 12, 34, 30 D) 28, 45, 53
Answer:
D, 28, 45, 53
Step-by-step explanation:
A Pythagorean triple consists of three positive integers a, b, and c, such that a2 + b2 = c2. ... The name is derived from the Pythagorean theorem, stating that every right triangle has side lengths satisfying the formula a2 + b2 = c2; thus, Pythagorean triples describe the three integer side lengths of a right triangle.
Can someone please answer this question please answer it correctly please show work please
Answer:
3) A
4) D
Step-by-step explanation:
3)
Find any four points where y intersects with x on graph A
(-4 , 2) (-2 , 1) (4, - 2) (2 , -1)
Find the relationship(s) between x and y
(-4 , 2) (-2 , 1) (4, - 2) (2 , -1)
x = -2y x = -2y x = -2y x = -2y
It is shown that there is a direct proportional relationship between x and y, (x = -2y)
Repeat for graphs B, C, and D
Find any four points where y intersects with x on graph B
(3 , 2) (5 , 4) (-3 , -4) (-4 , -5)
Find the relationship(s) between x and y
x = 1.5y x = 1.25y x = 0.75y x = 0.8y
As you can see, there is not a constant direct proportional relationship between x and y. B is not the correct answer
Graph C
Find any four points where y intersects with x on graph C
(-5 , -3) (-1 , -3) (1 , -3) (5 , -3)
Find the relationship(s) between x and y
x = 1.67y x = 0.3y x = -0.3y x = -1.67y
Once again, there is not a constant direct proportional relationship between x and y, so C is not the correct answer
Graph D
Find any four points where y intersects with x on graph D
(2 , 1) (2 , 2) (2 , 3) (2 , 4)
Find the relationship(s) between x and y
x = 2y x = y x = 0.67y x = .5y
There is no direct constant relationship between x and y in this graph either.
Graph A is the only graph with a direct proportional relationship between x and y.
4) Allison wants to know whom seventh-grade students will vote for in the upcoming class president election.
A) Every member of the student council
Asking every member of the student council who they are voting for will not help Allison, because it is likely that there are multiple grades involved in the student council, when Allison just wants to know the opinions of seventh-graders. The student council also might have a different opinion than the general seventh-graders.
B) All 12 girls in Allison's homeroom class
It is likely that the girls at Allison's school have a very different opinion than the boys at her school, making this a bad data set.
C) Every tenth student who arrives at school on Monday
Allison only want to know the opinions of seventh-graders, not every tenth student at her school, which would consist of more than seventh-graders.
D) Every tenth name on a roster of all of the seventh-graders at the school
This data set would be the most useful to Allison because it is multi-gender, only seventh-grade, and consists of the general seventh-graders (not part of a special group).
D
Hope this helps :)
Yesenia is 168 miles from Chicago. If she needs to be in Chicago in 3 hours, at what rate does she need to drive?
Answer:
56 per hour
Step-by-step explanation:
168 divided by 3
(-4.7)^0 Does anyone have the answers to the Exponents and Exponential Functions Unit test
The answer to (-4.7)⁰ is 1
How to solve the exponent
We have a rule of thumb that Any non-zero number raised to the power of 0 is equal to 1.
This is a fundamental rule in mathematics and is consistent across all real numbers.
So, when you have (-4.7)⁰, you're essentially raising the number -4.7 to the power of 0.
Even though -4.7 is negative, it's still a non-zero number. Therefore, according to the rule, (-4.7)⁰ equals 1.
Mike correctly found the slope and y-intercept of the line passing through the points (–5, –2) and (3, 14) as follows. m = StartFraction 14 minus (negative 2) Over 3 minus (negative 5) EndFraction = StartFraction 16 Over 8 EndFraction = 2. y = 2 x + b. Negative 2 = 2 (negative 5) + b. Negative 2 + 10 = negative 10 + b + 10. b = 8. What is the equation of the line in slope-intercept form?
Solution:
Mike correctly found the slope and y-intercept of the line passing through the points (–5, –2) and (3, 14)
To find: Equation of the line in slope - intercept form
The equation of line in slope intercept form is:y = mx + c ------ eqn 1
Where, "m" is the slope of line and "c" is the y intercept
Find the slope of line[tex]m = \frac{y_2-y_1}{x_2-x_1}[/tex]
From given,
[tex](x_1, y_1) = (-5, -2)\\\\(x_2, y_2) = (3, 14)[/tex]
Therefore,
[tex]m = \frac{14-(-2)}{3-(-5)}\\\\m = \frac{14+2}{3+5}\\\\m = \frac{16}{8}\\\\m = 2[/tex]
Thus slope of line is 2
Find the y intercept:Substitute m = 2 and (x, y) = (-5, -2) in eqn 1
-2 = 2(-5) + c
-2 = -10 + c
c = -2 + 10
c = 8
Thus the equation of line is:Substitute c = 8 and m = 2 in eqn 1
y = 2x + 8
Thus the equation of line is found
Answer:
the answer is y=2x+8
Step-by-step explanation:
Steve is buying a new car at the dealership. The price tag on the car reads $17,000.00. However, the salesman offers to sell him the car for 15% off the price tag. If sales tax is 8%, how much does the car cost after tax?
Answer:
$15606000
Step-by-step explanation:
17,000,000*.15 (15% off)
$ 2550000 off
17,000,000
-2550000
$14450000 (15% off)
*1.08 (sales tax)
$15606000
sorry if it is incorrect
Answer:
if the sale tax comes first before the 15% off,
Sales Tax: 1360
so the price became 18360
Price Off: 2754
18360 - 2754 = 15606
if the sale tax comes after the 15% off,
Price Off: 2550
17000-2550 = 14450
eight percent of 14450 is 1156
14450+1156 = 15606
anyways, the price is the same no matter if the sales tax comes first or the discount comes first
If 4/5x is -16, what is the value of -2x-15?
Answer:
25
Step-by-step explanation:
Given data
(4/5)x = -16
x = -16 (5/4)
x = -20
Now,
-2x-15 = ?
putting the value of x
-2(-20)-15 = 25
Substitute x+3y=4
2x-6y=8
Answer:
x=4, y=0. (4, 0).
Step-by-step explanation:
x+3y=4
2x-6y=8
------------
x=4-3y
2(4-3y)-6y=8
8-6y-6y=8
8-12y=8
12y=8-8
12y=0
y=0/12
y=0
x+3(0)=4
x+0=4
x=4-0
x=4
What is 4/9 as a decimal
Answer:
0.4 and alot of other 4's
Step-by-step explanation:
Answer:
.44444444 repeating
Any number less than 9 over 9 will be itself forever after a decimal point
Step-by-step explanation:
Solve the inequality for #5 Please! (Ignore my horrible handwriting im a lefty lol)
Question:
Solve the inequality:
5.1 – r ≥ 5
Answer:
r ≤ 0.1
Solution:
Step 1: Given inequality:
5.1 – r ≥ 5
Step 2: Subtract 5.1 from both sides of the inequality
⇒ 5.1 – r – 5.1 ≥ 5 – 5.1
⇒ – r ≥ – 0.1
Step 3: To reverse the inequality, multiply both sides by –1.
⇒ (–r) × (–1) ≥ – 0.1 × (–1)
Negative × Negative = Positive
⇒ r ≤ 0.1
Hence the solution to the given inequality is r ≤ 0.1.
Solve for x in the equation x squared minus 8 x + 41 = 0.
x = negative 4 plus-or-minus StartRoot 37 EndRoot i
x = –4 ± 5i
x = 4 plus-or-minus StartRoot 37 EndRoot i
x = 4 ± 5i
Answer:
Option 4 - The solution of x is [tex]x=4\pm5i[/tex].
Step-by-step explanation:
Given : Equation [tex]x^2-8x+41=0[/tex]
To find : Solve for x in the equation ?
Solution :
Solving a quadratic equation [tex]ax^2+bx+c=0[/tex] solution is given by, [tex]x=\frac{-b\pm\sqrt{b^2-4ac}}{2a}[/tex]
Here, a=1, b=-8 and c=41
[tex]x=\frac{-(-8)\pm\sqrt{(-8)^2-4(1)(41)}}{2(1)}[/tex]
[tex]x=\frac{8\pm\sqrt{-100}}{2}[/tex]
[tex]x=\frac{8\pm10i}{2}[/tex]
[tex]x=4\pm5i[/tex]
Therefore, the solution of x is [tex]x=4\pm5i[/tex].
Answer:
D
Step-by-step explanation:
trust meh
Which of the following equations would result in destructive interference with the wave y= -2sin(2pi/3 x)?
A.) y= -2sin(2pi/3 x)
B.) y=2sin(2pi/3 x)
C.) y= -2sin 3pi/2 x)
D.) y=2sin (3pi/2 x)
Answer:
B: 0.449 secs
A, C, and D: y=2sin(pi/5x), y=2sin(pi/10x), y=2sin(10pi x)
B: y=2sin(2pi/3x)
Answer:
B. y=2sin(2pi/3x)
Step-by-step explanation:
If a circle has a radius of 12 feet, whats its circumference most nearly?
Answer:24π feets
Step-by-step explanation:
Since the circumference C= 2πr and r=12
C= 2x12xπ
C=24π
mean of 10 observations is 2.5.if one observation 2.5 is deleted,determine the new mean
Answer: 2.5
Step-by-step explanation:
Mean m = Sum s ÷ no of observation n
m = 2.5, n = 10
2.5 = s ÷ 10
s = 10 × 2.5
s = 25.
If one observation 2.5 is deleted from sum 25, new sum equals 25 — 2.5 = 22.5.
The no of observation becomes 9.
Therefore new mean = 22.5 ÷ 9
m = 2.5.
How this helps?
The question is about calculation of a new mean after deleting an observation from a data group. Despite deleting an observation of 2.5 from a data set with a mean of 2.5, the mean remains the same.
Explanation:The subject matter is about computing the mean of a data set after a data point has been deleted. Starting off, we know we initially have 10 observations with a mean of 2.5. Therefore, the total sum of the observations can be calculated by multiplying the mean with the number of data points (2.5 * 10) to get 25.
Now, if we are to delete one observation worth 2.5, we have to subtract it from our total of 25 to get a new sum of 22.5. Then, we also decrease the number of data points by 1, which leaves us with 9 data points.
Finally, the new mean can be calculated by dividing the new sum by the new number of data points. In this case, it would be 22.5 divided by 9, which equals 2.5. Thus, the mean remains the same even after deleting the observation of 2.5.
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