Recall some identities:
sin²(x) + cos²(x) = 1
cos(2x) = cos²(x) - sin²(x)
and from the first identity, you can establish that
cos(2x) = 2 cos²(x) - 1
So, rewrite the integrand as much as possible in terms of cos(x) :
[tex]\sin^3(x) \cos(2x) = \sin(x) \sin^2(x) (\cos^2(x)-\sin^2(x)) \\\\ \sin^3(x) \cos(2x) = \sin(x) (1-\cos^2(x)) (2\cos^2(x) - 1)[/tex]
Then in the integral, substitute u = cos(x) and du = -sin(x) dx. With some rewriting, you end up with
[tex]\displaystyle \int_1^2 \sin^3(x) \cos(2x) \,\mathrm dx = -\int_{x=1}^{x=2} (1-\cos^2(x))(2\cos^2(x)-1) (-\sin(x)\,\mathrm dx) \\\\ = -\int_{u=\cos(1)}^{u=\cos(2)} (1-u^2)(2u^2-1)\,\mathrm du \\\\ = \int_{\cos(1)}^{\cos(2)} (1-3u^2+2u^4) \,\mathrm du \\\\ = \left(u-u^3+\frac25u^5\right)\bigg|_{\cos(1)}^{\cos(2)} \\\\ = \left(\cos(2)-\cos^3(2)+\frac25\cos^5(2)\right) - \left(\cos(1)-\cos^3(1)+\frac25\cos^5(1)\right) \\\\ = \boxed{\cos(2) - \cos^3(2) + \frac 25 \cos^5(2) - \cos(1) + \cos^3(1) - \frac25 \cos^5(1)}[/tex]
19. The product of two numbers is 120 and the sum of their squares is 289. The sum of the number is (a) 20 (b) 23 (c) 169 (d) None of these
Answer:
B
Step-by-step explanation:
Let the two numbers be a and b.
The product of the two is 120 and their sum of their squares is 289. In other words:
[tex]\displaystyle ab = 120 \text{ and } a^2 + b^2 = 289[/tex]
And we want to find the sum of the two numbers.
We can use the perfect square trinomial. Recall that:
[tex]a^2 + 2ab + b^2 = (a+b)^2[/tex]
From the first equation, multiply by two:
[tex]2ab = 240[/tex]
Add the two equations together:
[tex](a^2+b^2)+(2ab) = (289)+(240)[/tex]
Simplify and rewrite:
[tex]a^2+2ab+b^2 = 529[/tex]
Factor using the perfect square trinomial pattern:
[tex](a+b)^2 = 529[/tex]
And take the square root of both sides:
[tex]\displaystyle a + b = \pm\sqrt{529} = \pm23[/tex]
Hence, the sum of the two numbers is 23 or -23.
In conclusion, our answer is B.
HELP PLS!! PLS BRO LOLOLOLOL
Answer and Step-by-step explanation:
An easy way of doing this is to first square root the numerator and the denominator.
[tex]\sqrt{\frac{36}{81} } =\frac{\sqrt{36}}{\sqrt{81} } =[/tex] ±[tex]\frac{6}{9} =[/tex] ±[tex]\frac{2}{3}[/tex] <- This is the answer.
#teamtrees #PAW (Plant And Water)
I hope this helps!
PLEASE HELP
-10+7/4p=-38
Show your work in details if you can, I have a hard time understanding this.
Answer:
Here is your answer . Hope it helps you
Answer:
p = -49
Step-by-step explanation:
-10 + 7/4p = -38
+7/4 p = -38 + 10. ..........(-10 taken to other side)
+7/4 p = -28
p = -28 × 7/4 ........( now taken 7/4 to other side)
p = -196/4 .......(here we have multiplied . (-28×7 )
p = -196/4
p = -49
so p = -49
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Write the number in standard form.
ninety-two billion, seventy thousand,
one hundred eight
92.000.070.108
Step-by-step explanation:
Thousand has 3 zeros, a million has 6 and a billion has 9.
So for ninety-two billion, you take 92 and add 9 zeros behind it because a billion has 9 zeros
(92.000.000.000)
PLZ HELPPPPPPPPPPPPPPP
You have 24 quarts of brown stock you need .75 cups to make one serving of braised short ribs. How many can you make?
Answer:128
Step-by-step explanation:
1 quart= 4 cups
You have 24 quarts
24x4=96
96/0.75=128
A coin is flipped two times and the number of heads is counted. Which of the following is a valid assignment
of probabilities for the number of heads observed in two flips?
8.
Number of heads
Probability
0
1/10
1
5/10
2
4/10
b.
Number of heads
0
1
2
Probability
1/4
2/4
1/4
C.
0
1
Number of heads
Probability
2
1/3
1/3
1/3
d. All answer options are correct
Answer:
5/10, 1, 2/4,
Step-by-step explanation:
- Three times the quantity two decreased by x is nine.
Step-by-step explanation:
3(2-x)=9
this is the answer to the statement
The expression three times the quantity two decreased by x is nine is - 1
A quantity two decreased by x can be expressed thus :
(2 - x)Multiplying the quantity by 3 :
3(2 - x)The expression can be written Mathematically as thus : 3(2 - x) = 9
The expression can be expanded further as :
3(2 - x) = 9
6 - 3x = 9
Collect like terms
-3x = 9 - 6
-3x = 3
x = (3 ÷ -3)
x = - 1
Therefore, the expression equates to - 1
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Will mark as brainliest!
Answer:
[tex]f=12-\frac{t}{20}[/tex]
Step-by-step explanation:
[tex]-7+1[3+8(2^2-8)][/tex]-7+1[3+8(2^2-8)]
Answer:
-36
Step-by-step explanation:
-7+1[3-32]
= -7-29
= -36
PLS HELP :( i’ve been trying to understand this but i cant. please explain this if you know how to do it pleaseee
Answer:
[tex]\huge\boxed{\sf x = 17}[/tex]
Step-by-step explanation:
Since m is a straight line, its measure is 180 degrees.
So,
2x + 5 + 8x + 5 = 180
2x + 8x + 5 + 5 = 180
10x + 10 = 180
10x = 180 - 10
10x = 170
Divide both sides by 10
x = 17
[tex]\rule[225]{225}{2}[/tex]
Hope this helped!
~AH1807Peace!Chase thought that only 23 people would come to the banquet, but 48 people ended up coming. What was his percent error?
Answer:
The answer is 52%
ssmoga membantu
What is this number sentence equal to?
If you could help, thank you so much!
Answer:
18.63
Step-by-step explanation:
[tex] \frac{6}{10} = 0.6 \\ \frac{3}{100} = 0.03[/tex]
[tex]10 + 8 + 0.6 + 0.03 = 18.63[/tex]
solve the equation 6 = 5 x -2
Answer:
8/5 = x
Step-by-step explanation:
6 = 5x - 2
~Add 2 to both sides
8 = 5x
~Divide 5 to both sides
8/5 = x
Best of Luck!
Which sentence describes a two-column proof?
A.
The left column contains a series of definitions, and the right column states theorems.
B.
The left column contains a series of deductions, and the right column states reasons.
C.
The left column contains a series of reasons, and the right column states deductions.
D.
The left column contains a series of postulates, and the right column states definitions.
Order the numbers from least to greatest
Answer:
B
Step-by-step explanation:
It just is
One number exceeds another by 12 and their sum is 44 find the two numbers you
Answer:
Step-by-step explanation:
x - first
x + 12 - second
x + (x + 12) = 44
2x=44-12
x=16
first number is 16
second number is 16+12=28
two numbers are 16 and 28
16 and 28 are the required two numbers when one number exceeds another by 12 and their sum is 44
One number exceeds another by 12 and their sum is 44
We need to find the two numbers
What is equation?Two expressions with equal sign is called equation.
Let x be the first number
x + 12 be the second number
x + (x + 12) = 44
2x=44-12
x=16
Therefore first number is 16
second number is 16+12=28
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In ΔPQR, sin P = 0.2, sin R = 0.2 and r = 12. Find the length of p.
9514 1404 393
Answer:
p = 12
Step-by-step explanation:
The sines of the opposite angles are the same, so the lengths of the sides are the same.
p = 12
_____
The law of sines tells you ...
p/sin(P) = r/sin(R)
p/0.2 = 12/0.2 . . . . . . substitute given values
Multiply by 0.2 to get ...
p(0.2/0.2) = 12(0.2/0.2)
p = 12
Which describes the relationship between the distance and the difference?
Answer:
Which describes the relationship between the distance and the difference? The distance is always the opposite of the difference. ... The distance is the absolute value of the difference.
How is the expression 3 x 1/10 written as a decimal
A)3
B).3
C).03
D).003
Answer:
B)
Step-by-step explanation:
3.0 / .1 = 0.3
Answer:
3×1/10=0.3 in decimals.
Step-by-step explanation:
3×1÷10
3÷10
0.3
Help meeeeeeeeeeeeeeeee
Answer:
x-3
Step-by-step explanation:
the inverse is x-3
this is the answer
A spinner is divided into five colored sections that are not of equal size: red, blue, green, yellow, and purple. The spinner is spun several times, and the results are recorded below:
Spinner Results
Color Frequency
Red 8
Blue 8
Green 3
Yellow 7
Purple 16
Based on these results, express the probability that the next spin will land on red or green or purple as a decimal to the nearest hundredth.
Step-by-step explanation:
The probability is the number of desired observations (red, green, blue) divided by total number of observed outcomes.
twice a number added to the product of 14 and 2 is 30. find the number
Answer:
1
Step-by-step explanation:
2x + (14 • 2) = 30
2x + 28 = 30
2x = 30 - 28
2x = 2
x = 2/2
x = 1
Hope this helps you. Do mark me as brainliest.
Determine the range of the following graph:
y
12
11
10
9
8
7
2
1
-12-11-10-9-8-7-6-5-4-3-2-11
1 2 3 4 5 6 7 8 9 10 11 12
- 2
-3
-4
-8
-9
-10
-11
-12
Answer:
r = (-4,9]
Step-by-step explanation:
hope this will help
Answer:
-4<y≤2
Step-by-step explanation:
The range of a graph is the range of the y values.
HELP ME PLEASEEEEEEEE
Answer:
c
Step-by-step explanation:
i search it up
How many different sets of answers are possible on a true-false test that has 11
questions?
12. A jet stream can move a particle of air
11 miles in 5 minutes. At this rate, how far
will the particle move in 1 hour?
(5 minutes = 1 hour) (Example 1)
Answer:
132 particles
Step-by-step explanation:
5 minutes = 11 miles
1 hour (60 minutes)
11*x = 11 x 60 = 660
11x = 660/5=132
The distance of a jet stream can move a particle of air 132 miles in 1 hour.
Here,
A jet stream can move a particle of air 11 miles in 5 minutes.
We have to find, The distance of particles move in 1 hour.
What is distance?
The length along a line or line segment between two points on the line or line segment.
Now,
A jet stream can move a particle of air 11 miles in 5 minutes.
5 minutes = 11 miles
Then, for 1 minute a jet stream can move a particle of air = 11 / 5 miles
Hence, In 1 hour ( 60 minutes) a jet stream can move a particle of air;
⇒ [tex]\frac{11}{5} * 60 = 132[/tex]
So, The distance of a jet stream can move a particle of air 132 miles in 1 hour.
Learn more about the distance visit:
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multiply 18/-57 and 7/6
Answer:
-7/19
Step-by-step explanation:
First simplify the fractions. 18/-57= -6/19.
-6/19x7/6=-42/114
-42/114=-7/19
I hope this helps! I also put it into an online calculator to make sure it is right and it is!
Consider the sequence of numbers 50,48,46,44….
(A) What are the next three numbers in this sequence?
(B) Using n to represent the position of a number in this sequence, write an expression that allows you to calculate the nth number.
(C) use your expression to find the 100th number in this sequence
Step-by-step explanation:
(A) 42,40
(B) T(n) = 50 + (n-1)(-2)
= 50+2-2n
= 52-2n
(C) T(100) = 52 - 2(100)
= 52- 200
= - 148
Please Help Me I Will Give Brainliest
Answer:
Top left option
Step-by-step explanation:
There are a total of 10 boxes so
x + y = 10
The question states that x = 20 pound boxes and y = 30 pound boxes
20x + 30y = 280
