Recall the triple angle identity for sine:
sin(3x) = 3 sin(x) cos²(x) - sin³(x)
… = 3 sin(x) (1 - sin²(x)) - sin³(x)
… = 3 sin(x) - 4 sin³(x)
So the equation is equivalent to
sin²(x) - 2 sin(x) + 2 = sin²(3x)
sin²(x) - 2 sin(x) + 2 = (3 sin(x) - 4 sin³(x))²
sin²(x) - 2 sin(x) + 2 = 9 sin²(x) - 24 sin⁴(x) + 16 sin⁶(x)
16 sin⁶(x) - 24 sin⁴(x) + 8 sin²(x) + 2 sin(x) - 2 = 0
or, by replacing y = sin(x),
16y ⁶ - 24y ⁴ + 8y ² + 2y - 2 = 0
By inspection, y = 1 is a root, since
16 - 24 + 8 + 2 - 2 = 0
so we get one family of solutions,
sin(x) = 1 ===> x = arcsin(1) + 2nπ = π/2 + 2nπ
Now suppose y ≠ 1. Dividing both sides of the sextic equation by y - 1 then gives
(16y ⁶ - 24y ⁴ + 8y ² + 2y - 2)/(y - 1) = 0
16y ⁵ + 16y ⁴ - 8y ³ - 8y ² + 2 = 0
Unfortunately, the remaining roots can only be obtained through numerical methods. Using a calculator, you would find
y ≈ -1.127
y ≈ -0.475 ± 0.368i
y ≈ 0.539 ± 0.127i
But if you're solving over the reals, you can ignore these solutions. Remember that sin(x) is bounded between -1 and 1, and its value goes outside this interval only when x is complex.
2% of what amount is $200
Answer:
200×100÷2=10000
Step-by-step explanation:
i think that is right
Select all of the points which satisfy the linear inequality below.
-2x + 3y > 4
Answer:
x < − 2 + 3 y/ 2 linear inequality for x
y > 4/3 + 2x/3 linear inequality for y
Step-by-step explanation:
The point which satisfies the linear inequality is (-1,5) so option (C) is correct.
What is inequality?A difference between two values indicates whether one is smaller, larger, or basically not similar to the other.
In other words, inequality is just the opposite of equality for example 2 =2 then it is equal but if I say 3 =6 then it is wrong the correct expression is 3 < 6.
As per the given inequality,
-2x + 3y > 4
Put (-1,5) into the above inequality,
-2(-1) + 3(5) > 4
2 + 15 > 4
17 > 4 (True) thus (-1,5) will satisfy the inequality.
The rest points are not satisfying the inequality.
Hence " The point which satisfies the linear inequality is (-1,5)".
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The complete question is attached below,
suppose river has new treats that are 10 calories each. how many of the new treats can she have and remain in her calorie range?
She can have at least 3 and at most 6 dog treats in a day
River can have - n = 10/y number of treats to remain in her calorie range, where the calorie range is [y] calories.
What is the general equation of a Straight line?The general equation of a straight line is -
[y] = [m]x + [c]
where -
[m] is slope of line which tells the unit rate of change of [y] with respect to [x].
[c] is the y - intercept i.e. the point where the graph cuts the [y] axis.
The equation of a straight line can be also written as -
Ax + By + C = 0
By = - Ax - C
y = (- A/B)x - (C/A)
We have the following problem -
River has new treats that are 10 calories each. how many of the new treats can she have and remain in her calorie range.
Assume the calorie range to be [y] calories and assume she can have [n] number of treats. We can write the linear function as -
10n = y
OR
y = 10n
This this the equation of the straight line. She can have -
n = 10/y
number of treats to remain in her calorie range.
Therefore, River can have - n = 10/y number of treats to remain in her calorie range.
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1. Sam was driving to visit his friend in Texas. He will have to drive 550 miles to get to his friends home.
Sam drives 65 miles per hour and has already driven 325 miles. Write an equation to find how much
longer in hours Sam will need to drive to reach his destination.
Answer: 65x + 325y = 550
help fast
2
- 2
2
4
-2
4
N
Which equation could be solved using the graph above?
Ox2 + x - 2 = 0
Ox2 + 2x + 1 = 0
Ox2-1=0
Ox2 - 2x + 1 = 0
natilie walked 3/5 mile in 1/2 hour how fast did she walk
Answer:
1.2 mph
Step-by-step explanation:
3/5 = 0.6
Multiply 0.6 by 2 to find the distance gone in a whole hour
0.6*2 = 1.2
Which is greater? 2x²-2x + 6 - (-3x) or -(3-2x²) + 5 + 2x?
Answer:2x²-2x + 6 - (-3x)
Step-by-step explanation:
250000 divided by 100
Answer:
250,000÷100 is equal to 2,500
Answer:
2500 is the answer
hope it helps u
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A girl starts from Point A and walks to 285m to B on a bearing of 078°. She then walks due south to a point C which is 307m from A. What is the bearing of A from C and what is IBCI
Answer:
bearing of A from C is - 65.24°
the distance |BC| is 187.84 m
Step-by-step explanation:
given data
girl walks AB = 285 m (side c)
bearing angle B = 78°
girl walks AC = 307 m (side a)
solution
we use here the Cosine Law for getting side b that is
ac² = ab² + bc² - 2 × ab × cos(B) ...............1
307² = 285² + x² - 2 × 285 cos(78)
x = 187.84 m
and
now we get here angle θ , the bearing from A to C get by law of sines
sin (θ) =
sin (θ) = 0.5985
θ = 36.76°
and as we get here angle BAC that is
angle BCA = 180 - ( 36.76° + 78° )
angle BCA = 65.24°
and here negative bearing of A from C so - 65.24°
Step-by-step explanation:
Are the points O, M, and U collinear?
.
Choose 1 answer:
A. Yes
B. No
Answer:
It would be B because the letters are not in a straight line.
Step-by-step explanation:
Have a good day!!
The population of Miami-Dade County in Florida in 2010 was 2, 496, 435 and the land area was 1, 898 square miles. The population increased by 7.9% in 2015. Part 1: To the nearest whole number, find the population density, in people per square mile, for Miami- Dade County in 2015
The population density of the city which is the ratio of population size and the land area is 1419 people per square mile.
The initial population = 2496435The percentage increase in population = 7.9%Land Area = 1898 mi²The population of the city on 2015 can be calculated thus :
(1 + percent increase) × initial population (1 + 0.079) × 2496435 = 2693653.365 peopleThe population of a city is calculated using the relation :
Population density = Population / Land Area Population density = 2693653.365/1898 Population density = 1419.21Therefore, the population density is 1419 people per square mile.
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what is the midpoint of (5,-3) and (-1,-7) .
Slope intercept form
Answer: -2/6
Step-by-step explanation:
I really need help from you brainy users.
I don't expect you to write out a whole entire paragraph but give me some tips regarding moving out of my parents house before I turn 18.
Answer:
well, i guess ill tell you the way i did it. i rented a storage space, rented my apartment and bought all of my furniture before moving out of my parents house so i was totally ready to move out. then, when the day came to move out, i planned the entire day beforehand, planned where my furniture would go and did all the planning before actually leaving. anything else you wanna know? ill answer any questions
Step-by-step explanation:
In a collection of nickels, dimes, andquarters the value of the collection is$12.85. If the number of nickels is 6 lessthan twice the number of dimes and thenumber of quarters is 5 more than twicethe number of dimes, how many quartersare there?
pls answer the problem below
Answer:
9 hours
Step-by-step explanation:
just do 441/7 and then you get 63 miles per hour
and then you divide 567/63 and you get 9!
easy peasy hope I helped
NEED HELP! [URGENT] Submit the worksheet with your constructions to your teacher to be graded.
give me more details. that's a bit confusing
a. Draw a scaled copy of Polygon A with scale factor 1/2. b. Draw a scaled copy of Polygon B with scale factor 2. c. Draw a scaled factor of Polygon C with scale factor 1/4.
Given the f(x) =8(3-x) what is the value of f(-1)
Answer:
32 alternative forms: 2⁵
Step-by-step explanation:
Answer:
f(-1)=32
Step-by-step explanation:
f(x) =8(3-x)
f(-1)=8(3-(-1))
f(-1)=8(3+1
f(-1)=8*4
f(-1)=32
What is the reason for each step in the solution of the equation?
12−5x+6x=4
Drag and drop the reasons into the boxes to correctly complete the table.
12−5x+6x=4
12+x=4
x=−8
The correct reason for each step in the solution of the equation would be as follows:
12 − 5x + 6x = 4 - Given
12 + x = 4 - Combine Like Terms
x= −8 - Subtraction Property of Equality
Hope this answers the question. Have a nice day.
i need help with my math homework
Answer:
x=3
Step-by-step explanation:
g(x)=2x+1,
When g(x)=7
g(x)=2x+1,
7=2x+1
7-1=2x
x=6/2
x=3
This right circular cone has a volume of 75 cubic inches. The height of the cone is 9 inches. What is the length of the radius? A. 4 in. B. 5 in. C. 6 in. D. 8 in.
Answer:
B. 5 in.
Step-by-step explanation:
Answer:
either 8 or 4
Step-by-step explanation:
because it can't be 6 or 5
HELP PLEASE !!!!!!!!!!
Answer:
m < 2 = 38°
Step-by-step explanation:
The exterior angle with the measure of < 38° corresponds to m < 2 (as they are corresponding angles that lie on the same side of the transversal). Therefore, m < 2 = 38°.
77 x(3x+8)=107+88x
What is the answer
Answer:48
Step-by-step explanation: pretty easy
Determine whether the following statement is always, sometimes, or never true. Explain your reasoning.
Every complex number has both a real part and an imaginary part.
The statement "Every complex number has both a real part and an imaginary part" is always true.
Every complex number is defined as a number of the form "a + bi," where "a" represents the real part and "bi" represents the imaginary part.
The real part "a" is a real number, and the imaginary part "bi" is a multiple of the imaginary unit "i," where "i" is defined as the square root of -1.
Even if a complex number has a zero imaginary part, it is still considered to have both a real and an imaginary part.
In that case, "a" represents the real part, and the imaginary part "bi" becomes zero.
For example, the complex number 3 + 0i is a real number (3) and can be represented as having a real part of 3 and an imaginary part of 0i.
In conclusion, every complex number by definition consists of both a real and an imaginary part, making the statement "Every complex number has both a real part and an imaginary part" always true.
To learn more about the complex number;
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Adrian said that g(x) = |x + 4| is translated up 4 units from the parent function f(x) = |x|. Do you agree or disagree? Why?
Answer:
Below.
Step-by-step explanation:
Disagree.
The + 4 translates the parent function 4 units to the left ( not up).
PLEASE HELP MEEE ASAP
Step-by-step explanation:
first of all, g(x) has only negative functional result values (y) except for 0.
and a square is always positive.
so, the only possible right answers are the ones that include a minus ("-") sign.
the graph shows us that g(x) goes through the points (1, -3) and (-1, -3).
so, which equation turns an x = 1 into an y = -3 ?
therefore, the right answer must be D.
g(x) = -3x²
it works for both points :
-3 = -3×1² = -3 correct
-3 = -3×(-1)² = -3×1 = -3 correct
x + 3 + -3 = -6 please help me on this
Answer:-6
Step-by-step explanation:
X=-6
3 and -3 cancel out.
Answer:
-6
Step-by-step explanation:
The positives and negatives added cancel eachother out.
-6+3=-3
-3+-3=-6
someone please help. Giving brainliest!!
Answer:
A. None of these
Step-by-step explanation:
The largest perfect square under 100 would actually be 81 (9 x 9)
90 and 99 are not perfect squares and 64 is smaller than 81.
Answer:
a. none of these
Step-by-step explanation:
10 * 10 = 100, so 100 is a perfect square.
The next smaller perfect square is 9 * 9 = 81, but 81 is not in the choices.
Answer: a. none of these
Somebody help me please
Answer:
Step-by-step explanation: