Answers
a. The equivalent expression for cos(4x) in terms of just x is 8cos⁴x - 8cos²x + 1
b. The equivalent expression for sin(4x) in terms of just x is 4sinxcos³x - 2sinxcosx
a. The equivalent trigonometric expression for cos(4x)To find the trigonometric expression for cos(4x)
cos(4x) = cos[2(2x)]
Now from trigonometric identities
cos2Ф = 2cos²Ф - 1
Let 2x = Ф.
So, cos(4x) = 2cos²2x - 1
Also, cos2x = 2cos²x - 1
So, substituting cos2x into cos4x, we have
cos(4x) = 2cos²2x - 1
cos(4x) = 2[2cos²x - 1]² - 1
cos(4x) = 2[4cos⁴x - 4cos²x + 1] - 1
cos(4x) = 8cos⁴x - 8cos²x + 2 - 1
cos(4x) = 8cos⁴x - 8cos²x + 1
So, the equivalent expression for cos(4x) in terms of just x is 8cos⁴x - 8cos²x + 1
b. The equivalent trigonometric expression for sin(4x)To find the expression for sin(4x)
sin(4x) = sin2(2x)
From trigonometric identities sin2Ф = 2sinФcosФ
Let 2x = Ф
So, sin(4x) = sin2(2x)
= 2sin2xcos2x
Since
sin2x = 2sinxcosx and cos2x = 2cos²x - 1Substituting these into the equation, we have
sin(4x) = 2sin2xcos2x
sin(4x) = (2sinxcosx)(2cos²x - 1)
sin(4x) = 4sinxcos³x - 2sinxcosx
So, the equivalent expression for sin(4x) in terms of just x is 4sinxcos³x - 2sinxcosx
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Related Questions
Which statement below correctly depicts the expected value of a fair game?
E(X) = 0
E(X) = 1
E(X) > 0
E(X) >1
Answers
Answer:
e(x)=0 will be your answer
Step-by-step explanation:
First of all it is important to know about the fair game. A game is said to be fair if it has equal chances of winning for both the parties.
In Statistical terms, this can be said as expected value =0
If E(X) = 1 where X is from the player, then organizer of the game would have more chances of losing.
Similarly if E(X)>0where X is from the player, then organizer of the game would have more chances of losing.
Also if E(X)>1, where X is from the player, then organizer of the game would have more chances of losing.
Hence a fair game is one where
Answer: A. E(X) = 0
Step-by-step explanation:
Find the midpoints of A(-3,-9) and B (-8,-2) graphically
Answers
The midpoints of A(-3,-9) and B (-8,-2) is the coordinate points (-5.5, -3.5)
Midpoint of coordinatesCoordinate points are written in the form (x, y). The midpoint of two points (x1, y1) and (x2, y2) is expressed as:
M(x, y) ={(x1+x2)/2, (y1+y2)/2}
Given the following coordinates A(-3,-9) and B (-8,-2)
M(x, y) = {(-3+(-8))/2, (-9-(-2))/2}
M(x, y) = {(-3-8)/2, (-9+2)/2}
M(x, y) = (-11/2, -7/2)
M(x, y) = (-5.5, -3.5)
Hence the midpoints of A(-3,-9) and B (-8,-2) is the coordinate points (-5.5, -3.5)
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Find the volume of the prism. Round to the nearest tenth if necessary.
Answers
Answer:
14.7 (Because you asked for no units)
Step-by-step explanation:
The volume of this triangular prism is 1/2 BHL
0.5 * 3.5 * 2 * 4.2 = 14.7 ft^3
The range, a measure of variability, Group of answer choices is the difference between the largest (L) and smallest (S) value in a list of scores is the most informative when used to describe data sets without outliers includes only two values in its computation, regardless of the number of scores in a distribution all of the above
Answers
The correct answer is all of the above.
Why is the range the most convenient measure of variability?
Your data's spread from the lowest to the greatest value in the distribution is indicated by the range. The calculation of this variability index is the simplest. Simply subtract the lowest value from the highest value in the data set to determine the range.How do you find the range?
The difference between the lowest and highest values in a list or set is known as the range. Put all the numbers in order before determining the range. The lowest number should then be subtracted from the highest. The range of the list is provided in the response.Learn more about range
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[tex]y = 6x + 3 \\ y = 6x - 4[/tex]
state whether each pair of lines are parallel or perpendicular
pls help
Answers
Answer:
The lines are parallel, this is because they both have the same slope/gradient of 6.
Answer:
parallel lines
Step-by-step explanation:
the equation of a line in slope- intercept form is
y = mx + c ( m is the slope and c the y- intercept )
y = 6x + 3 ← is in slope- intercept form
with slope m = 6
y = 6x - 4 ← is in slope- intercept form
with slope m = 6
• Parallel lines have equal slopes
then the 2 given lines are parallel
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Warm Up
What is the exact distance between the points (4,-2) and (-6,4)?
Select the correct answer.
O 4√10
O2√34
O2√26
O 10√6
Submit
Answers
The distance between two points (4,-2) and (-6,4) exists [tex]2\sqrt{34}.[/tex]
How to estimate the distance between two points?Distance between two points exists the length of the line segment that joins the two given points. Distance between two points in coordinate geometry can be estimated by finding the length of the line segment merging the provided coordinates.
Let the points be (4,-2) and (-6,4).
The distance between two points [tex](x_{1} ,y_{1})[/tex] and [tex](x_{2} , y_{2} )[/tex] exists provided by the distance between the two points
[tex]= $\sqrt\\(x_{2} -x_{1} )^{2} +(y_{2} -y_{1})^{2}[/tex]
substituting the points in the equation, we get
[tex]$=\sqrt\\(-6-(4)^{2} +(4-(-2)^{2}[/tex]
simplifying the equation, we get
[tex]$=\sqrt{136}[/tex]
[tex]$=2\sqrt{34}[/tex]
The distance between two points (4,-2) and (-6,4) exists [tex]2\sqrt{34}.[/tex]
Therefore, the correct answer is option c. [tex]$2\sqrt{34}.[/tex]
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Shota invests $2000 in a certificate of deposit that earns 2, percent interest each year write a function that gives the total value v(t) in dollars, of the investment t years from now
Answers
V(t)=2000(1.02)^t function that gives the total value in dollars of the investment t years from now given that Shota invests $2000 in a certificate of deposit that earns 2 percent interest each year. This can be obtained by using the formula of compound interest formula.
What is the required function?Equation of compound interest compounded yearly,
A = P(1 + R/100)^t
where,
A = amount
P = principal amount
R = rate
t = number of years
Given that P = $2000, R = 2%, number of years = t,
V(t) = 2000(1+2/100)^t
V(t) = 2000(1+0.02)^t
V(t) = 2000(1.02)^t
Hence V(t)=2000(1.02)^t function that gives the total value in dollars of the investment t years from now given that Shota invests $2000 in a certificate of deposit that earns 2 percent interest each year.
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how do I solve for x? (the picture has the problem)
Answers
Answer:
x = 21.013155617496
Step-by-step explanation:
we have a right triangle then ,
[tex]\cos \left( 36\right) =\frac{17}{x}[/tex]
Then
[tex]x=\frac{17}{\cos \left( 36\right) } = 21.013155617496[/tex]
Then
x ≈ 21
Step-by-step explanation:
imagine this right-angled triangle is the sine and cosine scenario in a circle. just turn it in your mind or on the paper until it looks like this.
the point with the 36° angle is the center of this circle.
x is the radius of the circle, that is under 36° connecting the center of the circle with the arc of the circle.
17 is the cosine of the angle multiplied by the radius (as the standard cosine function is normed for the standard circle with radius 1).
and the other leg is the sine of the angle multiplied by the radius (this is FYI, we don't need that for the solution here).
so,
17 = cos(36) × x
x = 17/cos(36) = 21.01315562...
A rectangle measures 3.5 ft by 7 ft. It is enlarged by a scale factor of two. What is the area of the enlarged rectangle? T a m a
Answers
Answer:
98
Step-by-step explanation:
Solution 1, (quick)
When enlarging by a scale factor, the shape's area is multiplied by the scale factor squared.
3.5*7*2^2=98
This works because for a rectangle width x and length y, width 2x and length is 2y, area is 4xy compared to area xy originally.
Solution 2, (technical)
Scale factor of 2 means multiplying by 2
3.5^2=7
7*2=14
7*14=98
Zara is preparing a juice blend.In her recipe, the amount of water is twice the amount of strawberry juice, and amount of lemon juice is 1/5 the amount of strawberry juice. What is the ratio of the lemon water?
Answers
The ratio of the lemon water is 10:1
How to determine the ratioFrom the information, we have that;
The ratio of water to strawberry juice is 2 : 1
That of strawberry juice to lemon juice is 5: 1
The, we have that of water to lemon juice to be 10: 1 when compared
Hence, the ratio of the lemon water is 10: 1
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What is the solution to this system of equations x-2 y=15 2x+4y=-18
Answers
Answer:
(3, - 6 )
Step-by-step explanation:
x - 2y = 15 ( add 2y to both sides )
x = 2y + 15 → (1)
2x + 4y = - 18 → (2)
substitute x = 2y + 15 into (2)
2(2y + 15) + 4y = - 18 ← distribute parenthesis and simplify left side
4y + 30 + 4y = - 18
8y + 30 = - 18 ( subtract 30 from both sides )
8y = - 48 ( divide both sides by 8 )
y = - 6
substitute y = - 6 into (1)
x = 2(- 6) + 15 = - 12 + 15 = 3
solution is (3, - 6 )
Answer:
x is 3 y is -6 I really hope this helps
Joe's company sold insurance policies over the phone. Last month, 2 employees sold three hurricane policies, 6 employees sold two hurricane policies, and 9 employees sold a single hurricane policy. If a single employee is picked at random, what is the probability that the employee sold two hurricane policies or sold one hurricane policy
Answers
The probability that the employee sold two hurricane policies or sold one hurricane policy is 15/17.
According to the statement
Number of employees who sold three hurricane policies is 2
Number of employees who sold two hurricane policies is 6
Number of employees who sold 1 hurricane policies is 9
SO,
The total number of employees is 2+6+9=17.
The number that sold two hurricane policies or sold one hurricane policy is 6+9=15,
so the Probability = Total number of policies sold / total number of employees
Probability = 15/17
So, The probability that the employee sold two hurricane policies or sold one hurricane policy is 15/17.
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A comet is estimated to be about 960,000,000 km from Earth. Write this distance in scientific notation. [?] × 10 km Enter 13
Answers
Answer:
[tex](9*10^{9})( 6*10^{8})[/tex] or 9.6×10^8
Step-by-step explanation:
The answer is written above.
Hope this helps!
Answer:
9.6 * 10^8 km.
Step-by-step explanation:
960,000,000
= 9.6 * 10^8 km.
The value of the exponent = number of digits after the 9, which = 8.
Are the triangles similar? If so, state the similarity and the postulate or theorem that justifies your answer.
Answers
Answer: The triangles are not similar.
Step-by-step explanation:
Two figures are similar simply if they have the same shape, but not necessarily the same size. In a more mathematical sense, similar figures have the same angle measures and proportionate side lengths.
For reference, side lengths are proportionate when the ratios between corresponding/matching sides are the same.
Since we do not have any angles, we will try using the SSS Similarity Theorem, which states that if all three sides of both triangles are in proportion, then the triangles are similar. We will first list all the side lengths and match corresponding ones.
[tex]\triangle ABC \rightarrow 18,20,32\\ \triangle UVT \rightarrow 14,15,24[/tex]
Since we matched corresponding sides, we can now check whether they have the same ratio).
[tex]\frac{18}{14}=\frac{20}{15}=\frac{32}{24}[/tex]
[tex]\frac{9}{7}=\frac{4}{3}=\frac{4}{3}[/tex]
Not all of the side lengths have the same ratio, so these triangles aren't similar. you must multiply 14 by 9/7 to get to 18, but you need to multiply the other two sides by 4/3 to get to their
what is 12(w-3) if w= 10
Answers
Hey there!
12(w - 3)
= 12(10 - 3)
= 12(10) + 12(-3)
= 120 - 36
= 84
OR
12(w - 3)
= 12(10 - 3)
= 12(7)
= 84
Therefore, your answer should be:
84
Good luck on your assignment & enjoy your day!
~Amphitrite1040:)
This figure represents a design found in a glass panel. ABCD is a rectangle with
midpoints X, Y, Z, and W. Emily states that the quadrilateral formed by the segments
that join the midpoints of the sides is a rhombus. Do you agree with her? Explain why
or why not.
Answers
Answer: Yes
Step-by-step explanation:
Since ABCD is a rectangle, [tex]\angle AXY[/tex], [tex]\angle YBZ[/tex], [tex]\angle WCZ[/tex], and [tex]\angle WDX[/tex] are all right angles, and are thus all congruent because all right angles are congruent. Furthermore, because ABCD is a rectangle, we know that [tex]\overline{AB} \cong \overline{CD}[/tex] and [tex]\overline{AD} \cong \overline{BC}[/tex]. Because we are given that X, Y, Z, and W are midpoints, using the fact that halves of congruent segments are congruent, we can conclude that [tex]\overline{AY} \cong \overline{YB} \cong \overline{CW} \cong \overline{WD}[/tex] and that [tex]\overline{AX} \cong \overline{XD} \cong \overline{BZ} \cong \overline{ZC}[/tex]. As a result, we can conclude that [tex]\triangle AYX \cong \triangle DXW \cong \triangle CWZ \cong \triangle BYZ[/tex] by SAS, and thus by CPCTC, [tex]\overline{AY} \cong \overline{XW} \cong \overline{ZW} \cong \overline{YZ}[/tex]. Therefore, since the quadrilateral formed by the midpoints has four congruent sides, it must be a rhombus.
There are 5 different families of triplets at a triplets convention. Each triplet shook hands with all the other triplets, except his or her siblings. How many handshakes took place?
Answers
There is a total of 90 handshakes.
How many handshakes took place?
There are 5 sets of triplets.
If we select one of the sets, each of the triplets will have one handshake with the other 12 people. So we have 36 handshakes at this point.
Now we only care for the other 4 sets.
We select one set again, each triple on that set will have one handshake with the other 9 people, so for the 3 triplets, we have a total of 27 new handshakes.
For the remaining 3 sets we do the same, this time we add 18 handshakes.
Finally, there are 2 sets of triplets, this time we add 9 new handshakes.
To get the total number of handshakes we need to add them all:
36 + 27 + 18 + 9 = 90
There is a total of 90 handshakes.
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Which graph represents y= cos(2x - π) +5?
Answers
The graph that represents y= cos(2x - π) +5 is illustrated below.
What is a graph?It should be noted that a graph simply means a series of connections among variables given.
In this case the graph that represents y= cos(2x - π) +5 is illustrated below.
The graph is attached.
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Express sin S as a fraction in simplest terms. R S
Answers
Answer:
[tex]\sf Sin \ S = \dfrac{2}{9}[/tex]
Step-by-step explanation:
Trigonometry ratios:[tex]\sf \boxed{\bf Sin \ S = \dfrac{opposite \ side \ of \ \angle S}{hypotenuse}}[/tex]
[tex]\sf = \dfrac{RQ}{SQ}\\\\= \dfrac{2}{9}[/tex]
PLEASE I NEED HELP ON THIS QUESTION
Answers
Answer:
Option c is the correct answer
Step-by-step explanation:
distance of satrun - distance of jupiter
Write down two irrational numbers with a sum of 4
Answers
Answer:
π & 4-π
Step-by-step explanation:
Taking, a = π & b = 4-π ,a + b = π + 4 - π = 4A random sample of n = 16 scores is obtained from a normal population with m = 40 and s = 8. what is the probability that the sample mean will be within 2 points of the population mean?
Answers
Using the normal distribution, there is a 0.6826 = 68.26% probability that the sample mean will be within 2 points of the population mean.
Normal Probability DistributionThe z-score of a measure X of a normally distributed variable with mean [tex]\mu[/tex] and standard deviation [tex]\sigma[/tex] is given by:
[tex]Z = \frac{X - \mu}{\sigma}[/tex]
The z-score measures how many standard deviations the measure is above or below the mean. Looking at the z-score table, the p-value associated with this z-score is found, which is the percentile of X.By the Central Limit Theorem, the sampling distribution of sample means of size n has standard deviation [tex]s = \frac{\sigma}{\sqrt{n}}[/tex].For this problem, the parameters are given as follows:
[tex]\mu = 40, \sigma = 8, n = 16, s = \frac{8}{\sqrt{16}} = 2[/tex]
The probability that the sample mean will be within 2 points of the population mean is the p-value of Z when X = 40 + 2 = 42 subtracted by the p-value of Z when X = 40 - 2 = 38, hence:
X = 42:
[tex]Z = \frac{X - \mu}{\sigma}[/tex]
By the Central Limit Theorem
[tex]Z = \frac{X - \mu}{s}[/tex]
Z = (42 - 40)/2
Z = 1
Z = 1 has a p-value of 0.8413.
X = 38:
[tex]Z = \frac{X - \mu}{s}[/tex]
Z = (38 - 40)/2
Z = -1
Z = -1 has a p-value of 0.1587.
0.8413 - 0.1587 = 0.6826 = 68.26% probability that the sample mean will be within 2 points of the population mean.
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20 points!
i need help with these rational numbers
Answers
The rational numbers are explained below.
What are rational numbers?The complete question can't he found. Therefore, an overview will be given. Rational numbers are the numbers that can be represented by the quotient p/q of two integers.
In this case, p and q are integers and p isn't equal to 0.
The examples include 1/5, 1/3, -2, 5, etc.
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Ummm can y’all solve these problems I’ll give extra points for this
Pls solve step by step and label which is which
Answers
If sinx + sin^2x = 1 then, show that cos^12x + 3cos^10x + 3cos^8x + cos^6x = 1
Answers
This can be proved by using identities of trigonometry.
Prove that cos¹²x + 3cos¹⁰x + 3cos⁸x + cos⁶x = 1:Given that,
sin x + sin²x = 1 ⇒ sin x = 1 - sin²x
⇒ sin x = cos²x (∵sin²x + cos²x = 1)
We have to prove that,
cos¹²x + 3cos¹⁰x + 3cos⁸x + cos⁶x = 1
LHS = cos¹²x + 3cos¹⁰x + 3cos⁸x + cos⁶x
= sin⁶x + 3sin⁵x + 3sin⁴ + sin³x
= (sin²x)³ + 3(sin²x)²sin x + 3(sin²x) (sin x)² + (sin x)³
= (sin²x + sin x)³ (∵(a+b)³ = a³ + 3a²b + 3 ab² + b³)
= 1³ = 1 = RHS
LHS = RHS
Hence the equation is proved since both sides of the equation are equal.
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You must write your answer in fully simplified form.
2t= -12
Answers
Answer:
t=-6
Step-by-step explanation:
x/4 ÷ 2/3 = 1/5
solve for x
Answers
The value of x from the equation x/4 ÷ 2/3 = 1/5 is 8/15
Fractionx/4 ÷ 2/3 = 1/5
x/4 × 3/2 = 1/5
(x × 3) / (4 × 2) = 1/5
3x / 8 = 1/5
cross product3x × 5 = 8 × 1
15x = 8
x = 8/15
Therefore, value of x from the equation x/4 ÷ 2/3 = 1/5 is 8/15
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5
Sample Space and Venn Diagrams: Mastery Test
Type the correct answer in each box. If necessary, use / for the fraction bar.
A bag contains 5 blue marbles, 2 black marbles, and 3 red marbles. A marble is randomly drawn from the bag.
The probability of not drawing a black marble is
. The probability of drawing a red marble is
m. All rights reserved.
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Answers
The probability of not drawing a black marble is 4/5
The probability of drawing a red marble is 3/10
What are the probabilities?Probability determines the odds of the occurrence of a random event. The probability the event occurs is 1 and the probability that the event does not occur is 0.
Probability of not drawing a black marble = total marbles that are not black / total number of marbles
(5 + 3) / (5 + 3 + 2) = 8/10 = 4/5
The probability of drawing a red marble = total number of red marbles / total number of marbles
= 3/10
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Answer:
See image
Step-by-step explanation:
Plato
The gross domestic product of the United States in 2020 was $20,940,000,000. What is the gross domestic product of the United States in 2020 expressed in scientific notion?
Answers
Answer:
2.094 x 10¹⁰ I have to add more words soo
60÷8 explained well and not in a decimal form
Answers
Answer: 15/2
Step-by-step explanation: if you fraction it, it'll be 60/8 so you can simplify it by 4. So 60÷4/8÷4 will be 15/2. So the answer is 15/2
Answer:
15/2 or 7 1/2.
Step-by-step explanation:
60 / 8 divide top and bottom by 4:-
= (60/4) / (8/4)
= 15/2.