Prove that cos 4θ = 8 cos⁴ θ - 8 cos² θ + 1
Answers
Answer:
Step-by-step explanation:
Answer is in the pic.
Answer:
See below for proof.
Step-by-step explanation:
[tex]\boxed{\begin{minipage}{6 cm}\underline{Trigonometric Identities}\\\\$\cos (A \pm B)=\cos A \cos B \mp \sin A \sin B$\\ \\$\sin^2 \theta + \cos^2 \theta=1$\\\end{minipage}}[/tex]
Use the trigonometric identities to prove the given equation:
[tex]\begin{aligned} \implies \cos 4 \theta & = \cos (2 \theta +2 \theta)\\& = \cos 2 \theta \cos 2 \theta - \sin 2 \theta \sin 2 \theta\\& = \cos^2 2 \theta - \sin^2 2 \theta\\& = \cos^2 2 \theta - (1-\cos^2 2 \theta)\\& = 2\cos^2 2 \theta - 1\\ & = 2\left(\cos 2 \theta \right)^2-1\\& = 2\left(\cos\theta \cos\theta-\sin \theta\sin \theta\right)^2-1\\& = 2\left(\cos^2\theta-\sin ^2\theta\right)^2-1\\& = 2\left(\cos^2\theta-(1-\cos^2\theta)\right)^2-1\\& = 2\left(2\cos^2\theta-1\right)^2-1\\\end{aligned}[/tex]
[tex]\begin{aligned}& = 2\left(4\cos^4\theta-4\cos^2\theta+1\right)-1\\& =8\cos^4\theta-8\cos^2\theta+2-1\\ & =8\cos^4\theta-8\cos^2\theta+1 \end{aligned}[/tex]
Related Questions
Jean was working on a project involving linear regression. She obtained a predicted value of 8 when she entered 12 as her explanatory variable. Then Jean spilled coffee on her work and could no longer read her regression model. If she remembered that the y-intercept was 26, what was the value of the slope/
A -1.75
B 1.5
C -1.5
D 1.75
E 1.33
Answers
Based on the predicted value, the explanatory variable, and the y-intercept, the value of the slope for Jean's project is C. -1.5
What is the slope?A linear regression equation takes the form of:
y = mx + c
y = predicted value
m = slope
x = explanatory variable
c = y-intercept
Given these values, the slope:
8 = 12m + 26
12m = 8 - 26
m = -18 / 12
m = -1.5
In conclusion, option C is correct.
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1. Larry is in the market for a new lawn mower. He has found two on sale. The first lawn
mower is $275 with a 20 percent discount but must be assembled. The second lawn
mower is $299 with a 25 percent discount and is already assembled. Which of the
following describes the best decision for Larry?
Answers
The option that describes the best decision for Larry is Larry buys the second lawn mower because he thinks the extra $5 are worth not having to do the assembly.
How much does Larry spend?The cost of the first lawnmower after the discount is:
= 275 x (1 - 20%)
= 275 x 0.80
= $220
The cost of the second lawnmower is:
= 299 x 0.75
= $224.25
The difference between the lawnmowers is around $5. This $5 is the cost of the assembly on the second lawnmower and Larry might decide that this cost is worth it.
Options for this question include:
Larry buys the second lawn mower because he thinks the extra $5 are worth not having to do the assembly.Larry buys the second lawn mower because it is cheaper due to the better sale, and he doesn't want to assemble a lawn mower.Larry buys the first lawn mower because he likes the way it looks.Larry buys the first lawn mower because it is significantly cheaper.Larry buys the first lawn mower because he can take the day off of work to build it and will saveFind out more on economic decisions at https://brainly.com/question/27840103
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[Been stuck on it for weeks, help me or I will fail. I will give Brainlist to anyone who has a well written descriptive answer please.]
A bucket of paint has spilled on a tile floor. The paint flow can be expressed with the
function p(t) = 6t, where t represents time in minutes and p represents how far the paint is
spreading.
The flowing paint is creating a circular pattern on the tile. The area of the pattern can be
expressed as A(p) = лp².
Part A: Find the area of the circle of spilled paint as a function of time, or A[p(t)]. Show your
work. (6 points)
Part B: How large is the area of spilled paint after 8 minutes? You may use 3.14 to
approximate in this problem. (4 points)
Answers
Answer:
Part A: 28.26t²
Part B: 2,826 square unit
Explanation:
Part A:
A[p(t)] = π (3t)² = 3.14 * 9t² = 28.26t²
Part B:
p(10) = 3(10) = 30
A(p) = 3.14 * 30² = 3.14 * 900 = 2,826 square unit
The temperature in Anchorage,Alaska, dropped over four consecutive hours from 0 to -44F. If the temperature dropped the same amount each hour, how much did the temperature change each hour?
Answers
The temperature change in Anchorage,Alaska, each hour is; -11°F per hour
What is the rate of change of the temperature?When we talk of rate of change with respect to two values, we simply mean the rate of change of the output value with respect to the input value.
Now, we are told that the temperature went from 0°F to -44°F over a four hour period. Thus;
Rate of change = Change in temperature/change in time
Thus;
Rate of change = (-44 - 0)/(4 - 0)
Rate of change = -44/4
Rate of change = -11°F per hour
The temperature change in Anchorage,Alaska, each hour is; -11°F per hour
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PLEASE HELP ASAP!!!!!!!!!!!!!!!!
Initially, there were only 3 weeds at a park. The weeds grew at a rate of 5% each week. The following function represents the weekly weed growth: f(x) = 3(1.05)x. Rewrite the function to show how quickly the weeds grow each day and calculate this rate as a percentage.
f(x) = 3(1.007)7x; spreads at a rate of approximately 0.7% daily
f(x) = 3(1.05)7x; spreads at a rate of approximately 0.5% daily
f(x) = 3(0.15)x; spreads at a rate of approximately 0.15% daily
f(x) = 3(1.007)x; spreads at a rate of approximately 0.07% daily
Answers
Based on the calculation below, the rate at which the weeds grow each day is 0.7%.
How do we calculate the daily percentage growth rate?Given:
f(x) = 3(1.05)x = 3(1 + 0.05)x ................... (1)
Since there are 7 days in a week, the rate at which the weeds grow each day can be calculated from equation (1) by dividing 0.05 (i.e. 5% weekly growth rate) as follows:
f(x) = 3(1 + (0.05 / 7))x
f(x) = 3(1 + 0.007)x
f(x) = 3(1.007)x
Since 0.007 is the same as 0.7%, this implies that the rate at which the weeds grow each day is 0.7%.
The correct option is "f(x) = 3(1.007)7x; spreads at a rate of approximately 0.7% daily".
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Answer: Option 1
Step-by-step explanation: Correct for my test.
[tex]\frac{-9}{5}+ \frac{7}{2}[/tex]
Answers
Answer:
exact form: 17/10
decimal form: 1.7
mixed number form: 1 7/10
Step-by-step explanation:
A store sells a package of 50 paper plates for $2.50 and a package of 80 paper plates for $4.00. Which of the following statements is correct? Multiple choice question. A) The unit prices are the same. B) The package of 50 paper plates has the lower unit price. C) The package of 80 paper plates has the lower unit price. D) The package of 50 paper plates has the higher unit price.
Answers
Answer:
A) The unit prices are the same.
Step-by-step explanation:
First convert dollars to cents to make the calculation easier
Package of 50 paper plates costs $2.50
$2.50 = 2.50 x 100 = 250¢
So the unit price, which is the price per paper plate = 250/50 = 5¢
Package of 80 paper plates costs $4.00
$4.00 = 4.00 x 100 = 400¢
So the unit price, which is the price per paper plate = 400/80 = 5¢
So the unit prices of both packages are the same
for each positive integer $n$, the mean of the first $n$ terms of a sequence is $n$. what is the $2008^{\text{th}}$ term of the sequence?
Answers
B(4015) is the [tex]2008^{th[/tex]term of the sequence.
What is the sequence?
A sequence is an enumerated group of particulars in mathematics where reiterations are permitted and order is important. analogous to a set, it has members( also called rudiments, or terms). The length of the series is the number of rudiments( potentially horizonless). A list of particulars in a particular order, generally figures, is called a sequence.Since the mean of the first terms is, the sum of the first terms is n². Thus, the sum of the first 2007 terms is 2007² and the sum of the first 2008 terms is 2008² Hence, the 2008th term of the sequence is
2008² - 2007² = (2008 + 2007) (2008 - 2007) = 4015
Note that n² is the sum of the first n odd integers.
From inspection, we see that the sum of the sequence is n². We also notice that n² is the sum of the first n odd integers.
Because 4015 is the only odd integer, B is the answer.
B(4015) is the [tex]2008^{th[/tex]term of the sequence.
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Write 'six more than four times a number equals 18' as an equation.
6x + 4 = 18
4x + 6 = 18
6x(4) = 18
4x(6) = 18
Answers
The required equation is 4x+6=18.
Option (B) is correct.
What is algebra?A part of mathematics in which letters and other general symbols are used to represent numbers and quantities in formulae and equations.
Given:
Let the number be 'x'
In mathematics, an equation is a formula that expresses the equality of two expressions, by connecting them with the equals sign.
We have to find the equation..
According to given question we have
i.e
four times a number=4*x
'six more than=4x+6
equals 18'=18
i.e
4x+6=18
Therefore, the required equation is 4x+6=18.
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4) Mr. Carter loves underwater diving. His house
boat is 250 feet above sea level. After jumping from
that level, he ended up at 650 feet below sea level.
What was the total distance that Mr. Carter traveled
from his boat to where he ended up?
Answers
Answer:
he traveled 900 feet
Step-by-step explanation:
250 + 650 = 900
Mr. Carter traveled a total distance of 900 feet from his boat to where he ended up.
Here, we have to find the total distance that Mr. Carter traveled from his boat to where he ended up, we need to calculate the sum of the distances above and below sea level.
The distance above sea level is 250 feet (his house boat's level), and the distance below sea level is 650 feet (where he ended up after jumping).
Total distance traveled = Distance above sea level + Distance below sea level
Total distance traveled = 250 feet + 650 feet
Total distance traveled = 900 feet
Mr. Carter traveled a total distance of 900 feet from his boat to where he ended up.
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a number is chosen at random from the positive, even integers from 2 to 50. find the probability that the number chosen is divisible by 5 or 8.
Answers
The probability that the number chosen is dividible by 5 or 8 is 0.306.
According to the given question.
A number is chosen at random from the positive even integers from 2 to 50.
Now the total positive intergers from 2 to 50 which are divisible by 5 or 8 are {5, 8, 10, 15, 16, 20, 24, 25,30, 32, 35, 40, 45, 48, 50}
So, there are total 15 positive intergers from 2 to 50 which are divisible by 5 or 8.
⇒ Total number of favourable outcomes = 15
And the total positive integers from 2 and 50 is 49.
⇒ Total number of outcomes is 49.
As we know that, proabability is calculated by taking the ratio of favourable outcomes to the total number of outcomes.
Therefore, the probability that the number chosen is divisible by 5 or 8
= 15/49
= 0.306
Hence, the probability that the number chosen is dividible by 5 or 8 is 0.306.
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HELP!!!!!!!!!!!!! 30 points that is all my points
Answers
1) Unit Rate of the given options are;
$0.05 for 1 ounce
24 bottles per case
16 ounces per bottle
2) Rate of the given options are;
5 bottles for $5
$1 per 16 ounces
What is the Unit Rate?A unit rate (or unit ratio) describes how many units of the first type of quantity corresponds to one unit of the second type of quantity. For example, if you ran 70 yards in 10 seconds, you ran on average 7 yards in 1 second.
Meanwhile, a rate is the ratio between two different units of measure.
Thus;
1) Unit Rate of the given options are;
$0.05 for 1 ounce
24 bottles per case
16 ounces per bottle
2) Rate of the given options are;
5 bottles for $5
$1 per 16 ounces
3) The option that is neither a unit rate nor a rate is; 36 bottles
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resolver esto
2x - 3y = 7
4x + y = 7
Ayuda es para hoy ;-; si t sale el resultado deben de cambiarlo y ver si salio ;-;
Answers
y=-1
as an ordered pair it is (2,-1)
f(x)= 4x - 2
find x if f(x)=2
Answers
Answer:
1
Step-by-step explanation:
a) Flip the equation
4x - 2 = 2
b) Add 2 to both sides
4x - 2 + 2 = 2 + 2
4x = 4
c) Divide both sides by 4
4x/4 = 4/4
x = 1
The coordinates of A and B are (3k, 8) and (k, -3) respectively. Given that the gradient of the line segment AB is 3, find the value of k.
Answers
Step-by-step explanation:
gradient = slope or several other words.
it describes how strongly a line (or tangent to a bent curve) is going up or down or ... if it is changing at all.
it is represented by the ratio
(y coordinate change / x coordinate change)
when going from one point on the line to another.
in our case, when going from A to B we have
x changes by -2k (from 3k to k).
y changes by -11 (from 8 to -3).
so, the gradient or slope is
-11/-2k = 3
11/2k = 3
11 = 3×2k = 6k
k = 11/6
A = (33/6, 8) = (11/2, 8)
B = (11/6, -3)
21. The cost to mail a package is $5.00. Brian has postcard stamps that are worth $0.34 each and first class
stamps that are worth $0.49 each.
If Brian puts 7 first class stamps on the package, how many postcard stamps will be needed.
Answers
Answer:
Okay so a couple key points
$5.00 to ship it
Postcard stamps = $0.34
First class stamps= $0.49
And lastly he has 7 first class stamps
So with that being said you take .49 and times it by 7 since he has seven of them and that would be $3.43
And now you take $3.49 and subtract it from the $5.00 you'd be left with $1.57
Now you take $.34 and divide it by $1.57 and you would get a total of 4.61 so that means you would need five postcard stamps
So to be able to ship the package you'd need 7 first class stamps and 5 postcard stamps
(a) The marked price of a bicycle is Rs 2,000. If the shopkeeper allows some discount and a customer bought it for Rs 1921 including 13% VAT, how much amount was given as the discount?
Answers
The amount of the discount that was given is $300.
What is the discount given?A discount reduces the price of an item. When a discount is given, the item becomes cheaper. VAT is value added tax. VAT increases the price of an item.
The equation that can be used to represent the total cost of the bicycle is:
Total cost after VAT and discount = (1 + VAT) x (marked price - discount)
1921 = (1 + 0.13) x (2,000 - d)
1921 = 1.13 x (2,000 - d)
In order to determine the value of discount, take the following steps:
Divide both sides of the above equation by 1.13
1921 / 1.13 = 2000 - d
1,700 = 2,000 - d
Combine similar terms: 2000 - 1700 = d
Add similar terms together: d = 300
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Rewrite one fourth times x cubed times y plus three fourths times x times y squared using a common factor.
one fourth times x times y times the quantity x squared plus 3 times y end quantity
one fourth times x cubed times y squared times the quantity y plus 3 times x end quantity
one half times x times y times the quantity 2 times x squared plus 6 times y end quantity
one half times x times y times the quantity x squared plus 3 times y end quantity
Answers
The expression 1/4x³y + 3/4xy² can be rewritten as: A. 1/4xy(x² + 3y).
How to Rewrite an Expression?We can rewrite an expression by using a factor that is common in an expression.
Given the expression, 1/4x³y + 3/4xy², the common factor in the expression that we can factor out is 1/4xy.
To rewrite the expression, we have the following:
1/4xy into 1/4x³y will give us x²
1/4xy into 3/4xy² will give us 3y.
This will give us: 1/4xy(x² + 3y).
Therefore, the expression 1/4x³y + 3/4xy² can be rewritten as: A. 1/4xy(x² + 3y).
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Answer:
A. one fourth times x times y times the quantity x squared plus 3 times y end quantity
Step-by-step explanation:
Hope this helps you :)
If A:B = 7:8 an B:C =12:7,find A:C in its simplest form.
Answers
Answer:
84:56
Step-by-step explanation:
a:b:c
(7:8)
( 12:7)
Answer:
A:C = 3:2
Step-by-step explanation:
A:B = 7:8 is a proportion relationship and can be represented using fractions:
[tex]\displaystyle{\frac{A}{B} = \frac{7}{8}}[/tex]
Similarly the proportion relation B:C = 12:7 can be represented as:
[tex]\displaystyle{\frac{B}{C} = \frac{12}{7}}[/tex]
Multiply the first equation by the second
[tex]\displaystyle\frac{A}{B}\;\mathsf{x}\;\frac{B}{C} = \frac{7}{8}\;\mathsf{x}\;\frac{12}{7}[/tex]
This evaluates to
[tex]\displaystyle\frac{A}{C} = \frac{12}{8}[/tex]
Simplifying gives us
[tex]\displaystyle\frac{A}{C} = \frac{3}{2}[/tex]
So A:C = 3: 2
ya Williams-
set to the inequality
Solve the inequality.
-1.5(4x+1) > 45-25(x+1)
1. For parentheses:
Distribute
2. If necessary:
Combine Terms
3. Apply properties:
Add
Multiply
Subtract
Divide
4. To start over:
Reset
English
V
KE'NAIYA
Answers
The solution to the inequality using William's template is x > 21.5/19
How to determine the solution to the inequality using William's template?The inequality expression is given as
-1.5(4x + 1) > 45 - 25(x + 1)
For parentheses: Distribute
So, we have
-6x - 1.5 > 45 - 25x - 25
2. If necessary: Combine Terms
So, we have
19x > 21.5
3. Apply properties: Divide both sides by 19
x > 21.5/19
Hence, the solution to the inequality using William's template is x > 21.5/19
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Can y’all please help me on question 23 please
Answers
Answer:
x<-5/4 or x>2
Step-by-step explanation:
-4x -11 > 5
+11 +11
-4x>5
/ /
-4 -4 Switch the sign when divided by a negative.
x<-5/4
8x - 7 > 9
+7 +7
8x>16
/ /
8 8
x>2
WILL GIVE BRAINLIEST need help asap
What will be the location of the y value of R' after using the translation rule (x + 4, y - 7), if the pre-image R is located at ( -17, -3)
Answers
The location of the y value of R' after using the translation rule is -10
What will be the location of the y value of R' after using the translation rule?The translation rule is given as:
(x + 4, y - 7)
The pre-image of R is located at (-17, -3)
Rewrite as
R = (-17, -3)
When the translation rule is applied, we have:
R' = (-17 + 4, -3 - 7)
Evaluate
R' = (-13, -10)
Remove the x coordinate
R'y = -10
Hence, the location of the y value of R' after using the translation rule is -10
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A restaurant serves breakfast, lunch, anddinner. The restaurant offers a daily special.To estimate the percentage of customers who order the daily specials, the restaurant selected a random sample of customers who had lunch during 2 month period. From the results, the restaurant determined that 25% of these customers ordered the daily
special. Which statement about the sample it true?
1) no generalizations can be made because the sample size is too small.
2)the same shows that exactly 25% of the customers ordered the daily specials
3) the sample might not be representative of the population behave it only survey lunch customers.
4) the sample percentage is the percentage of customers who order daily specials
Answers
Answer:
nnk;'/mnhjjfn;h;;vhxpycoyyoly,
Step-by-step explanation:
I]o]ogi-gogoig]gogh]o]v]b]bob]b]k k k lgg]goghjppkngghhjjl
No generalizations can be made because the sample size is too small. Therefore, the correct option is option A.
What is restaurant?A restaurant is an institution of business which prepares and provides clients with food and beverages. Although many restaurants now provide take-out as well as food delivery services, meals are often served and consumed on the premises.
A restaurant serves breakfast, lunch, and dinner. The restaurant offers a daily special. To estimate the percentage of customers who order the daily specials, the restaurant selected a random sample of customers who had lunch during 2 month period. From the results, the restaurant determined that 25% of these customers ordered the daily special. No generalizations can be made because the sample size is too small.
Therefore, the correct option is option A.
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The next model of a sports car will cost 4.6% more than the current model. The current model costs $34,000. How much will the price increase in dollars? What will be the price of the next model?
Answers
Based on the price of the current model of a sports car, the price increase for the next model will be $1,564
The price of the next model of the sports car will be $35,564.
How to find the price?The increase in price in dollar terms for the next model of the sports car can be found by the formula:
= Current cost of current model x Price increase in percentage
= 34,000 x 4.6%
= $1,564
The price of the next model is therefore:
= Current price of current model + Price increase
= 34,000 + 1,564
= $35,564
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SILL GOVE BRIANLIST TO BEST ANSWER
WHATS the volume of the prism ?
Answers
Answer:
629.3 cm3
Step-by-step explanation:
This is known as a rectangular prism aka cuboid.
The formula for the volume is Length x width x height
so 14.5x7x6.2 = 629.3 cm cubed
Answer the 2 questions in photo [ Brainliest + 20 points ]
( you don’t have to answer both just at least one of them please)
Answers
Can I see the full photo? With the segments too? I think I have an idea but just in case I need to see the full paper please
Which is equivalent to the fraction represented by point M on the number line?
Answers
From the calculation below, the equivalent to the fraction represented by point M on the number line is [tex]\frac{1}{4}[/tex].
How do we determine the fraction equivalent?On the number line, it can be observed that there are eight points starting from the first point after 0 up to 1.
Since M is on the second point on the number line, we can have:
P = Point at which M is located on the number line = 2
T = Total number of points on the number line = 8
Therefore, we have:
[tex]M =\frac{P}{T}[/tex]
[tex]M=\frac{2}{8}[/tex]
[tex]M =\frac{1}{4}[/tex]
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Please answer please
Answers
answer
ex 17
2 3/4 ×8
=11/4×8/1
=11/24
ex 18
9/2× 5/4
=36/10
4
find the rise and the run
Answers
Answer:
Rise is 4
Run is 5
Slope is 4/5
Step-by-step explanation:
The 2 points on the line (4,-3) and (9,1)
Rise is 4
Run is 5
Slope is 4/5
A sidewalk that is 2 ft wide surrounds a rectangle that is 4 ft by 6 ft what's the area of the sidewalk in square feet
Answers
The area of a sidewalk that surrounds a rectangle of 4 ft by 6 ft is 56 ft²
The given situation can be depicted in the attached picture.
Recall the formula to calculate the area of a rectangle:
A = l ⨉ w
Where:
l = length
w = wide
In the attached picture, the area of the sidewalk is the one with green color.
Notice that:
Area of sidewalk = area of bigger rectangle – area of smaller rectangle
Parameters of the bigger rectangle:
l = 6 + 2 + 2 = 10 ft.
w = 4 + 2 + 2 = 8 ft.
Area of bigger rectangle = 10 x 8 = 80 ft²
Parameters of the smaller rectangle:
l = 6 ft.
w = 4 ft.
Area of bigger rectangle = 6 x 4 = 24 ft²
Hence,
Area of sidewalk = 80 - 24 = 56 ft²
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