Given:
[tex]\cos^4 \alpha+\sin^4\alpha=\dfrac{1}{4}(3+\cos 4 \alpha)[/tex]
To prove:
The given statement.
Proof:
We have,
[tex]\cos^4 \alpha+\sin^4\alpha=\dfrac{1}{4}(3+\cos 4 \alpha)[/tex]
[tex]LHS=\cos^4 \alpha+\sin^4\alpha[/tex]
[tex]LHS=(\cos^2 \alpha)^2+(\sin^2 \alpha)^2[/tex]
[tex]LHS=(\cos^2 \alpha+\sin^2\alpha)^2-2\sin ^2\alpha\cos^2 \alpha[/tex] [tex][\because a^2+b^2=(a+b)^2-2ab][/tex]
[tex]LHS=(1)^2-2(1-\cos^2 \alpha)\cos^2 \alpha [/tex] [tex][\because \cos^2 \alpha+\sin^2\alpha=1][/tex]
[tex]LHS=1-2\cos^2 \alpha+2\cos^4 \alpha [/tex]
Now,
[tex]RHS=\dfrac{1}{4}(3+\cos 4 \alpha)[/tex]
[tex]RHS=\dfrac{1}{4}[3+(2\cos^2 2\alpha-1)][/tex] [tex][\because \cos 2\theta=2\cos^2\theta -1][/tex]
[tex]RHS=\dfrac{1}{4}[2+2\cos^2 2\alpha][/tex]
[tex]RHS=\dfrac{1}{4}[2+2(2\cos^2 \alpha-1)^2][/tex] [tex][\because \cos 2\theta=2\cos^2\theta -1][/tex]
[tex]RHS=\dfrac{1}{4}[2+2(4\cos^4 \alpha-4\cos \alpha+1)][/tex] [tex][\because (a-b)^2=a^2-2ab+b^2][/tex]
[tex]RHS=\dfrac{1}{4}[2+8\cos^4 \alpha-8\cos \alpha+2][/tex]
[tex]RHS=\dfrac{1}{4}[4+8\cos^4 \alpha-8\cos \alpha][/tex]
[tex]RHS=1+2\cos^4 \alpha-2\cos \alpha[/tex]
[tex]RHS=1-2\cos^2 \alpha+2\cos^4 \alpha [/tex]
[tex]LHS=RHS[/tex]
Hence proved.
what is the ratio for the volumes of two similar pyramids, given that the ratio of thier edge lengths is 8:3
Answer:
512:27
Step-by-step explanation:
We just cube the edge length ratio since a pyramid is a 3-dimensional figure and volume is 3-dimensional:
8:3 -> 8^3:3^3 -> 512:27
Therefore, the correct answer is 512:27
Answer:
512:27 the answers is that one.
please help!!! ASAP
Answer:D
Step-by-step explanation:
It's on a computer
Answer:
simulation
Step-by-step explanation:
im in 8th grade
$12,000 at 2.2% for 5 years.
what’s the interest?
Answer:
264.
Step-by-step explanation:
12,000X2.2%=264.
Your father gave you $8.82 to buy a present. This covered 7/9 of the cost. How much did the present
cost?
Answer:
$6.86
Step-by-step explanation:
Answer:
$11.34
Step-by-step explanation:
8.82 x 9/7 = 11.34
This means 8.82 is 7/9 of 11.34
The present costs 11.34 dollars
What is the answer please only if you know?
Answer:
Step-by-step explanation:
to find the length of a diagonal of a square, you would multiply the length of a side by the square root of 2. so 5[tex]\sqrt{2}[/tex] ≈7.07 inches, but since it's the nearest tenth, the answer would be 7.1 inches
Mr. Sullivan is organizing teams for the middle school’s annual field day. There are eight classes at the school and 21 students in each class.
Answer:
8 people in every team
Step-by-step explanation:
Answer:
14 teams
Step-by-step explanation:
We know that the students at the school=number of classes * the number of students in each class. This will equal 8 * 21 = 168 students. Now we know that the number of teams = total students divided by the number of students in each team which would be 168/12 = 14. There will be 14 teams.
Enter an inequality that represents the description, and then solve.
Dave has $11 to spend on an $8 book and two birthday cards (c) for his friends. How much can he spend
on each card if he buys the same card for each friend?
Answer:
$6
Step-by-step explanation:
Solve the inequality.
4(x +3) -72x+3(x + 1)
O A. X<5
O B. X> 3
C.
no solution
D.
all real numbers
Can someone help me with this 7th-grade math problem it is ratios If it is correct I will give brainliest so make sure it is correct
Answer:
B
Step-by-step explanation:
denominators are related
40 x 2.5=100
48 x 2.5=120
so they are the same
hope this helps
take care
ASAP PLZZ Helllllpppp!!!! Figure N is the result of a transformation on Figure M. Which transformation would accomplish this?
Directions in the first screenshot.
Answer:
b
Step-by-step explanation:
Identify an equation in point-slope form for the line parallel to y =-2/3x+8 that
passes through (4,-5).
Investors buy a studio apartment for $140,000. Of this amount, they have a down payment of $21,000. Their down payment is what percent of the purchase price? What percent of the purchase price would a $7,000 down payment be?
Answer:
a. Down payment of $21,000
= 21,000 / 140,000
= 0.15
= 15%
b. Down payment of $7,000
= 7,000/ 140,000
= 0.05
= 5%
CAN ANYONE FIND THE SLOPE OF THIS LINE WITHIN THE NEXT COUPLE OF MINUTES PLSS
Answer:
Hello.
Slope= y=-0.5x + 1.5
Step-by-step explanation:
PLSSS HELP MEE!!!
How many ice cream servings can the girls buy if they only want to spend $10.00 on ice cream?
Answer:
depends on how much each serving is
The number of servings of ice cream to buy with $ 10 is A = 5 servings if the cost per serving is $ 2
What is an Equation?Equations are mathematical statements with two algebraic expressions flanking the equals (=) sign on either side.
It demonstrates the equality of the relationship between the expressions printed on the left and right sides.
Coefficients, variables, operators, constants, terms, expressions, and the equal to sign are some of the components of an equation. The "=" sign and terms on both sides must always be present when writing an equation.
Given data ,
Let the equation be represented as A
Now , the value of A is
Substituting the values in the equation , we get
The number of ice cream servings the girls can buy depends on the price of each serving of ice cream. Without this information, we cannot determine the exact number of servings they can buy.
On simplifying the equation , we get
For example, if each serving of ice cream costs $1.00, the girls can buy 10 servings of ice cream with $10.00. On the other hand, if each serving of ice cream costs $2.00, the girls can buy only 5 servings of ice cream with $10.00.
Hence , we need to know the price of each serving of ice cream to determine the number of servings the girls can buy with $10.00.
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Plz need help like rn What is m
Answer:
measure of angle p=
94
Step-by-step explanation:
ok 5 sided polygon= 540
so put all of those equal to 540
need to find x first
540=132+(4x-10)+(5x-11)+128+(4x+2)
simplify and combine like terms
x=23
now plug in for angle p
(4(23)+2)
94
HEPPPPPPPPPPPPPP MEEEEEEEEE
19. (3,5), (2,-2)
write an equation in slope-intercept form for the line that passes through each pair of points
Answer:
y=7x−16 the answer ;)
What is the distance between the points (-2, 4) and (7, -5)?
Solve for x: 7(9 + x) = 84
x= 3
x= 21
x=16
x=64
PLEASE HURRY WILL GIVE BRAINLIEST!!!
Answer:
my answer is 2 . 3 I think
Find the slope of the line going through the points (0, -7) and (8, 1).
Answer:
8,1
Step-by-step explanation:
maybe i think this will help you
When exercising, the maximum heart rate (r ) in beats per minute is modeled by the equation shown in the box, where (a ) represents the age of the person in years. r = 220 - a If Sanchez is 34 years old, what is his maximum heart rate?
Answer:
186 beats per minute
Step-by-step explanation:
The maximum heart rate (r ) in beats per minute is modeled by the equation :
r = 220 - a ...(1)
Where
a is age of the person in years
If Sanchez age is, a = 34 years
Put a = 34 in equation (1)
r = 220 - 34
= 186 beats per minute
Hence, the maximum heart rate is 186 beats per minute.
y - 4 × x - 6 is?
Answer:
y-4x-6
Step-by-step explanation:
I'm not 100% sure but I think this is correct!
Answer:
y - 4x - 6
Step-by-step explanation:
y - 4 • x - 6 = y - 4x - 6
#3 A keyboarding instructor at a community college collected data comparing a student's age and their typing speed. The equation for the line of best fit is given as y = -1.4x + 117.8, where x is the "age in years" and y is the "typing speed. If you are 25 years of age, what is your typing speed?
Answer:
82.8
Step-by-step explanation:
In the above question, we are given the following information:
The equation for the line of best fit is given as y = -1.4x + 117.8, where
x is the "age in years"
y is the "typing speed. If you are 25 years of age, what is your typing speed?
This is calculated as:
y = -1.4x + 117.8
x = 25 years
y = -1.4(25) + 117.8
y = -35 + 117.8
y = 82.8
Hence, If you are 25 years of age, what is your typing speed is 82.8
please help me I have no idea
Answer:
where a is in the equation put 6.05 and where b is put 3.611
Step-by-step explanation:
-6.05 + (-3.611)
-6.05 - 3.611
-9.661
Answer:
-9.661
Step-by-step explanation:
First of you want to plug in a and b making the equation
-6.05 + (-3.611)
or
-6.05 - 3.611
*I hope your teacher will allow you to use a calculator because of the decimals
Then you will add -6.05 + (-3.611) together
to get a negative answer of...
-9.661
Pls help stuck on this question. Point-slope line
(11 points) pls show work
Answer:
Hope this can help u ;)
Step-by-step explanation:
What is the equation of the line that passes through the point (2,-5) and is perpendicular to the line y = -2x + 1? A y = 2x - 1 B y = -x-4 C y = { x - 4 D y = į x-6
Answer:
sorry
Step-by-step explanation:
im so sorry
Find the area of the isosceles triangle. please help thanks
Answer:
1080 m^2 Don't submit m^2 in your answer.
Step-by-step explanation:
Givens
The catch is to find h
To do that, use a^2 + b^2 = c^2
a b and c are in the same 1/2 triangle.
a = 48/2 = 24 m
b = h = ?
c = 51 meters
Solution
a^2 + b^2 = 51^2 Substitute for b^2 = h^2
24^2 + h^2 = 51^2 Expand 24^2 and 51^2
576 + h^2 = 2601 Subtract 576 from both sides
h^2 = 2601 - 576
h^2 = 2025 Take the square root of both sides
h = 45
Area
Area = 1/2 b * h
Area = 1/2 48 * 45
Area = 1080
Remark
Notice that to find h you only use 1/2 of 48 because that is the base of the right triangle.
To find the area, you need to use all of 48 because 48 is the full length of the base.
help pleasee look the picture
Answer:
-19
Step-by-step explanation:
3^2= 9
(9-16)(4)= -28
-28+9= -19
Answer:
-19
Step-by-step explanation:
At the equation together then
Multiply 9-7 x 4
Calculate 9-28
That equals -19 so -19 is the answer
Help please!
Bart had an income of $38,250 last year. If the inflation rate is 4.5%, what is his purchasing power?
A. $33,614.85
B. $34,543.75
C. $34,998.15
D. $36,528.75
Answer: D) $36,528.75
=======================================================
Explanation:
Positive inflation eats away at the purchasing power of money.
If inflation is 4.5%, then the purchasing power goes down by 4.5% meaning his purchasing power is 100% - 4.5% = 95.5% of what it used to be.
95.5% of $38,250 = 0.955*38250 = 36,528.75
So his purchasing power is $36,528.75
The purchasing power of Bart by the income of last year is option D. $36,528.75.
What is Inflation?Inflation can simply be defined as the increase in the rate of the prices over a period of time. In other words, inflation describes how much more expensive has a commodity or service becomes in a period of time.
It is clear that, inflation decreases the purchasing power of money.
Given that, the inflation rate is 4.5%.
Also, Bart had an income of $38,250 last year.
Purchasing power will get decreased by 4.5%.
Purchasing power = $38,250 - ($38,250 × 4.5%)
= $38,250 - (1721.25)
= $36,528.75
Hence the purchasing power of Bart is $36,528.75.
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I need the answer to this, with the explanation