Answers
Recall the half-angle identity for sine,
sin²(x/2) = (1 - cos(x))/2
Then the given equation is identical to
sin²(θ) = 1 - cos(θ)
Also recall the Pythagorean identity,
sin²(θ) + cos²(θ) = 1
Then we rewrite the equation as
1 - cos²(θ) = 1 - cos(θ)
Factoring the left side, we have
(1 - cos(θ)) (1 + cos(θ)) = 1 - cos(θ)
and so
(1 - cos(θ)) (1 + cos(θ)) - (1 - cos(θ)) = 0
and we factor this further as
(1 - cos(θ)) (1 + cos(θ) - 1) = 0
which gives
cos(θ) (1 - cos(θ)) = 0
Then either
cos(θ) = 0 or 1 - cos(θ) = 0
cos(θ) = 0 or cos(θ) = 1
[θ = arccos(0) + 2nπ or θ = -arccos(0) + 2nπ]
… or [θ = arccos(1) + 2nπ or θ = -arccos(1) + 2nπ]
(where n is any integer)
[θ = π/2 + 2nπ or θ = -π/2 + 2nπ] or [θ = 0 + 2nπ]
In the interval 0 ≤ θ < 2π, we get three solutions:
• first solution set with n = 0 ⇒ θ = π/2
• second solution set with n = 1 ⇒ θ = 3π/2
• third solution set with n = 0 ⇒ θ = 0
So, the first choice is correct.
Related Questions
Dave and Kyle are 40 miles apart on a path when they start moving toward each other. Dave runs at a constant speed of 8 miles per hour, while Kyle walks at a constant speed of 5 miles per hour. How long does it take until they meet?
Answers
Answer:
About 3 hours
Step-by-step explanation:
together, they go 13 miles in one hour. 40/13 = 3 with remainder 1
The scale on a map is 2 in. =250 miles. If Toledo and Kansas City are 2.25 in. How far are they apart in real life?
Answers
Answer:
281.25 miles
Step-by-step explanation:
2=250
1=125
0.5=62.5
0.25=31.25
250+31.25=281.25
Hassan made a vegetable salad with 2 3/8 pounds of tomatoes, 1 1/4 pounds of asparagus, and 2 7/8 pounds of potatoes. How many pounds of vegetables did he use altogether?
Btw please show work.
Answers
Step-by-step explanation:
OK so
Tomatoes = 2 3/8 = 19/8 (turned into improper fraction)
Asparagus = 1 1/4 =5/4
Potatoes = 2 7/8 = 23/8
Total vegetables used =
19/8 + 5/4 + 23/8 = 42/8 + 5/4 = 42+10/8 =52/8
Answer is 52/8 = 6.5
Answer : 52/8 pounds were used altogether
See picture for solution.
What is the purpose of constructing a confidence interval?
O In order to find a range of values to estimate the population
proportion.
o In order to find a range of values to estimate the sample proportion.
o To find the exact value of the population proportion.
o In order to find the exact sample proportion.
Answers
Using the interpretation of a confidence interval, it is found that the correct option is:
In order to find a range of values to estimate the population proportion.x% confidence interval:
A confidence interval is built from a sample, has bounds a and b, and has a confidence level of x%. It means that we are x% confident that the population mean is between a and b. From the above bullet point, we have that the interval is taken from a sample, and is used to estimate a population parameter.Hence, the correct option is:
In order to find a range of values to estimate the population proportion.You can learn more about the interpretation of a confidence interval at https://brainly.com/question/25822483
If the sum of squares for a sample containing 51 items equals 200, what is the standard deviation?
A) 1.98
B) 2.00
C) 3.92
D) 4.00
Answers
The standard deviation is 1.98. Option A is correct.
The standard deviation formula is given as
[tex]\sigma = \sqrt{\frac{SS}{n} }[/tex]where:
SS is the sum of squares = 200n is the sample size = 51Substitute the given parameters into the formula to have:
[tex]\sigma = \sqrt{\frac{200}{51} }\\\sigma = \sqrt{3.922}\\\sigma = 1.980[/tex]
Hence the standard deviation is 1.98
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what is integration ....nd differentiation
Answers
Answer:
Differentiation is used to break down the function into parts, and integration is used to unite those parts to form the original function. Geometrically the differentiation and integration formula is used to find the slope of a curve, and the area under the curve respectively.
Answer:
integration is the opposite of differentiation
Step-by-step explanation:
integration requires addition of 1 to the power and then division by the value eg
the integral of
[tex]{x}^{2} = \frac{ {x}^{2 + 1} }{2 + 1} \\ = \frac{ {x}^{3} }{3} [/tex]
while in differentiation, u subtract one and multiply by the value. eg
the differential of
[tex] \frac{ {x}^{3} }{3} = 3( \frac{ {x}^{3 - 1} }{3} ) \\ = 3( \frac{ {x}^{2} }{3} ) \\ = {x}^{2} [/tex]
In the rectangular trapezoid ABCD AC is driven by a CD (see figure).
Find the width of the trapezoid if AC = 3, AD = 5.
Answers
Answer:
solution given:
let's see only in a right-angled triangle Δ ACD.
AC=3 units
AD=5 units
since Δ ACD is a right-angled triangle. It satisfies Pythagoras law
[tex]AD^{2}=AC^{2}+CD^{2}[/tex]
25=9+[tex]CD^2[/tex]
[tex]CD^2[/tex]=25-9=16
[tex]CD=\sqrt{16} =4 units[/tex]
Now
Area of rectangle Δ ACD=[tex]\frac{1}{2} CD*AC=\frac{1}{2}*4*3=6\: square \:units[/tex]
similarly,
[tex]\frac{1}{2} AD*CE=6\: square \:units\\ 5*CE=6*2\\CE=\frac{12}{5}=2.4 units[/tex]
since AB=CE=2.4 units
Δ ACE is a right-angled triangle. It satisfies Pythagoras law.
[tex]AC^{2}=AE^{2}+CE^{2}\\3^2=AE^2+2.4^2\\9-5.76=AE^2\\AE and BC=\sqrt{1.24} =1.11 units[/tex]
now
area of trapezoid=area ofΔACD+Area of ΔABC
=6+[tex]\frac{1}{2}AB*BC=6+\frac{1}{2}*2.4*1.11=6+1.33 =7.33\: square \:units[/tex]
Joe wants to build a doll house for his daughter. He wants the doll house to look just like his house. His house is 29 feet wide and 32 feet tall at the highest point of the roof. If the dollhouse will be 2.5 feet wide, how many feet will its highest point be? (Round your answer to one decimal place.)
Answers
Answer:
3.5 feet
Step-by-step explanation:
The highest point of the dollhouse will be 2.76 feet.
What is proportion?A mathematical comparison of two numbers is called a proportion. According to proportion, two sets of provided numbers are said to be directly proportional to one another if they increase or decrease in the same ratio.
Given that,
The width of house = 29 feet.
And the height of the house = 32 feet.
Also, the width of dollhouse = 2.5 feet
Let the height of the doll is x feet.
To find the value of x,
Use proportion property,
29 / 32 = 2.5 / x
x = (32 × 2.5) / 29
x = 80 / 29
x = 2.758
the height of the doll is 2.76 feet.
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Solids X and Y are similar.
X has volume 64 cm3
Y has volume 343 cm3
The surface area of X is 176 cm2
Work out the surface area of Y.
Answers
Answer:
i think surface area of Y is 539 cm²
use the result (a+b)²=a²+2ab+b² to find the value of 49²+98+1
pls help me
I will give brainliest
Answers
Answer:
2500
Step-by-step explanation:
a² + 2ab + b²
49² + 98 + 1
Comparing terms
a²= 49²
a= 49
2ab = 98
2× 49 × b = 98
98b = 98
b= 98/98
b = 1
or
b²= 1
b=√1
b= 1
a=49 and b= 1
Hence (a+b)²= (49+1)²
50²= 2500
Given the following computer output, select the correct least-squares regression line:
Answers
The correct form of the regression line is [tex]\log time = 0.699 + 0.500\cdot \log length[/tex] (Correct choice: D).
According to the information presented in the figure, we know the following function:
The independent variable is 'length'.The dependent variable is 'time'.The regression line is of the form: [tex]\ln y = A + B\cdot \ln x[/tex].Hence, the correct form of the regression line is:
[tex]\log time = 0.699 + 0.500\cdot \log length[/tex] (Correct choice: D)
The correct form of the regression line is [tex]\log time = 0.699 + 0.500\cdot \log length[/tex] (Correct choice: D).
To learn more on regression lines, we kindly invite to check this verified question: https://brainly.com/question/7656407
Answer:
D: logTime=0.699 + 0.500
Step-by-step explanation:
0.699 is the x value which can be found in data under Coef and Constant. The y value can be found under coef and log(length)
3 + 12 + 48 + 192 + ...
Whats the answer
Answers
3 12 48 192 768 3072 12288
add them all up and you get 16383
Answer:
3, 12, 48, 192, 768, 3072, 12288 these are the next seven numbers
Add them all up and you get
= 16383 Answer CStep-by-step explanation:
[tex]3, 12, 48, 192[/tex]
So from 3 to 12 is multiply by 4
So we have checked that it works for all of them
Rule:
Each term is: Two to the power of the number3/4 times the number squared plus 0 times the number
I hope this has helped :)
This is a special array.
1,2,2,5,2,2,9,0,...
what is next number?
Answers
Answer:
2
Step-by-step explanation:
the first number is one .then you write number two twice . The next number is the summation of the three previous numbers .then you write number two twice and the next number is the summation of the three previous numbers .and you do that regularly
What is the value of x in the equation 4
.
+
2
Answers
Answer:
x=-3
Step-by-step explanation:
Sophie has $4.20 worth of dimes and quarters. She has twice as many quarters as dimes. Write a system of equations that could be used to determine the number of dimes and the number of quarters that Sophie has.
Answers
Answer:
0.25q +0.10d = 4.20q = 2dStep-by-step explanation:
Let q represent the number of quarters, and d represent the number of dimes.
0.25q +0.10d = 4.20 . . . . . the value of the coins
q = 2d . . . . . . . . . . . . . . . the relation between the numbers of coins.
__
Additional comment
Using substitution, the equation simplifies to ...
0.60d = 4.20
d = 7
q = 14
Note that the coins can be grouped into groups of 2 quarters and 1 dime. The value of the group is $0.60, and the number of dimes is equal to the number of groups. Thinking about the problem in this way gets you immediately to the single equation 0.60d = 4.20.
Please help I don't understand
| X = 2 | Y = 3 | Z = 4 |
Evaluate the expression using the solutions above
5x + 6z + 2y
Answers
Answer:
The expression can be written as:
5(2) + 6(3) + 2(4).
if I'm wrong, please tell me
Suppose the sample space for a probability experiment has 48 elements. If items from the sample space are selected without replacement, how many different ways can you select all of the items?
Answers
Answer:
56study hard watch your book for 20 minute and look....this is a hint
URGENT PLS HELP
John is driving around town. When he reaches the gas station, he notes that he has traveled 18 miles. He reaches home 2 hours later and notes that he has traveled 34 miles. If d represents the distance and t represents the time, in hours, John has traveled since the gas station, which of the following equations can be used to model this situation?
1. d = 18t + 52
2. d = 16 + 18t
3. d = 8t + 18
4. d = 8t - 18
Answers
Use the exponential regression equation that best fits the data (10,4). (12,20), (13,35), and (16,300) to estimate y when x = 15.
A 53.1
В. 97.2
C. 146.7
D. 254.8
Answers
Answer:
c 146.7
Step-by-step explanation:
cuz
Which graph or graphs show that a 5 minute phone call costs $4?
Answers
The graph should show an increase of $ 0.80 for every minute that the call lasts.
A 5 minute phone call costs $ 4, with which, to determine the information that the graph should show, the following calculation must be performed:
4/5 = X 0.80 = X
Therefore, the graph should show an increase of $ 0.80 for every minute that the call lasts.
Learn more about maths in https://brainly.com/question/25834039
Answer:
Step-by-step explanation:
A couple can afford to make a monthly mortgage payment of $700. If the mortgage rate is 2% and the couple intends to secure a 30-year mortgage, how much can they borrow?
Answers
========================================================
Explanation:
r = 0.02 = decimal form of 2% annual interesti = monthly interest ratei = r/12 = 0.02/12 = (2/100)/12 = 1/600.n = number of months = 12*30 = 360P = monthly payment = 700We'll plug those items into the formula below.
L = loan amount
L = P*( (1+i)^n - 1)/( i*(1+i)^n )
L = 700*( (1+1/600)^360 - 1 )/( 1/600*(1+1/600)^360 )
L = 189,383.961539639
L = 189,383.96
The couple can afford to borrow at most $189,383.96
Determine if the two lines are parallel. Explain your answer for brainliest.
3x+4y=8
-12x-9y=27
Answers
Below!
Explanation:The lines are not parallel because the slope is not the same for both equations. If a slope is the same with different y-intercepts, the two line will eventually be parallel. However, if a line has same slope and y-intercept, then there will be infinite solutions to the equation.
Hoped this helped
[tex]BrainiacUser1357[/tex]
Help help help help math math
Answers
When a number is divided by 5, it has a remainder of 4.
The number is between 35 and 40.
What is the number?
Answers
Answer:
37
Step-by-step explanation:
I plugged in the numbers between 35 and 40 and got this. No problem. :)
The price of a CD decreased from $15 to $10. What is the percent of decrease
Answers
Answer:
Step-by-step explanation:
amount of decrease is $15 - $10 = $5
as a percentage of the original price
100($5/$15) = 33⅓ %
What is the LCD/LCM of this equation
-12.2 + 2.8W - 8.8 = 1.05w
Answers
Answer: 2.8 W. 2.6 W. 2.4 W. 2.2 W. 2.0 W. 1.9 W. 1.7 W
Step-by-step explanation:
Answer:
LCD/ LCM = 100
Step-by-step explanation:
- 12.2 + 2.8W - 8.8 = 1.05W
- 12 [tex]\frac{2}{10}[/tex] + 2 [tex]\frac{8}{10}[/tex] W - 8 [tex]\frac{8}{10}[/tex] = 1 [tex]\frac{5}{100}[/tex] W
- [tex]\frac{122}{10}[/tex] + [tex]\frac{28}{10}[/tex] W - [tex]\frac{88}{10}[/tex] = [tex]\frac{105}{100}[/tex] W
the LCM/ LCD is 100
A newborn child receives a $6000 gift toward a college education from her grandparents. How much will the $6000 be worth in 18 years if it is invested at 7.6% compounded quarterly?
Answers
Answer:
333.333333333
Step-by-step explanation:
you have to divide
Please answer the question on the image.
Answers
Answer:
both isosceles???
Step-by-step explanation:
Answer:
They are similar.
Step-by-step explanation:
There might be other ways but
<X(Angle of x) is used for both triangle XAB and XYZ
And lines AB and YZ are parallel, From this we know that a line intersected by parallel lines it forms same angles for both parallel angles.
The similarity statement is: AAA Because angle(A=B , B=Z)
Jake ran 4% miles in % of an hour. How fast did he run in one hour?
Answers
I WILL GIVE BRAINLY!!!
-x - 8 ≤ 16
Please answer! Is the answer 24 or -24?
My last inequality before I got my answer was : -x ≤ 24.
But as we know if there is a varible that has no number associated with it we assume that the number for it is just one, but in this case -x would be -1x. So if I divide -1x/-1 that would be x, but if I divide 24/-1, that would be -24. So in the end someone just tell me if it's 24 or -24?!
Answers
Answer: The answer is X ≥ -24.
Explanation in the image.