Answers
Answer:
Option A: [tex]\{0,\frac{\pi}{2},\frac{3\pi}{2}\}[/tex]
Step-by-step explanation:
[tex]sin^2(\theta)=2sin^2(\frac{\theta}{2}), [0,2\pi)[/tex]
[tex]sin^2(\theta)=2sin^2(\frac{\theta}{2})[/tex]
[tex]sin^2(\theta)=2sin(\frac{\theta}{2})sin(\frac{\theta}{2})[/tex]
[tex]sin^2(\theta)=2(\sqrt{\frac{1-cos(\theta)}{2}})(\sqrt{\frac{1-cos(\theta)}{2}})[/tex]
[tex]sin^2(\theta)=2(\frac{1-cos(\theta)}{2})[/tex]
[tex]sin^2(\theta)=\frac{2-2cos(\theta)}{2}[/tex]
[tex]sin^2(\theta)=1-cos(\theta)[/tex]
[tex]1-cos^2(\theta)=1-cos(\theta)[/tex]
[tex]-cos^2(\theta)=-cos(\theta)[/tex]
[tex]cos^2(\theta)=cos(\theta)[/tex]
[tex]cos^2(\theta)-cos(\theta)=0[/tex]
[tex]cos(\theta)[cos(\theta)-1]=0[/tex]
[tex]cos(\theta)=0[/tex]
[tex]\theta=\frac{\pi}{2},\frac{3\pi}{2}[/tex]
[tex]cos(\theta)-1=0[/tex]
[tex]cos(\theta)=1[/tex]
[tex]\theta=0[/tex]
Therefore, the solutions contained within the interval are [tex]\{0,\frac{\pi}{2},\frac{3\pi}{2}\}[/tex]
Helpful Tips:
Half-Angle Formula: [tex]sin(\frac{\theta}{2})=\pm\sqrt{\frac{1-cos(\theta)}{2}}[/tex]
Pythagorean Identity: [tex]sin^2(\theta)+cos^2(\theta)=1,sin^2(\theta)=1-cos^2(\theta),cos^2(\theta)=1-sin^2(\theta)[/tex]
[tex]\sin^2 \theta = 2 \sin^2 \left(\dfrac{\theta}2 \right)\\\\\implies \sin^2 \theta = 1- \cos 2 \cdot \dfrac{\theta}2\\\\\implies \sin^2 \theta = 1- \cos \theta \\\\\implies 1-\cos^2 \theta = 1 - \cos \theta \\\\\implies -\cos^2 \theta - \cos \theta = 0\\\\\implies \cos^2 \theta - \cos \theta = 0\\\\\implies \cos \theta( \cos \theta -1) = 0\\\\\\\text{Now,}\\\\\cos \theta = 0\\\\\implies \theta = n\pi + \dfrac{\pi}2\\\\\text{For n = 0,1 and}~ [0.2\pi)\\\\[/tex]
[tex]\theta = \dfrac{\pi}2, \dfrac{3\pi}2[/tex]
[tex]\text{Again,} \\\\\cos \theta -1= 0\\\\\implies \cos \theta = 1\\\\\implies \theta = 2n\pi\\\\\text{For n= 0 and}~ [0,2\pi)\\\\\theta = 0\\\\\text{Combine solutions,}\\\\\theta = 0, \dfrac{\pi}2, \dfrac{3\pi}2[/tex]
Related Questions
Solve the following exponential equation:
5^x=2
e^x=10
Answers
Number 1.
[tex]5^x=2[/tex], first we will take the [tex]log[/tex] of both of these numbers getting us,
[tex]x=\frac{log(2)}{log(5)}[/tex], which then gives us that [tex]x=0.4306777777....[/tex]
Number 2
[tex]e^x=10[/tex], solve for our exponent, [tex]2.718282^x=10[/tex], use the logarithmic formula, and we get,
[tex]x=\frac{log(10)}{log(2.718282)}[/tex], which then we get that [tex]x=2.302585[/tex]
CAN YOU GUYS HELP PLEASE, I HAVE A LIMIT ON THIS!!!!
Answers
Answer:
the 2nd choice
Step-by-step explanation:
hope this helps
may i get braineist pls
Answer:
A. or y=-1/3x-3
Step-by-step explanation:
Marsha is driving 630 total miles on a trip. She has already driven 2/5 of the distance. How far has Marsha driven?
Answers
Answer:
Marsha has driven 252 miles
Step-by-step explanation:
In order to figure out the problem, you take the total distance and multiply it by the fraction she has gone. This would be 630*(2/5) which would equal 252. Thus, Masha has driven 252 total miles.
Hope this helps!
Answer: 252 miles
Step-by-step explanation:
Which graph represents (x,y)(x,y)left parenthesis, x, comma, y, right parenthesis-pairs that make the equation y = 2xy=2xy, equals, 2, x true?
Choose 1 answer:
Choose 1 answer:
(Choice A)
A
(Choice B)
B
(Choice C)
C
Answers
Step-by-step explanation:
The graph has a slope of 0.5 and a y-intercept of 5
The equation is given as:
A linear equation is represented as:
Where m represents the slope, and c represents the y-intercept
So, by comparison:
m = 0.5
c = 5
This means that:
The slope is 0.5, and the y-intercept is 5
Hence, the graph of has a slope of 0.5, and a y-intercept of 5
See attachment for the graph of
A
i know its a because the other kids'answer is wrong.
After sitting in a refrigerator for a while, a turkey at a temperature of 34^\circ34
∘
F is placed on the counter and slowly warms closer to room temperature (67^\circ67
∘
F). Newton's Law of Heating explains that the temperature of the turkey will increase proportionally to the difference between the temperature of the turkey and the temperature of the room, as given by the formula below:
T=T_a+(T_0-T_a)e^{-kt}
T=T
a
+(T
0
−T
a
)e
−kt
T_a=T
a
= the temperature surrounding the object
T_0=T
0
= the initial temperature of the object
t=t= the time in minutes
T=T= the temperature of the object after tt minutes
k=k= decay constant
The turkey reaches the temperature of 45^\circ45
∘
F after 15 minutes. Using this information, find the value of kk, to the nearest thousandth. Use the resulting equation to determine the Fahrenheit temperature of the turkey, to the nearest degree, after 60 minutes.
Enter only the final temperature into the input box.
Answers
The temperature of the body after 60 minutes is 60.5° F.
Using the Newton's law of cooling;
T(t) =[tex]Ts + (To - Ts)e^-kt[/tex]
T(t) = temperature at time t
Ts = temperature of the surrounding
To = Temperature of object
t = time taken
k = cooling constant
Now;
[tex]T(t) -Ts=(To - Ts)e^-kt\\45 - 67 = (34 - 67)e^-15k\\-22 = -33e^-15k\\-22/-33 = e^-15k\\0.667 = e^-15k\\ln(0.667) = ln( e^-15k)\\-0.405 = -15k\\k = -0.405/ -15\\k = 0.027[/tex]
At t = 60 min
[tex]T(t) = Ts + (To - Ts)e^-ktT(60) = 67 + (34 - 67)e^-( 0.027*60)\\T(60) = \\67 + (-33)e^-( 0.027 * 60)\\T(60) = \\60.5 F[/tex]
T(60) = 60.5° F
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Answer:
T≈169
Step-by-step explanation:
Enter only the final temperature into the input box.
T_a = 67\hspace{50px}T_0=210
T
a
=67T
0
=210
\text{Write Formula:}
Write Formula:
T=67+(210-67)e^{-kt}
T=67+(210−67)e
−kt
Plug in givens
\color{blue}{T}=67+143e^{-k\color{green}{t}}
T=67+143e
−kt
Simplify
\text{Plug in given time and temperature:}
Plug in given time and temperature:
\color{blue}{193}=67+143e^{-k(\color{green}{1.5})}
193=67+143e
−k(1.5)
Simplify
\text{Solve for }k\text{:}
Solve for k:
193=
193=
\,\,67+143e^{-1.5k}
67+143e
−1.5k
-67\phantom{=}
−67=
\,\,-67
−67
126=
126=
\,\,143e^{-1.5k}
143e
−1.5k
\frac{126}{143}=
143
126
=
\,\,\frac{143e^{-1.5k}}{143}
143
143e
−1.5k
Divide by 143
0.8811189=
0.8811189=
\,\,e^{-1.5k}
e
−1.5k
\ln\left(0.8811189\right)=
ln(0.8811189)=
\,\,\ln\left(e^{-1.5k}\right)
ln(e
−1.5k
)
Take ln of both sides
\ln\left(0.8811189\right)=
ln(0.8811189)=
\,\,-1.5k
−1.5k
\frac{\ln\left(0.8811189\right)}{-1.5}=
−1.5
ln(0.8811189)
=
\,\,\frac{-1.5k}{-1.5}
−1.5
−1.5k
Divide by -1.5
0.0843751=
0.0843751=
\,\,k
k
k\approx
k≈
\,\,\color{blue}{0.084}
0.084
Round k to nearest thousandth
\text{Complete Formula:}
Complete Formula:
T=67+143e^{-\color{blue}{0.084}t}
T=67+143e
−0.084t
Plug in k
\text{Find temperature after }\color{green}{4}\text{ minutes:}
Find temperature after 4 minutes:
T=67+143e^{-0.084(\color{green}{4})}
T=67+143e
−0.084(4)
Plug in time
T=67+143e^{-0.336}
T=67+143e
−0.336
Multiply
T=169.1911041
T=169.1911041
Plug into the calculator
T\approx 169
T≈169
please help with this!!!!
Answers
Answer:
6311000
Step-by-step explanation:
The first thing we do is calculate the total amount of people in both states. 6.89 x10 raised to the 6th power is equal to 6890000. Then we multiply 5.79 by 10 to the fifth power, which equals to 579000. Then we subtract the numbers, and get 6311000. They want you to answer in standard notation, which is the regular way of writing numbers such as 10, instead of 10^1.
The difference in population
[tex] = (6.89 \times {10}^{6} ) - (5.79 \times {10}^{5} ) \\ = (6.89 \times 10 \times {10}^{5} ) - (5.79 \times {10}^{5} ) \\ = (68.9 \times {10}^{5} ) - (5.79 \times {10}^{5} ) \\ = (68.9 - 5.79) \times {10}^{5} \\ = 63.11 \times {10}^{5} [/tex]
Answer:
[tex]63.11 \times {10}^{5} [/tex]
Hope you could get an idea from here.
Doubt clarification - use comment section.
What is f(3) for the function f(x) = 5x + 7?
Answers
What is 2-b/3=-5/2
Please explain thoroughly and only explain the 2-step version
Answers
Multiply both sides of the equation
by 3
6 - b = -5
Move the constant to the right-hand side and change its sign
-b= -5 - 6
Calculate the difference
-b=-11
Change the signs
b=11
Answer:
b = 13.5
Step-by-step explanation:
2 -b/3 = -5/2
im assuming we want to find b. therefore we want b = something. so because b is on top of a fraction right now, I will multiply both sides by 3 so -b is not on top of a fraction anymore.
6 -b = -15/2
we should get rid of the divided by 2 on the other side too, to make it easier. we can do this by multiplying 2 on both sides.
12 - 2b = -15
remember, we want to isolate b. so first we should get rid of the 12 next to the 2b. we can do this by subtracting 12 on both sides!
-2b = -27
finally, we can divide both sides by -2 and b will be isolated!
b = 13.5
ask me if you have anymore questions!Measure the paper clip to the nearest 1/8 inch.
A. 1 2/8 inches
B. 1 1/2 inches
C. 1 4/8 inches
D. 1 3/8 inches
Answers
Answer:
1 3/8 D
Step-by-step explanation:
The first and most obvious part of the answer is that the paper clip is at least 1 inch long. The question is how much longer?
It is between 1/4 of an inch and 1/2 of an inch. In fact it is exactly half way between them.
So that resembles an average.
(1/4 + 1/2)/2 = (1/4 + 2/4) / 2 = (3/4)/2
[tex]\frac{\frac{3}{4} }{2} = \frac{\frac{3}{4} }{\frac{2}{1} }[/tex]
Now the trick is to invert the denominator (make it 1/2) and multiply
3/4 * 1/2 = 3/8
The area of Asia is approximately 4.46 × 107 square kilometers. Its population is approximately 3.70 × 109 people. What is the approximate population density (people per square kilometer) of Asia? Write your answer in standard form. If necessary, round your answer to the nearest hundredth.
- 120.54 people/km2
- 82.96 people/km2
- 8.30 x 10 (15th power) people/km2
- 8.30 x 10 (1st power) people/km2
Answers
Step-by-step explanation:
you have to be careful what you are typing.
the way you wrote the numbers Asia would be a tiny place with very few people.
but I think you meant to say :
area is 4.46 × 10⁷ km²
with 3.70 × 10⁹ people.
people / km² = 3.70/4.46 × 10⁹/10⁷ =
= 0.829596413... × 10² = 82.9596413... =
= 82.96 people/km²
The population density will be equal to 82.96 people/km2. The correct option is B.
What is an expression?Expression in maths is defined as the collection of numbers variables and functions by using signs like addition, subtraction, multiplication, and division.
Numbers (constants), variables, operations, functions, brackets, punctuation, and grouping can all be represented by mathematical symbols, which can also be used to indicate the logical syntax's order of operations and other features.
Given that the area is 4.46 × 10⁷ km² with 3.70 × 10⁹ people. The population density will be calculated as below:-
Density = people / km² = ( 3.70 × 10⁹ ) / ( 4.46 x 10⁷ )
Density = 0.829596413... × 10² = 82.9596413
Density = 82.96 people/km²
Therefore, the population density will be equal to 82.96 people/km2. The correct option is B.
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b) Find the square root of 66.4225.
Answers
Answer:8.15
Step-by-step explanation: 8.15
what is the length of FD? Need help do tomorrow!!!
Answers
Answer:
Step-by-step explanation:
The tenure of an FD can vary from 7, 15 or 45 days to 1.5 years and can be as high as 10 years. These investments are safer than Post Office Schemes as they are covered by the Deposit Insurance and Credit Guarantee Corporation (DICGC).
The length of FD in a right-angled triangle with EC as 18 units and CF as 26 units is 20.2 units.
The length of FD can be found using the Pythagorean Theorem. The Pythagorean Theorem states that in a right triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides. In this case, the hypotenuse is FD, and the other two sides are EC and CF.
EC = 18 units
CF = 26 units
Plugging these values into the Pythagorean Theorem, we get:
[tex]FD^2[/tex] = [tex]EC^2[/tex] + [tex]CF^2[/tex]
[tex]FD^2[/tex] = [tex]18^2[/tex] + [tex]26^2[/tex]
[tex]FD^2[/tex] = 324 + 676
[tex]FD^2[/tex] = 1000
FD = [tex]\sqrt{[/tex](1000)
FD = 10 * [tex]\sqrt{[/tex](10)
FD = 20.2 units
Therefore, the length of FD is 20.2 units.
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Correct Question:
In a right-angled triangle with EC as 18 units and CF as 26 units, what is the length of FD.
what is the eight term of the arithmetic sequence defined by the rule A(n)=-12+2(n-1) RIGHT ANSWER ONLY
Answers
Answer:
a₈ = 2
Step-by-step explanation:
Substitute n = 8 into the nth term rule
A(8) = - 12 + 2(8 - 1) = - 12 + 2(7) = - 12 + 14 = 2
One number is eight more than twice another. If their difference is 25, what is the larger
number?
35
42
17
Answers
Answer:
17
Step-by-step explanation:
Let the smaller number = x
Let the larger number = y
y = 2x + 8
y - x = 25 Add x to both sides
y = x + 25 Substitute this into the first equation
x + 25 = 2x + 8 Subtract 8 from both sides
x + 25 - 8 = 2x Combine
x + 17 = 2x Subtract x from both sides.
17 = 2x - x Combine
17 = x
Based on the graph, what is true about polygon ABCD?
a. The polygon is a hexagon
b. The area of the polygon is 96 square units
c. The perimeter is 20 unit
d. Side AD is congruent to side DC
Answers
Answer:
B
Step-by-step explanation:
The Width of the Rectangle (read Polygon) is 8 units and the Length of the Rectangle is 12 units.
The area of the polygon is 96 square units. Therefore, the correct answer is option C.
From the given graph, we have polygon ABCD.
From the graph, we have AB=DC=12 units and BC=AD=8 units.
Here, the given polygon is rectangle.
Area of a rectangle = Length×Breadth
Now, area of a rectangle = 12×8
= 96 square units
Perimeter of a rectangle = 2(Length+Breadth)
= 2(12+8)
= 2(20)
= 40 units
Therefore, the correct answer is option C.
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lost on this a little
Answers
Answer:
the answer would be 7/5
Step-by-step explanation:
2 of them would already make a whole, and if you add another one, it's going to be more than just one pitcher of lemonade.
2x +6y = 2
х= 8 - Зу
What is X and Y?
Answers
Answer:
no solutions exist
Step-by-step explanation:
We can solve this by using system of equations and using substitution.
2(8-3y)+6y=2 Substitute x=8-3y into the first equation.
16-6y+6y=2 Distribute.
16=2 Simplify.
16 does not equal 2, so there are no solutions.
is the following relation a function {(-6,-3),(-6,-1),(-6,5),(-6,8)
Answers
Answer:
Step-by-step explanation:
This relation (this group of points) is NOT a function. In order to be a function, every x-value (the first number in the pair) can only be paired up with one y-value (only one partner for the x)
In this relation, -6 is paired up with four different y-values (four different partners!)
If you plotted these points on a graph, you would see they are all lined up in a vertical line. If you have learned about the vertical line test, you will see that a vertical line goes through all four points. This set of points fails the vertical line test. To be a function, a vertical line can only touch the points one at a time.
To be a function, every x can only have one y partner.
Hope everyone has a happy holiday for whatever you celebrate :)
Answers
Answer:
Happy holidays to everyone!
Step-by-step explanation:
What is an equation of the line that passes through the points (-4, -5) and (-3, -2)
Answers
Answer: The equation of the line is y = 3/2 x + 1
Step-by-step explanation:
Slope 3/2; The Equation of the line passing through (-4, -5) having a slope of 3/2 is y + 5 = 3/2 (x + 4) or y = 3/2 times X + 6 - 5 or y = 3/2 times X + 1[Ans] graph {3/2*x+1[-10, 10, -5, 5]}
I hope this makes sense to you :)
1) Use the distributive property to write an expression that is equivalent to 8(a + 4).
8(a + 4) = _a + _
Answers
Distribute the 8 to each term in the parentheses.
Hello could I please get a straightforward answer I need to catch up pelsse
Answers
Answer:
Step-by-step explanation:
For the school play, tickets cost $9.50 for adults and $3 for kids under 12. What would be the total cost for 8 adult tickets and 21 kids tickets? What would be the total cost for aa adult tickets and kk kids tickets?
Answers
Answer:
$1399.5aa +3kkStep-by-step explanation:
a)8 adult tickets cost 8($9.50) = $76
21 kids tickets cost 21($3) = $63
The total cost of these tickets is $76 +63 = $139.
__
b)aa adult tickets cost 9.50aa
kk kids tickets cost 3.00kk
The total cost of these tickets is 9.50aa +3.00kk.
_____
Additional comment
We ordinarily expect a variable to be a single letter. This problem statement defines each variable using a double letter, so that is what we have used. (Sometimes Brainly doubles up variable content in copied text, so we suspect that may have happened here.)
Find an equation of the line having the given slope and containing the given point. Slope 7 through (6,3)
Answers
[tex](\stackrel{x_1}{6}~,~\stackrel{y_1}{3})~\hspace{10em} \stackrel{slope}{m}\implies 7 \\\\\\ \begin{array}{|c|ll} \cline{1-1} \textit{point-slope form}\\ \cline{1-1} \\ y-y_1=m(x-x_1) \\\\ \cline{1-1} \end{array}\implies y-\stackrel{y_1}{3}=\stackrel{m}{7}(x-\stackrel{x_1}{6}) \\\\\\ y-3=7x-42\implies y = 7x-39[/tex]
Answer:
[tex]y-3=7(x-6)[/tex]
Step-by-step explanation:
Pre-SolvingWe are given that a line has a slope (m) of 7 and passes through the point (6,3).
We want to write the equation of this line.
There are three ways to do it:
Slope-intercept form, which is y=mx+b where m is the slope and b is the value of y at the y-intercept.Point-slope form, which is [tex]y-y_1=m(x-x_1)[/tex] where m is the slope and [tex](x_1,y_1)[/tex] is a point. Standard form, which is ax+by=c where a, b, and c are free integer coefficients.As the question doesn't specify, either form should work, but let's write the equation in point-slope form as that's the easiest to do.
SolvingAs we are already given both the point and the slope, we can plug them both into the equation.
Starting with the slope, substitute 7 for m.
[tex]y-y_1=7(x-x_1)[/tex]
Now, substitute 6 as [tex]x_1[/tex] and 3 as [tex]y_1[/tex].
[tex]y-3=7(x-6)[/tex]
a dogs hair grew 2 1/2 inches in 12 1/2weeks how long will the hair grow in 1 week?
Answers
Answer:
0.2 inches
Step-by-step explanation:
12.5 weeks = 2.5 inches
1 week = p inches
Cross multiply
12.5p = 2.5
Divide both sides by 12.5
p = 2.5 / 12.5
p = 0.2 inches
Hence, the length grown in 1 week is 0.2 inches
Hope I Helped You:)
What is the value of the expression? 75÷5−[(9−7)×3]×2
Answers
Answer:
3
Step-by-step explanation:
75÷5-[(9-7)×3]×2
do innermost inclusion symbols first, which would be 9-7
75÷5-[2×3]×2
now multiply 2 and 3
75÷5-(6)×2
now divide 75 by 5
15-6×2
now multiply
15-12
3
The required simplified value of the given expression is 13.
What is simplification?The process in mathematics to operate and interpret the function to make the function or expression simple or more understandable is called simplifying and the process is called simplification.
Here,
Given expression,
= 75÷5−[(9−7)×3]×2
Following the PEMDAS rule,
= 25 - [2×3]×2
= 25 -[6]×2
= 25 - 12
= 13
Thus, the required simplified value of the given expression is 13.
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I need help with this ASAP
Answers
Answer:
32+24=8(4+3)
20+8=4(5+2)
100+75=25(3+4)
49+28=7(7+4)
32+24=12(3+2) I don't know if this is right because 12 times 3 is 36 but that was the last option
Step-by-step explanation:
it's distributive property you multiply the numbers on the inside by the numbers on the outside.
The following stock prices reflect the closing price for the past five days. Find the 3 day SMA
(Simple Moving Average) for days 2-4.
$44.25, $44, $41.75, $40, $39.75
Answers
Answer:
solution
Step-by-step explanation: total=209.75
p/18+7=9 what is the value of p
Answers
Answer:
p=36
Step-by-step explanation:
Rearrange the equation by subtracting what is to the right of the equal sign from both sides of the equation:
p/18+7-(9)=0
Simplify p/18
(p/18+7)-9=0
7= 7/1= 7*18/18
p+7*18/18=p+126/18
(p+126)/18-9=0
9=9/1=9*18/18
(p+126)-(9*18)/18=p-36/18
p-36/18=0
p-36=0
p=36
Hope this helped :)
Can you please help me with number 15
Answers
Answer:
142.25 miles
Step-by-step explanation:
239- 96.75
142.25