Answers
[tex]\\ \rm\rightarrowtail 2-sin^2x=2cos^2\left(\dfrac{x}{2}\right)[/tex]
[tex]\\ \rm\rightarrowtail 1+1-sin^2x=2cos^2\left(\dfrac{x}{2}\right)[/tex]
1-sin^2x=cos^2x[tex]\\ \rm\rightarrowtail 1+cos^2x=2cos^2\left(\dfrac{x}{2}\right)[/tex]
[tex]\\ \rm\rightarrowtail cos^2x=2cos^2\left(\dfrac{x}{2}\right)-1[/tex]
[tex]\\ \rm\rightarrowtail cos^2x=cosx[/tex]
[tex]\\ \rm\rightarrowtail cosx=1[/tex]
[tex]\\ \rm\rightarrowtail x=n\pi[/tex]
As x lies in between 0 and 2π
The answers are 0,2πAnswer:
[tex]x=0,\dfrac{\pi}{2}, \dfrac{3\pi}{2},2\pi[/tex]
Step-by-step explanation:
Given equation:
[tex]2-\sin^2x=2 \cos^2(\frac{x}{2})[/tex]
Subtract [tex]2 \cos^2(\frac{x}{2})[/tex] from both sides:
[tex]\implies 2-\sin^2x-2 \cos^2(\frac{x}{2})=0[/tex]
Pythagorean identity
[tex]\sin^2x+\cos^2x =1[/tex]
[tex]\implies \sin^2x =1-\cos^2x[/tex]
Substituting into the equation:
[tex]\implies 2-(1-\cos^2x)-2 \cos^2(\frac{x}{2})=0[/tex]
[tex]\implies 1+\cos^2x-2 \cos^2(\frac{x}{2})=0[/tex]
Cosine double angle identity:
[tex]\cos(2x)=2 \cos^2x-1[/tex]
[tex]\implies 2 \cos^2x=\cos(2x)+1[/tex]
Substituting [tex]\frac{x}{2}[/tex] for [tex]x[/tex]:
[tex]\implies 2 \cos^2(\frac{x}{2})=\cos(2\cdot\frac{x}{2})+1[/tex]
[tex]\implies 2 \cos^2(\frac{x}{2})=\cos x+1[/tex]
Substituting into the equation:
[tex]\implies 1+\cos^2x- (\cos x +1)=0[/tex]
[tex]\implies \cos^2x- \cos x =0[/tex]
Factoring:
[tex]\implies \cos x(\cos x- 1)=0[/tex]
Therefore,
[tex]\cos x=0[/tex]
[tex]\implies x=\dfrac{\pi}{2}, \dfrac{3\pi}{2}[/tex]
and [tex]\cos x-1=0[/tex]
[tex]\implies \cos x=1[/tex]
[tex]\implies x=0, 2\pi[/tex]
Related Questions
help with math pls its easy i think
Answers
State all translations for the following
f(x)=2(1/5)^x+4-8
Answers
Step-by-step explanation:
please mark me as brainlist please
Can someone help me on this question? I’ll give brainliest if you answer soon!!!
Answers
Answer:
I believe the answer would be 122 degrees due to them being a consecutive angle on the same transverse!
Answer:
68
Step-by-step explanation:
Knowing that;
∠EHD and ∠CHF are vertical angles
Since ∠EHD = 68 Thus, ∠CHF = 68
Look at picture below for more information;
~Learn with Lenvy~
what is the surface area of a square pyramid with side length 2 yd and slant height 4 yd.
Answers
Answer:
20 yd^2.
Step-by-step explanation:
Area of base = 2^2 = 4 yd^2.
It has 4 side faces so:
Area of each triangular side = 1/2 * 2 * 4 = 4 yd^2
Total surface area = 4 + 4(4).
= 20 yd^2.
please help all of my piont help me
pleeeeeeeeeeeeeeeeeeeeeeeeeaaaaaaaaaaaaaaaaaasse
Answers
Answer:
3) -82.4)-70.5) -15.6)-14.7)58)159)1610 )1611)6412)21.13)1500 in the middle (not above not below)14)450 mAnswer:
3) -82 4)-70 5) -15 6)-14 7)5 8)15 9)16 10)16 11)64 12)21 13)1500 in the middle 14)450
Step-by-step explanation:
#4 please help me ty if u do
Answers
-3/2
8x - 6 = 12x
- 6 = 4x
-1.5 = x
- 1.5 = -3/2 = x
The inverse of the logarithmic function f(x) = log0.5x is f−1(x) = 0.5x. what values of a, b, and c complete the table for the inverse function? x −2 −1 0 1 2 y 4 a b 0.5 c a = b = c =
Answers
The value of a, b and c can be calculated using the inverse function.The calculated the value of a= 2, b=1 and c= 0.25
What is an inverse function?The inverse function of a function is defined if f is a function that undoes the operation [tex]f^{-1}[/tex]
The inverse of f exists if and only if f is bijective, and if it exists, is denoted by [tex]f^{-1}[/tex] .
It is given that the function [tex]\rm F(x)=Log_{0.5}x[/tex] is and its inverse is
[tex]f^{-1}(x)=0.5^x[/tex] .
Substitute the value of x= -1 in a given inverse function and calculate the value of a.
[tex]f^{-1}(x)=0.5^x[/tex]
[tex]f^{-1}(-1)=0.5^{-1}[/tex]
[tex]f^{-1}(-1)=2[/tex]
Substitute the value of x is 0 in the given inverse function.
[tex]f^{-1}(x)=0.5^x[/tex]
[tex]f^{-1}(0)=0.5^0[/tex]
[tex]f^{-1}(0)=1[/tex]
Substitute the value of x is 2 in the given inverse function.
[tex]f^{-1}(x)=0.5^x[/tex]
[tex]f^{-1}(2)=0.5^2[/tex]
[tex]f^{-1}(2)=0.25[/tex]
Thus the value of a, b and c can be calculated using the inverse function.The calculated the value of a= 2, b=1 and c= 0.25
To know more about the Inverse function follow
https://brainly.com/question/20538141
Answer:
a = 2
b = 1
c = 0.25
Which graph shows the logarithmic function
f(x) = log0.5x ?
B.
Step-by-step explanation:
did it on ED :)
Mr. Gallagher purchased snacks at the theater. If
he handed the cashier a 20-dollar bill and received
$1.83 in change, what was his total?
Answers
Answer: $18.17
Step 1: Subtract 1 dollar.
$20 - $1 = $19
Step 2: Subtract 83 cents.
$19.00 - $0.83 = $18.17
Answer = $18.17
Divide 180kg in the ratio 5:3:2
Answers
Answer:
90 : 54 : 36
Step-by-step explanation:
Add the ratios: 5 + 3 + 2 = 10
Divide 180 kg by 10: 180 ÷ 10 = 18
Multiply each number of the ratio by 18:
5 × 18 : 3 × 18 : 2 × 18
⇒ 90 : 54 : 36
What is an right angle and abtuce angle
Answers
Answer:
An right angle is an angle which is 90 degrees and obtuse angle which is greater than 90 degrees. There’s also an acute angle which is shorter than 90 degrees.
Step-by-step explanation:
What is a right angle and obtuse angle?
Answer: A right angle is defined as a 90-degree angle.
Step-by-step explanation:
A right angle in trigonometry is an angle of approximately 90 degrees or /2 radians, which corresponds to a quarter turn. Right angles are formed when the endpoint of a ray is on a line and the adjacent angles are equal.
Answer: An angle that is greater than 90 degrees but less than 180 degrees is called an obtuse angle.
Step-by-step explanation:
The smaller angle is the obtuse angle. It is greater than 90 degrees but less than 180 degrees.
Which of the following is not a unit rate?
A. 18 horses to 7 cats
B. 2 teachers per bus
C. $20.00 for one hour
D. 4 boys per class
Answers
Which situation could be represented by the expression c−5?
A
Jovanni has 5 pieces of candy. He gave some candy to Elias.
B
Jill has some candy. Diego gave her 5 more pieces of candy.
C
Isabelle has 5 fewer pieces of candy than Destiny.
D
Derrick shared some candy among his 5 friend
Answers
Answer:
C
Step-by-step explanation:
candy = c. and Isabelle has five fewer meaning to subtract from the amount destiny has (c)
There are 36 people in a gym class. What fraction represents the number
of people that are running? *1/4 of the people are jumping *1/2 of the
people are lifting weights *the remaining are running
Answers
Answer:
9
Step-by-step explanation:
Jumping = 36/4 = 9
Lifting weights = 36/2 = 18
Running = 36-(9+18) = 9
Answer:
9 people
Step-by-step explanation:
First, we need to find out what fraction of the class is running. If 1/4 of the class is jumping, and 1/2 of the class are lifting weights, and the remaining people are running, this can be written like this. 4/4 - 1/4 - 2/4 = 1/4
1/4 of the class is running. Now we need to find out how many students is 1/4 of the class. To find this, we need to divide 36 by 4, which is 9. This means 9 people are running in gym.
I hope this helps :)
7. In an orchestra, the ratio of trombones to violas is 4:6. There are 9 violas. Write and solve a proportion to
determine how many trombones are in the orchestra.
Answers
Answer: 6 trombones
Step-by-step explanation:
We can set up a proportion.
Let x be the number of trombones there are.
[tex]\frac{4 trombones}{6violas} = \frac{x trombones}{9 violas}[/tex]
We can cross multiply:
4*9 = 6x
6x = 36
Divide both sides by 6:
x = 6
An experienced teaching assistant can grade all exams for a calculus class 2 hours faster than a new teaching assistant. Working together, the teaching assistants can grade all the exams for the calculus class in 7 hours. How long would it take for the new teaching assistant to grade all of the exams for the calculus class alone?
Answers
Answer:
15.07 hours
Step-by-step explanation:
The given relationship lets us write an equation in that uses rates in terms of tasks per hour. Let 'a' represent the number of hours it takes the assistant to grade exams alone. Then (a-2) is the number of hours it takes the experienced teacher. Working together, their total rate is ...
1/a + 1/(a-2) = 1/7 . . . . . . . tasks per hour
7(a -2) +7(a) = a(a -2) . . . . . multiply by 7a(a -2)
14a -14 = a^2 -2a . . . . . . . eliminate parentheses
a^2 -16a = -14 . . . . . . . . . subtract 14a
a^2 -16a +65 = 50 . . . . add 64 to complete the square
(a -8)^2 = 50 . . . . . . . write as a square
a -8 = ±5√2 . . . . . . take the square root
a = 8 +5√2 ≈ 15.072
It would take the teaching assistant about 15.07 hours to grade the exams alone.
_____
Additional comment
The quadratic also gives an extraneous solution of about 0.93 hours. This results in a negative task time for the experienced teacher. That solution makes no sense in the given scenario, so we ignored it.
How many solutions does an equation have? Is there always a solution?
Answers
BODMAS
Step-by-step explanation:
brackets,of,dividing, multiplication, subtraction
a tree in dante's neighborhood grew 18 inches in the first 2 years after it was planted. if the tree continues to grow at this same rate, which graph best represents the growth of the tree in inches per year ?
Answers
Answer:
The third one would be the option.
Step-by-step explanation:
If the tree is already at 18 inches on the 2nd year, then it would be at 9 inches on the first year, and at 27 on the third year. And so on.
Graph [3] represents the growth of the tree in inches per year.
What is direct proportionality?Direct proportion or direct variation is the relation between two quantities where the ratio of the two is equal to a constant value. Mathematically, direct proportionality is represented by a straight line and it equation is of the form -
y = kx
k = y/x
where [k] is the constant
Given is a tree in Dante's neighborhood grew 18 inches in the first 2 years after it was planted. The tree continues to grow at this same rate.
Since the tree grew 18 inches in the first 2 years and continues at the same rate, then it will grew by (18/2) or 9 inches per year. The equation that will represent this relation between growth [y] and time [x] is given by -
y = 9x
Graph number [3] represents the relation.
Therefore, graph [3] represents the growth of the tree in inches per year
To solve more questions on direct proportionality, visit the link below-
brainly.com/question/20400984
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Determine whether the statement is true or false . |-4|=|4|
True
False
Answers
Answer:
I thinks it's true but I'm not 100 %
Express this number in scientific notation.
0.0013?
Answers
Answer:
1.3*10^-3
Step-by-step explanation:
When using scientific notation, if you are moving to the left it is a positive exponent. If you are moving to the right it is a negative exponent. since we moved the decimal point to the right, the exponent will be negative.
Answer:
1.3x10^-3
Step-by-step explanation:
if you make the number bigger then you add a negative sign to your answer and since we moved the decimal 3 spaces to left to make it bigger, it is 10^-3
explain of how you got the work
Answers
Brianna's car
Step-by-step explanation:
Looking at the first equation the car decreases by 15% each year.
So our depreciation rate is 15%
The second equation represents an exponential depreciation with a depreciation rate given by;
1-0.9=0.1×100=10%
So our depreciation rate is 10%
Of course Brianna's car is decreasing in value quicker, since the car loses more value as the depreciation rate is higher.
There are 500 students at a middle school. Of these students, 350 will attend a
dance,
What percentage of the students at this school will attend the dance?
A. 15%
B.
30%
C. 35%
D. 70%
ОА
ОВ
OD
Answers
I did step by step math
Answer:
D
Step-by-step explanation:
If there are 500 students and only 350 are going then just divide 350 by 500. (350 / 500 = 0.7
PLEASE HELP WILL GIVE BRAINLIEST AND THANKS!!!
Answers
Answer:
Part A
1) Line
2) Angle
3) Point
4) Line Segment
5) Ray
6) Plane
Part B
1) Line Segment
2) Draw number 3 in part a times 2 with different letters
Some pls help me with number 6
Answers
Answer:
m∠CBD = m∠CDB ⇒ proved
Step-by-step explanation:
Let us solve the question
∵ AB ⊥ BD ⇒ given
→ That means m∠ABD = 90°
∴ m∠ABD = 90° ⇒ proved
∵ ED ⊥ BD ⇒ given
→ That means m∠EDB = 90°
∴ m∠EDB = 90° ⇒ proved
∵ ∠ABD and ∠EDB have the same measure 90°
∴ m∠ABD = m∠EDB ⇒ proved
∵ m∠ABD = m∠ABC + m∠CBD
∵ m∠EDB = m∠EDC + m∠CDB
→ Equate the two right sides
∴ m∠ABC + m∠CBD = m∠EDC + m∠CDB
∵ m∠ABC = m∠EDC ⇒ given
→ That means 1 angle on the left side = 1 angle on the right side, then
the other two angles must be equal in measures
∴ m∠CBD = m∠CDB ⇒ proved
→
Riley is building an ant farm for his school's science fair. He
has enough materials to build a farm that is 343 square inches
in total, 10 square inches of which will be taken up for food
and water sources. Of the remaining available space, each ant
needs at least 3 square inches to themselves. What is the
maximum amount of ants Riley's ant farm can house?
Answers
Answer:
111
Step-by-step explanation:
you need to subtract the amount of land taken up by the water and food sources and divide the rest by 3.
PLEASE HELP AND I WILL GIVE BRAINLIEST
A brand new grocery store sells cantaloupes for $3 each. Also, for their grand opening
celebration, each customer gets a $2 coupon towards their purchase as they enter the store.
Write a function equation to represent "x" as the number of cantaloupes purchased and "f(x)"
being the total cost in dollars for the cantaloupes including the coupon. *
(10 Points)
O a. F (x) = 2x + 3
O b. F (x) = 3x - 2
O c. F (x) = 2x - 3
O d. F (x) = 3x
Answers
Answer:
b
Step-by-step explanation:
every cantaloup costs $3 so they would pay x × $3
but they got a $2 coupon so you have to deduct that.
Mexicos flag has a height-to-length ratio is 4:7.If a Mexican flag is 28 feet long, what is its height?
Answers
Divide the length of the flag by its portion of the ratio, then multiply that by the ratio portion for the width:
28/7 = 4
4 x 4 = 16
The height is 16 feet
Answer:
16 feet
Step-by-step explanation:
If x:y=a:b, then xb=ya
4*7=x:28
4*28=7x
x=16
please explain throughly
Answers
Answer:
36≥ -6x≥18
6≥-x≥3
so x is
-3,-4,-5&-6
Hope This Helps!!!
Aubree invested $8,300 in an account paying an interest rate of 2.2% compounded quarterly. Assuming no deposits or withdrawals are made, how much money, to the nearest cent, would be in the account after 13 years?
Answers
Answer:
A≈11039.41
Step-by-step explanation:
Plug in values
A=8300(1.0055)^{52}
A=8300(1.0055)
52
Simplify
A=11039.4131948
A=11039.4131948
Round)
A≈11039.41
After a 13-year span, the money in the account would be $11.038.4.
As mentioned in the question,
Aubree invested $8,300 in an account paying an interest rate of 2.2% compounded quarterly.
Compound interest is the interest on deposits computed on both the initial principal and the interest earned over time.
Here,
principal(P) = 8300
rate(r) = 2.2% = 0.022
Time(a) = 13 years,
n = 4
Apply compound interest identity,
Amount = principal[1 + r/n]ⁿᵃ
Amount = 8300[ 1 + 0.022/4]⁴ˣ¹³
Amount = $11.038.4
Thus, After a 13-year span, the money in the account would be $11.038.4.
Learn more about compound interest here: https://brainly.com/question/14295570
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A school group visited a Festival of Lights. Each ticket cost $12 per person plus a parking fee of $50 per group. If the group paid a total of $890, how many people went?
Answers
Answer: 70 people went.
Step-by-step explanation:
12p + $50 = $890
890-50= 840
840/12= 70
Name the property that each statement illustrates. You buy a sandwich for $2.95, an apple for $.45, and a bottle of juice for $1.05. What is the total cost?
Answers
Answer:
The total cost is $4.45
Step-by-step explanation:
I just added them
Step-by-step explanation:
let a= cost of sandwich=2.95dollars
b=cost of apple= 0.45dollars
c= cost of juice= 1.05 dollars
then t= total cost is calulated as:
t=a+b+c
t=2.95+0.45+1.05
t=4.45