Consider the following.
sin(x) + sin(3x) = 4 sin(x) cos2(x)
Prove the identity.
Answers
Proved the identity sin(x) + sin(3x) = 4sin(x) cos²(x)
Given,
sin(x) + sin(3x) = 4 sin(x) cos²(x)
We have to prove the above given identity:
So,
sin(x) + sin(3x)
sin(x) + sin(2x + x)
By using the identity sin (A + B) we get,
sin(x) + sin (2x) cos(x) + cos (2x) sin(x)
Now use the identities:
sin 2x and cos 2x
We get,
sin(x) + 2sin(x) cos(x) cos(x) + (2cos²(x) - 1) sin(x)
Then,
sin(x) + 2sin(x) cos²(x) + 2sin(x) cos²(x) - sin(x)
We get,
4 sin(x) cos²(x)
Hence proved the identity sin(x) + sin(3x) = 4sin(x) cos²(x)
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Related Questions
determine the measure of angle j.
Answers
The measure of the value of ∠j is 10°.
What are a few properties of a triangle considering its angles?
A triangle's outer angle is the sum of its internal angles' two opposite sides.
The sum of two linear angles is supplementary.
Given, the three angles of the triangle are measured as (5x - 3)°,
(4x - 2)° and j°.
Also, the exterior angle of the triangle is (10x - 20)°.
Following the literature established above, considering that a triangle's outer angle is the product of its internal angles' two opposite sides, we have: (5x - 3) + (4x - 2) = 10x - 20 ⇒ 9x - 5 = 10x - 20 ⇒ 10x - 9x = 20 - 5
⇒ x = 15 --(i)
Again, since the sum of two linear angles is supplementary, we have using (i):
j + (10x - 20) = 180 ⇒ j + 150 - 20 = 180 ⇒ j = 10°
Therefore, the measure of the value of ∠j is 10°.
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Triangle TAB has a perimeter of 40 cmeters. Could the measures of the sides as shown actually represent the measures of the sides of the triangle? Justify your answer.
Answers
EXPLANATION
We can apply the Obtuse Triangle Theorem to check if the the measures of the sides appropiately represents the measures of the triangle:
Perimeter = 40 cm = 4x + 2x + 2 + x + 3
Adding like terms:
40 = 7x + 5
Subtracting 5 to both sides:
40 - 5 = 7x
Adding like terms:
35 = 7x
Dividing both sides by 7:
35/7 = x
Simplifying:
[tex]5=x[/tex]Then, by the obtuse triangle theorem, the following relationship should be fulfilled.
URGENT DUE TODAY!!!!!
49. Which of the following is the solution of
0.125x +1 -0.25x < -3?
a. x < -0.5
b. x < 0.5
C. x > 0.5
d. x < 32
e. x > 32
Answers
The solution of 0.125x +1 -0.25x < -3 is x > 32
Difference between equality and inequality equations
Both mathematical phrases, equations and inequalities, are created by connecting two expressions.The equal sign (=) indicates that two expressions in an equation are believed to be equivalent. The symbols show that the two expressions in an inequality are not always equal: >, <, ≤ or ≥. Or in simple words the equation which has '=' sign is an equality equation while the inequality equation has the signs are >, <, ≤ or ≥.
The given expression is,
0.125x +1 - 0.25x < -3
subract the x value,
0.125x - 0.25x + 1 < -3
-0.125x + 1 < -3
-0.125x < -3 - 1
-0.125x < -4
Now, if on both the sides the signs are -ve then change the inequality sign, like this
x > 4 / 0.125
x > 32
Hence, The solution of 0.125x +1 -0.25x < -3 is x > 32
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A candy box is made from a piece of cardboard that measures by inches. Squares of equal size will be cut out of each corner. The sides will then be folded up to form a rectangular box. What size square should be cut from each corner to obtain maximum volume?.
Answers
Using the concept of Application of Derivatives(A.O.D),we got 4.79 inches will be the square size for making the volume of candy box maximum.
A candy box is made from a piece of a cardboard that measures 43 × 23 inches.
Let squares of equal size will be cut out of each corner with the measure of x inches.
Therefore, measures of each side of the candy box will become
Length = (43 - 2x)
Width = (23 - 2x)
Height = x
Now we have to calculate the value of x for which volume of the box should be maximum.
Volume (V) = Length×Width×Height
=>V = (43 -2x)×(23 - 2x)×(x)
=>V= [(43)×(23) - 46x - 86x + 4x²]×x
=>V= [989 - 132x + 4x²]×x
=>V= 4x³- 132x² + 989x
Now we find the derivative of V and equate it to 0
[tex]\frac{dV}{dx}[/tex]= [tex]12x^{2} -264x+989[/tex]=0
Now we get values of x by quadratic formula
x=(264±[tex]\sqrt{264^{2}-4.12.989 }[/tex] )/24
=>x=(264±[tex]\sqrt{69696-47472}[/tex])/24
=>x=(264+√22224)/24, and x=(264-√22224)/24
=>x=(264+149.07)/24 and x=(264-149.07)/24
=>x=17.212 and x=4.79
Now we test it by second derivative test for the maximum volume.
[tex]\frac{d^{n}V }{dx}[/tex]=24x-264
For x = 17.212
[tex]\frac{d^{n}V }{dx}[/tex]=(24×17.212)-264=413.088-264=149.088
This value is > 0 so volume will be minimum.
For x = 4.79
[tex]\frac{d^{n}V }{dx}[/tex]=(24×4.79)-264= -149.04
-149.04 < 0, so volume of the box will be maximum.
Therefore, for x = 4.79 inches volume of the box will be maximum.
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(Complete Question) is
A candy box is made from a piece of cardboard that measures 43 by 23 inches. Squares of equal size will be cut out from each corner. The sides will then be folded up so that to form a rectangular box. What size square should be cut from each corner so that to obtain maximum volume?
Which inequality statements are true? Select all that apply.-0.7<-0.40.5 -0.6-0.8<-0.70.6 > 0.4
Answers
Answer:
-0.7 < -0.4
0.5 > -0.6
-0.8 < -0.7
0.6 > 0.4
Explanation:
The symbol < means less than and the symbol > means greater than.
On the other hand, if we identify each number on a number line, we get:
Therefore, the true statements are:
-0.7 < -0.4
0.5 > -0.6
-0.8 < -0.7
0.6 > 0.4
Because, for example, -0.7 is on the left of -0.4 which means that -0.7 is less than -0.4.
0.5 in on the right of -0.6 which means that 0.5 is greater than -0.6.
Find the mean for the data set. 6, 14, 7, 4, 12, 8, 13, 4, 18, 14
Answers
Answer:
Concept:
Mean is just another name for average. To find the mean of a data set, add all the values together and divide by the number of values in the set. The result is your mean!
The values are given below as
[tex]6,14,7,4,12,8,13,4,18,14[/tex]The image below shows how to calculate the mean
By substituting values, we will have
[tex]\begin{gathered} \bar{x}=\frac{\sum ^{}_{n\mathop=0}x}{n} \\ n=10 \end{gathered}[/tex][tex]\begin{gathered} \bar{x}=\frac{6+14+7+4+12+8+13+4+18+14}{10} \\ \bar{x}=\frac{100}{10} \\ \bar{x}=10 \end{gathered}[/tex]Hence,
The mean = 10
To calculate the variance, we will use the formula below
[tex]^{}\sigma^2=\frac{\sum ^{\infty}_{n\mathop=0}(x-\bar{x})^2}{n}[/tex][tex]\begin{gathered} \sigma^2=\frac{\sum ^{\infty}_{n\mathop{=}0}(x-\bar{x})^2}{n} \\ (x-\bar{x})^2=(6-10)^2+(14-10)^2+(7-10)^2+(4-10)^2+(12-10)^2+(8-10)^2+(13-10)^2+(4-10)^2+(18-10)^2+(14-10)^2 \\ (x-\bar{x})^2=16+16+9+36+4+4+9+36+64+16 \\ (x-\bar{x})^2=210 \end{gathered}[/tex][tex]\begin{gathered} \sigma^2=\frac{\sum ^{\infty}_{n\mathop{=}0}(x-\bar{x})^2}{n} \\ \sigma^2=\frac{210}{10} \\ \sigma^2=\frac{210}{10} \\ \sigma^2=21 \end{gathered}[/tex]Hence
The variance = 21
To calculate the standard deviation,
[tex]\begin{gathered} \sigma=\sqrt[]{variance} \\ \sigma=\sqrt[]{21} \\ \sigma=4.58 \end{gathered}[/tex]Hence,
The standard deviation is = 4.58
Question 2 of 10
Multiply the following complex numbers:
(4-6) (6-8)
OA. -24-68i
OB. 72 +68i
O C. -24 +68/
OD. 72-68i
Answers
-24-68i is solution of complex number.
What is the best definition of a complex number?
Complex numbers are those that are expressed as a+ib, where a, b are actual numbers, and I is a fictitious number termed a "iota."Real and imaginary components are split into two separate numbers, which are referred to as complex numbers. The foundation of more complex mathematics, like algebra, are complex numbers. They have numerous practical applications, particularly in the fields of electronics and electromagnetism.(4-6i)(6-8i)
= 24-32i-36i+48i²
= 24 + -68i - 48
= -24-68i
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The complete question is -
Multiply the following complex numbers (4-6i)(6-8i)
2/7×7/10 reduced to the smallest fraction
Answers
Answer:
1/5
Step-by-step explanation:
I multiplied 2 * 7, which is 14. Then, I multiplied 7 * 10 which equals 70. Then, I divided 14/70 by 14. The answer is 1/5.
Evaluate cos 150° without using a calculator.Ο Α.√32B. 2O C. -1/2OD. -32
Answers
Note that:
[tex]\begin{gathered} cos(A+B)=cosAcosB-sinAsinB \\ cos(150^0)=cos(90+60) \end{gathered}[/tex]Applying the addition formula given above to cos 150:
[tex]\begin{gathered} cos(150)=cos(90)cos(60)-sin(90)sin(60) \\ \\ cos(150)=0(\frac{1}{2})-1(\frac{\sqrt{3}}{2}) \\ \\ cos(150)=0-\frac{\sqrt{3}}{2} \\ \\ cos \end{gathered}[/tex]ashley’s house and the public Library are plotted on the coordinate plane as shown.What is the distance between Ashely’s house and the public library to the nearest tenth
Answers
The distance between Ashely’s house and the public library is of 7.8 miles.
What is the distance between two points?Suppose that we have two points with coordinates [tex](x_1,y_1)[/tex] and [tex](x_2,y_2)[/tex]. The shortest distance between these two points is given by the following rule:
[tex]D = \sqrt{(x_2-x_1)^2+(y_2-y_1)^2}[/tex]
This formula is derived from the Pythagorean Theorem, as the points form a right triangle in the Cartesian plane, with the hypotenuse representing the distance between them.
In the context of this problem, the coordinates of her house and of the library are given as follows:
House: (4,2).Library: (10,7).Hence the distance is calculated as follows:
[tex]D = \sqrt{(x_2-x_1)^2+(y_2-y_1)^2}[/tex]
[tex]D = \sqrt{(10 - 4)^2+(7 - 2)^2}[/tex]
D = sqrt(61)
D = 7.8 miles, as each unit on the plane represents one mile.
Missing informationThe problem states that each unit on the plane represents one mile and the image gives their position on the plane.
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Last year, Milan had $10,000 to invest. He invested some of it in an account that paid 9% simple interest per year, and he invested the rest in an account that
paid 7% simple interest per year. After one year, he received a total of $780 in interest. How much did he invest in each account?
Answers
Answer:
$4000 at 9% and $6000 at 7%Step-by-step explanation:
Let the amount invested at 9% be x.
Then the amount invested at 7% is 10000 - x.
The amount of interest after one year is $780.
Set up equation to represent this:
0.09x + 0.07(10000 - x) = 7800.09x + 700 - 0.07x = 7800.02x = 780 - 7000.02x = 80x = 80/0.02x = 4000Amount invested at 7% is:
10000 - 4000 = 6000Answer:
Milan invested:
$4,000 into the account earning 9% interest.$6,000 into the account earning 7% interest.Step-by-step explanation:
Given information:
Total amount invested = $10,000.Account A = 9% simple interest per year.Account B = 7% simple interest per year. Total interest earned after one year = $780.Let x be the amount invested in Account A.
Therefore, the amount invested in Account B is (10000 - x).
Simple Interest Formula
I = Prt
where:
I = Interest earned.P = Principal invested.r = Interest rate (in decimal form).t = Time (in years).Create two equations using the given information:
[tex]\begin{aligned}\textsf{Interest: Account A} &= x \cdot 0.09 \cdot 1\\& = 0.09x\end{aligned}[/tex]
[tex]\begin{aligned}\textsf{Interest: Account B} &= (10000 - x) \cdot 0.07 \cdot 1 \\& = 0.07(10000 - x)\\& = 700-0.07x\end{aligned}[/tex]
As the total interest earned was $780, set the sum of the two found equations to 780 and solve for x:
[tex]\begin{aligned}\implies 0.09x+700-0.07x&=780\\0.02x+700&=780\\0.02x&=80\\x&=4000\end{aligned}[/tex]
Therefore, Milan invested:
$4,000 into the account earning 9% interest.$6,000 into the account earning 7% interest.Write the sentence as an equation.
y decreased by 283 is equal to 48
Answers
Answer:
y - 283 = 48
Step-by-step explanation:
y decreased by 283 means that 283 is subtracted from y. All set equal to 48.
283 is being subtracted from y therefore y-283 is equal to 48
Hakeem is the oldest of three siblings who ages are consecutive integers. If the sum of their ages are 69, find Hakeem age
Answers
Answer:
24
Step-by-step explanation:
Hakeem is the oldest of three siblings who ages are consecutive integers. If the sum of their ages are 69, find Hakeem age.
It should be noted that:
= 22 + 23 + 24
= 69
Hakeem's age is 24
Ten bags of oranges were sold to a family of eight.each family member ate six oranges, what fraction of the oranges were eaten?
Answers
The fraction of oranges eaten by the family is 3/5
Given,
Number of oranges in a bag = 8 oranges
Number of bags sold to a family = 10 bags
Number of members in the family = 6 members
Number oranges ate by each member of the family = 6 oranges
We have to find the fraction of the oranges eaten by the family members;
Here,
8 oranges in 1 bag and 10 bags sold.
So, total number of oranges sold = 8 x 10 = 80 oranges
Each member ate 6 oranges and there are 8 members.
So, total number of oranges eaten by them = 6 x 8 = 48 oranges
Now,
The fraction of oranges eaten by the family = 48/80 = 6/10 = 3/5
That is,
3/5th of oranges were eaten by the family.
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if square one is the largest of the three squares in the model,which statement is true?
Answers
_______________
I'm reading your question
___________________
According to the Pythagorean theorem
Hypotenuse ^2 = Side 1 ^2 + Side 2 ^2
The hypotenuse is the case (Square 1 )
area of the sqaure= side of the square^2
________________________
That means the area of 1 is = area of square 2 + area fo square3 (J is false)
_______________________________
We are not sure about the relation between square 2 and 3 just the addition is the square 1 (F and H we have no certainty)
______________________________
G is true
______________________________
Answer
G
The image of the point (1, 1) under a translation is (0, 3). Find the coordinates of the image of the point (-4, "-1) under the same translation.
Answers
The image of the point (-4, -1) after the translation is:
(-3, 1)
How to identify the translation?
A translation:
Tᵃ'ᵇ applied to a point (x, y) gives:
Tᵃ'ᵇ(x, y) = (x + a, y + b).
In this case, we know that:
Tᵃᵇ(1, 1) = (1 + a, 1 + b) = (0, 3)
Then:
1 + a = 0
1 + b = 3
Solving these we get:
a = 1
b = 3 - 1 = 2
The translation is:
T¹'²
Applying this to the point (-4, -1) gives:
T¹'²(-4, -1) = (-4 + 1, -1 + 2) = (-3, 1)
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Which is the better buy? 2-gallon container of laundry detergent for $23.36. or 17-cup container of laundry detergent for $20.06
Answers
we gotta know the conversion between cups and gallons
We know
1 cup (US) = 0.0625 gallons
So, 2 gallon would be 2/0.0625 = 32 cups
So, the problem in cups is:
32 cups = $23.36
17 cups = $20.06
Definitely 32 cups for 23.36 is better buy
2 gallons for $23.36 is the better buy
Note:
23.36/32 = $0.73 per cup
20.06/17 = $1.18 per cup
The average price of a certain model of pickup truck in 1992 was $12,500. In 2013, the average price of the pickup truck was $22,000. What is the percentage increase in the average price of the pickup truck?
Answers
The average price in 1992 and 2013 was given. It is required to find the percentage increase in the average price over these years.
Recall that the formula for Percentage Increase is:
[tex]\text{percent increase}=\frac{\text{final price-initial price}}{\text{initial price}}\times100\%[/tex]Substitute initial price=12500 and final price= 22000 into the formula:
[tex]\begin{gathered} \text{percent increase}=\frac{22000-12500}{12500}\times100\%=\frac{9500}{12500}\times100\%=0.76\times100\% \\ \Rightarrow\text{percent increase}=76\% \end{gathered}[/tex]Hence, the percent increase in the average price is 76%.
Answer:
depends on whether or not the dealerships are scammers
Step-by-step explanation:
What is the solution to the inequality?
17<9+x
OA. x<8
OB. x>8
OC. x > 26
OD. x< 26
SUBMIT
Answers
Answer:
B) x > 8
Step-by-step explanation:
1-Step: Combine like terms by subtracting 9 on both sides : 17-9 < X
8 < X
2- Step: flip the signs (Since 8 is less than X, Then X is greater than X)
X > 8
3. 40 ounces of granola costs $6.52
a. How much does granola cost per ounce? (Give your answer to the nearest cent.) pleasee help
Answers
Answer: $0.2
Step-by-step explanation:
To find the answer you divide 6.52 by 40 which will get 0.163. To check the answer and make sure its correct you can multiply 0.163 by 40 to get 6.52 making the answer correct so 0.2 is the answer to the nearest cent. Hope this helps :)
Here is the answer again
Nick was thinking of a number. Nick adds 11 then divides by 11 to get an answer of 28. Form an equation with
x
from the information.
Answers
Answer:
??
Step-by-step explanation:
Which is greater 12.5 or |-13|?
Answers
From the question, we are asked to find which is great between 12.5 or |-13|
recall:
|-13|
The absolute value is the distance between a number and zero. The distance between -13 and 0 is 13.
The absolute symbol makes any number positive.
So, |-13| is the greater because |-13| = 13.
So, between 12.5 and |-13|, |-13| is greater since it is same as 13.
Factorise 4x^2+13x-15
Answers
The factors of the given quadratic equations are [tex]\frac{-13+\sqrt{409} }{8}[/tex] and [tex]\frac{-13-\sqrt{409} }{8}[/tex]
What is factorization?
When we try to write the given equation in terms of linear values from which we calculate the roots of the given equation then we say that we have factorized them. Root of a equation means a value which satisfies a equation.
We are given a quadratic equation
[tex]4x^2+13x-15[/tex]
We use quadratic formula to compute the factor of the equation
[tex]x=\frac{-b}{2a}[/tex]±[tex]\frac{\sqrt{b^2-4ac} }{2a}[/tex]
[tex]x=\frac{-13}{8}[/tex]±[tex]\frac{\sqrt{169+240} }{8}[/tex]
[tex]x=\frac{-13}{8}[/tex]±[tex]\frac{\sqrt{409} }{8}[/tex]
Therefore the factors are [tex]\frac{-13+\sqrt{409} }{8}[/tex] and [tex]\frac{-13-\sqrt{409} }{8}[/tex]
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Paul biked 7 miles on Saturday and 15 miles on Sunday. What was the total distance, in miles, Paul biked during those 2 days?
Answers
We have that:
Paul biked 7 miles on Saturday.
Paul bike 15 miles on Sunday.
The total distance, in miles, the Paul biked during Saturday and Sunday is:
[tex]T=7+15=22[/tex]Kala earns 45 dollars each week working part-time at a bookstore. She earns one additional dollar for each book that she sells.
Let A be the amount (in dollars) that Kala earns in a week if she sells
B books.
Write an equation relating to A to B Then use this equation to find the amount of money Kala earns if she sells 21 books.
Answers
The equation relating to A to B is A = 45 + b and the amount is $66.
What is an equation?An equation is used to show the relationship between the variables that are illustrated.
In this case, Kala earns 45 dollars each week working part-time at a bookstore and she earns one additional dollar for each book that she sells.
Let the books be represented as b. The equation will be:.
A = 45 + 1b
A = 45 + b
where A = amount earned.
The amount earned when 21 books are sold will be:
A = 45 + b
A = 45 + 21
A = 66
The amount is $66.
This illustrates the concept of equations.
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can you show how to solve this proplem
Answers
In the given figure : lines l and m are parallel and a, b are transversal
Since, the lines a and b are parallel and l act as a transversal.
Then,
[tex]\begin{gathered} \angle BAC=\angle ACb\text{ (alternate interior angles)} \\ \text{ Substitute the values} \\ 2x=46 \\ x=\frac{46}{2} \\ x=23 \end{gathered}[/tex]x = 23
Answer : x = 23
Which equation is true when x=10 but not when x= – 10? Select all that apply.
Answers
Applying power rules, the equations that is true when x = 10 but false when x = -10 is given as follows:
x³ = 10000.
Exponents of 10The behavior of the exponents of 10 is different for even and for odd exponents, as follows:
Even exponents generate even functions, that is, 10^n = (-10)^n if the exponent n is even.Odd exponents generate odd functions, that is 10^n = -(-10)^n if the exponent is odd.In this problem, we have two exponents, as follows:
Exponent 2 is even.Exponent 3 is odd.We want different results for the power when x = 10 and when x = -10, hence the correct option will involve the odd exponent 3.
The third power of 10 is calculated as follows:
10³ = 10 x 10 x 10 = 100 x 10 = 1000.
Hence the equation that is true for x = 10 and false for x = -10 is:
x³ = 10000.
As when x = -10, the result is:
(-10)³ = (-10) x (-10) x (-10) = 100 x (-10) = -1000.
Missing informationThe options are given by the image at the end of the answer.
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Write an explicit formula for an, the nth term of the sequence
64, -16, 4,....
Answers
The formula for the given arithmetic series for the term aₙ will be aₙ=a₁+(n-1)d.
What is arithmetic series?An arithmetic series is the sum of a sequence like 2,..., where each term is calculated by adding (or subtracting) a constant from the previous one. An ordered group of numbers with a shared difference between each succeeding term is known as an arithmetic sequence. For instance, the common difference in the arithmetic series 3, 9, 15, 21, and 27 is 6.So, the possible sequence of -64, -16, 4:
We can notice the sequence as -(4³), -(4²), 4, 4², 4³.So, the formula for the aₙ term will be:
aₙ = a₁ + (n - 1)dTherefore, the formula for the given arithmetic series for the term aₙ will be aₙ = a₁ + (n - 1)d.
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The correct question is given below:
Write an explicit formula for an, the nth term of the sequence
-64, -16, 4,....
What is 1 1/4 Divided by 10
Answers
Answer: 0.125
Step-by-step explanation:
1 1/4 is equal to 5/4
5/4 is equal to 1.25
then you would do 1.25 divided by 10
once you do that you would get your answer of 0.125!
Answer:
11/4÷10 =
11/4 ×1/10=
11/40=
0.275
Step-by-step explanation:
you can ask questions for clarification
a train increased its prices from £663 to £895.05 how much has this increased in a percentage
Answers
Answer:
The price has increased 35%
Step-by-step explanation:
GivenInitial price = £663,New price = £895.05.Find the percent increaseFind the amount of increase:
£895.05 - £663 = £232.05Find the percent value of the increase over the initial price:
232.05/663 × 100% = 35%