Answers
Answer: see proof below
Step-by-step explanation:
Use the following Sum/Difference Identities:
sin(A + B) = sin A · cos B + sin B · cos A
sin(A - B) = sin A · cos B - sin B · cos A
Use the following Half-Angle Identities:
[tex]\sin\bigg(\dfrac{\theta}{2}\bigg)=\dfrac{\sqrt{1-\cos \theta}}{\sqrt2}\\\\\\\cos\bigg(\dfrac{\theta}{2}\bigg)=\dfrac{\sqrt{1+\cos \theta}}{\sqrt2}[/tex]
[tex]\text{Use the Unit circle to evaluate:}\ \cos\dfrac{\pi}{4} = \sin\dfrac{\pi}{4} = \dfrac{\sqrt2}{2}[/tex]
Use the following side work:
[tex]\sin\bigg(\dfrac{\pi}{8}\bigg)=\sin\bigg(\dfrac{\frac{\pi}{4}}{2}\bigg)=\dfrac{\sqrt{1-\cos \frac{\pi}{4}}}{\sqrt2}=\dfrac{\sqrt{2-\sqrt2}}{2}\\\\\\\cos\bigg(\dfrac{\pi}{8}\bigg)=\cos\bigg(\dfrac{\frac{\pi}{4}}{2}\bigg)=\dfrac{\sqrt{1+\cos \frac{\pi}{4}}}{\sqrt2}=\dfrac{\sqrt{2+\sqrt2}}{2}[/tex]
Proof LHS → RHS
[tex]\text{LHS:}\qquad \qquad \qquad \qquad \qquad \sin^2\bigg(\dfrac{\pi}{8}+\dfrac{A}{2}\bigg)-\sin^2\bigg(\dfrac{\pi}{8}-\dfrac{A}{2}\bigg)\\\\\text{Sum/Difference Identity:}\qquad \bigg(\sin\dfrac{\pi}{8}\cdot \cos \dfrac{A}{2}+\sin \dfrac{A}{2}\cdot \cos \dfrac{\pi}{8}\bigg)^2\\\\.\qquad \qquad \qquad\qquad \qquad \quad -\bigg(\sin\dfrac{\pi}{8}\cdot \cos \dfrac{A}{2}-\sin \dfrac{A}{2}\cdot \cos \dfrac{\pi}{8}\bigg)^2[/tex]
[tex]\text{Expand and Simplify:}\qquad \quad 4\sin \dfrac{\pi}{8}\cdot \cos\dfrac{A}{2}\cdot \sin \dfrac{A}{2}\cdot \cos \dfrac{A}{2}\\\\\\\text{Substitute:}\qquad \qquad \qquad 4\bigg(\dfrac{\sqrt{2-\sqrt2}}{2}\bigg)\cdot \cos \dfrac{A}{2}\cdot \sin \dfrac{A}{2}\bigg(\dfrac{\sqrt{2+\sqrt2}}{2}\bigg)\\\\\\\text{Simplify:}\qquad \qquad \qquad \sqrt2\cos \bigg(\dfrac{A}{2}\bigg)\cdot \sin \bigg(\dfrac{A}{2}\bigg)[/tex]
[tex]\text{Half-Angle Identity:} \quad \sqrt2\bigg(\dfrac{\sqrt{1+\cos A}}{\sqrt2}\bigg)\bigg(\dfrac{\sqrt{1-\cos A}}{\sqrt2}\bigg)\\\\\\\text{Simplify:}\qquad \qquad \qquad \dfrac{\sqrt{1-\cos^2 A}}{\sqrt2}\\\\\\.\qquad \qquad \qquad \qquad =\dfrac{\sqrt{\sin^2 A}}{\sqrt2}\\\\\\.\qquad \qquad \qquad \qquad =\dfrac{\sin A}{\sqrt2}[/tex]
[tex]\text{LHS = RHS:}\quad \dfrac{\sin A}{\sqrt2} = \dfrac{\sin A}{\sqrt2}\quad \checkmark[/tex]
Related Questions
h - 20/j = k solve for j
Answers
Answer: j = 20/k-h
Step-by-step explanation: Multiply to remove your variable from your denominator.
I hope this helps you out.
i need help with this math question please :)
Answers
Answer: 9/16
Step-by-step explanation:
Answer: 9/16
Step-by-step explanation: (1/4)^2 turns into 1/16. 5/8 - 1/16 so make them have the same denominator. 5/8 x 2 = 10/16 - 1/16 = 9/16
Suppose statement p is true but statement q is false. Which of these compound statements are true? Check all that
apply.
p ^ q
p V q
p -> q
q -> p
p<-> q
q <-> q
Answers
Answer:
b and d
Step-by-step explanation:
Answer:
Your answer would be p V q and q->p. Hope I helped :D
The number that has the exponent over it is called the what?
Answers
Answer: An expression that represents repeated multiplication of the same factor is called a power .
I hope this is the answer you were looking for if it's not I'm really sorry:(
can you please help me
Answers
Answer:
12
Step-by-step explanation:
Write an algebraic expression for 1 more than the quotient of n and 2
Answers
Answer:
1 + n/2 or n/2 + 1
Answer:2/n
Explanation:
Quotient just means "divide", so this would just be equal to
2/n
Hope this helps!
find the equation of the line containing the point (-3,5) and having slope: 4
a) y=4x-7
b) y=4x
c) y=-4-24
d) y=4x+17
e) none of the above
Answers
Answer: d) y=4x+17
remember y=mx+b
good luck
Justin wants to cover the top of a circular swimming pool to keep insects out of the pool. If the diameter of the pool is 20 feet, what will be the area of the top of the cover (the portion of the cover that covers the surface of the pool)?
Answers
Answer: The area is approximately 314 ft^2
Step-by-step explanation:
Well if the diameter is 20 feet then the radius will be 10 because the radius is half the diameter, and to find the area of a circle, you use the formula
A= [tex]\pi r^{2}[/tex] where A is the area and pi times the radius squared.
We will represent pi by 3.14
A= 3.14* 10^2
A= 3.14 * 100
A = 314
What’s the answer for number 9
Answers
Steven earns extra money babysitting. He charges $33.75 for 5 hours and $54.00 for 8
hours.
Enter an equation to represent the relationship. Let x represent the number of hours
Steven babysits and y represent the amount he charges.
The equation is y =
Answers
Answer:
y = 6.75x
Step-by-step explanation:
These 2 pieces of information can be used to create 2 points on the graph, where x is the number of hours and y is the amount he charges:
(5, 33.75) and (8, 54) are the 2 points
Use rise/run (y2 - y1) / (x2 - x1) to find the slope
(54 - 33.75) / (8 - 5)
20.25/3
= 6.75
Now, plug this and another point into y = mx + b to find b, the y intercept
54 = 6.75(8) + b
54 = 54 + b
0 = b
So, the equation is y = 6.75x
I need help with this geometry question.
Answers
Answer:
Step-by-step explanation:
a). There are 8 points in the figure attached.
b). There are 9 lines in the given figure.
c). There are 5 planes in the figure attached.
d). Three collinear points are D,G and F.
e). Four co-planar points are G, F, H and C.
f). Intersection of planes ABC and ABE is the common line AB.
g). Intersection of planes BCH and DEF is the common line EF.
h). Intersection of AD and DF is a point D.
a florist sells bouquets that use Rose's and carnations in a ratio of 2 to 7. she receives a shipment of 343 carnations. how many Rose's will the florist need?
Answers
Answer: 98 roses
Step-by-step explanation: 2:7 is like x:343
343 divided by 7 is 49, so that is what you multiply 2 by to get the roses. 49 x 2 = 98
Answer:
343÷7×2=98
98 roses
The ratio of flowers is 7 to 2. The answer is question 98
What is the speed of a train that travels 52 kilometers in 3 hours?
Answers
Answer:
17 km/h
Step-by-step explanation:
52:3=17.3≈17 km/hour
Answer:
17 km/hr
Step-by-step explanation:
Speed=
[tex] \frac{distance}{time} [/tex]
so, since the units are the same, kilometer per hour and hour, we simply divide
that's
[tex] \frac{52}{3} [/tex]
and that's 17.333 approximately 17km/hr
Charlie and Clare are playing a number-guessing game. Charlie picked two numbers between 1 and 5. To win the game, Clare must guess both his numbers in three tries. Her guesses, simulated using a random-number generator, are shown in the table. If Charlie’s numbers are 1 and 3, what is the experimental probability that Clare won?
Answers
The experimental probability is 3/10 that Clare won if Charlie's numbers were 1 and 3
What is probability?The probability is defined as the possibility of an event is equal to the ratio of the number of favorable outcomes and the total number of outcomes.
Charlie picked two numbers between 1 and 5
So the numbers should be 2, 3, and 4
There is the number of favorable outcomes is 3
To win the game, Clare must guess both his numbers in three tries
The experimental probability that Clare won if Charlie's numbers were 1 and 3 = P(E)
There are 10 trials given
There are total number of possible outcomes is 10
So the required experimental probability P(E) = favorable outcomes/total outcomes
⇒ P(E) = 3/10
Hence, the experimental probability is 3/10 that Clare won if Charlie's numbers were 1 and 3.
Learn more about the probability here:
brainly.com/question/11234923
#SPJ5
Answer: 3/10
Step-by-step explanation:
a town Pharmacy in hospital is doing it as a part of format of the town has a scale 18840.35 miles what is the distance between the pharmacy in the hospital on the map in inches pharmacy in hospital is doing it as a part it’s a map of the town has a scale where 1inch = 0.25 miles what is the distance between the pharmacy in the hospital on the map in inches
Answers
Answer:34344e4365655.6
Step-by-step explanation:
Write the phrase as an algebraic expression. A number less than eight and two thirds HELP ME
Answers
The picture is the answer
The algebric expression of the given phrase will be x < 8 + 2/3.
What is an expression?A mixture of variables, numbers, addition, subtraction, multiplication, and division are called expressions.
A statement expressing the equality of two mathematical expressions is known as an equation.
As per the given phrase,
A number less than eight and two thirds
Suppose that number is x then,
x < 8 + 2/3
Hence "The algebric expression of the given phrase will be x < 8 + 2/3".
To learn more about expression,
https://brainly.com/question/14083225
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Given that 110ₓ = 40₅ , find the value of x.
Answers
Answer:
is therr more to that question
Step-by-step explanation:
Answer:
x = 4
Step-by-step explanation:
110ₓ = 40₅ --------- change all to base ten.
the system called binary
1x²+1x ¹+0x⁰ = 45¹+0*5⁰
x² + x + 0 = 20
x² + x – 20 = 0
solve x using Quadratic equation
-1 ± [tex]\sqrt{1^2 - 4 * 1 * (-20)}[/tex]
x = ------------------------------------
2 * 1
x = -5; x = 4
therefore, x = 4
5xy – 3z when x = -2, y = 4, z = 6
Answers
Answer:
-58
Step-by-step explanation:
If we have the expression [tex]5xy-3z[/tex] and we know the values of x, y, and z, we can substitute inside the equation to find it's value.
[tex]5\cdot-2\cdot4 - 3\cdot6\\\\-40 - 18\\\\-58[/tex]
Hope this helped!
Answer: -58
Step-by-step explanation:
Fill it in to get (5)(-2)(4) - (3)(6) which is -40 - 18 = -58
The formula for average speed when traveling a distance of ddd over a time ttt is \dfrac dt
t
d
start fraction, d, divided by, t, end fraction.
Jeff and Cameron are arguing about which one of them is faster. Jeff says "I can run 777 kilometers per hour!" and Cameron says "I can run 100100100 meters per minute!"
Answers
Answer:
Jeff is faster
Step-by-step explanation:
kilometers are faster than meters
and 7 kilometers is 700 meters
so 100 meters isn't fast enough compared to Jeff's speed
Jeff is faster than Cameron as his average speed is more than that of Cameron's average speed.
How to calculate the average speed of a body?The average speed of a body is the ratio of the distance covered by the body to the time it takes to cover that distance.
How to solve the given question?In the question, we are given that Jeff says that he can run 7 kilometers per hour while Cameron says that he can run 100 meters per minute.
We are asked to tell who is the faster among them.
We need to know that, 1 kilometer = 1000 meters, and 1 hour = 60 minutes.
Jeff's average speed = 7 km/hr. = 7km/1hr = (7*1000) m / (1*60) min
or, Jeff's average speed = 7000/60 meters per minute = 116.67 meters per minute.
We are given that Cameron's average speed is 100 meters per minute.
So, we can say that Jeff is faster than Cameron, as
Jeff's average speed > Cameron's average speed.
∴ Jeff is faster than Cameron as his average speed is more than that of Cameron's average speed.
Learn more about the average speed of a body at
https://brainly.com/question/4931057
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If x-2=2^(2/3) + 2^(1/3), show that x3 - 6x3 + 6x-2=0
Answers
so x - 2 = 2^(2/3) + 2^(1/3)
=(x-2)^3 = [2^(2/3)+2^(1/3)]^3
use (a-b)^3= a^3 - b^3 - 3ab(a-b)
and (a+b)^3= a^3 + b^3 + 3ab(a+b) formulae.
or x^3 - 8 - 6x(x-2) = 2^2 + 2^1 + 3*[2^{(2/3)+(1/3)}[2^(2/3)+2^(1/3)
or x^3 - 8 - 6x^2 + 12x = 4 + 2 + 6(x-2)
or x^3 - 8 -6x^2 + 12x = 6 + 6x - 12
or x^3 - 6x^2 +6x = 2
If G is a 3 x 4 matrix and H is a 4 x 3 matrix, what is the dimension of GH?
Answers
3. Find the midpoint between the following points. (-2,-5) and (-3,1)
Answers
Answer:
Step-by-step explanation:
(x₁, y₁) = (-2 , -5) & (x₂ , y₂) = (-3 , 1)
Midpoint = [tex](\frac{x_{1}+x_{2}}{2},\frac{y_{1}+y_{2}}{2})[/tex]
[tex]=(\frac{-2 + [-3]}{2} ,\frac{-5 + 1}{2})\\\\\\=(\frac{-5}{2} , \frac{-4}{2})\\\\=(-2.5, -2)[/tex]
The width of a rectangle is 2 units shorter than half the length x write an expression that represents the perimeter of the rectangle
Answers
Answer:
3x-4
Step-by-step explanation:
Length = x
width = 2 units shorter than half the length x
width = x/2 - 2
For rectangle
perimeter = 2(length + width)
Perimeter for given rectangle = 2(length + width)
Perimeter for given rectangle = 2(x + x/2 - 2) = 2x + 2x/2 -4
Perimeter for given rectangle = 2x+x -4 = 3x -4
Expression that represents the perimeter of the rectangle is 3x-4.
a box shaped like a rectangular prism has a length of 8 inches and a width of 8 inches. its volume is 512 cubic inches. what is the height of the box
Answers
Answer:
8 inches
Step-by-step explanation:
the volume is length x width x height
so the equation would be 8 x 8 x (height) = 512
8x8 is 64, and you divide 512 by 64 which is 8
Answer:
16
Step-by-step explanation:
How are operations with variables like operations with real numbers?
Answers
Answer:
The relation between inputs and outputs is given by an equation called the equation of function. Since the outputs are also real values, hence, reasoning by analogy, you may use the arithmetical operation between functions (addition, subtraction, multiplication, division) to produce new functions.
A man weighing 340 pounds is admitted into a hospital for a special diet to lose no more than 6 pounds per week. What is the shortest time in weeks it could take the man to reduce his weight to 243 pounds?
Answers
Answer:
Step-by-step explanation:
340-243 =97
97/6 = 16
16 weeks to lose 6 pounds per week.
Answer:
Let's think.
If we want the man to be in the best case scenario in which he loses the most pounds that his limit lets him, we should always make the man lose the highest he could go: 6 pounds.
This is just a simple multi step problem if you break it down.
If his goal is to get to 243 pounds, we firstly subtract 243 from 340 to see how much he needs to lose.
340 - 243 = 97
The man wants to lose 97 pounds.
Let's divide this number by 6 with remainder.
I'll let you think for a bit.
97 / 6 = 16 remainder 1.
This means that in 16 weeks, he needs to lose 6 pounds while on the 17th week, he needs to lose 1 pound.
I'm sorry if i misunderstood.
Good luck Mate :)
Don't forget to try everyday !
The table shows the time Merrida spent driving and the number of miles she drove. She drove the same number of miles each hour. How many miles did she travel each hour? • 23 miles per hour • 35 miles per hour • 42 miles per hour • 53 miles per hour
Answers
Answer:
35 miles per hour
Step-by-step explanation:
175 -140 = 35
35/1 = 35
Answer:
35
Step-by-step explanation:
A chef is going to use a mixture of two brands of Italian dressing. The first brand contains 8% vinegar, and the second brand contains 13% vinegar. The chef wants to make 270 milliliters of a dressing that is 9% vinegar. How much of each brand should she use
Answers
Answer:
Amount of 8% vinegar = 216 mL
Amount of 13% vinegar = 54 mL
Step-by-step explanation:
First, of all, let x be the amount of 8% vinegar used
Since, the scientist needs 270 mL of vinegar, then;
Let 270 - x be the amount of 13% vinegar used
Now, we are told that she needs 270 mL of a 9% vinegar.
This means that;
8%x + 13%(270 - x) = 9%(270)
Simplifying this gives;
0.08x + 0.13(270 - x) = 0.09 × 270
Expanding the bracket gives;
0.08x + 35.1 - 0.13x = 24.3
0.13x - 0.08x = 35.1 - 24.3
0.05x = 10.8
x = 10.8/0.05
x = 216 mL
Thus, she will need to use 216 mL of the 8% vinegar.
Since 270 - x is the amount of 13% vinegar used , then;
the amount of 13% vinegar = 270 - 216 = 54 mL
43. Can you place ten lumps of sugar in three empty cups so that
there is an odd number of lumps in each cup?
Answers
Answer: No.
Step-by-step explanation:
We have 10 lumps of sugar, and we want to divide them into 3 cups, in such a way that there is an odd number of lumps in each cup.
This only can happen if we have 3 odd numbers such that the addition is equal to 10.
Now 10 is an even number, remember that even numbers can be written as:
2*k
where k is an integer number.
And odd numbers can be written as:
2*n + 1
where n is an integer.
Then we have 3 odd numbers, let's call them:
(2*n + 1), (2*k + 1) and (2*p + 1).
Now let's add them:
(2*n + 1) + (2*k + 1) + (2*p + 1).
2*(n + k + p) + 1 + 1+ 1
2*(n + k + p) + 2 + 1 =
2*(n + k + p + 1) + 1.
Now, the number n + k + p + 1 is an integer number, let's call it X, then we have that the addition of the 3 odd numbers is:
2*X + 1
This is an odd number
So for any 3 odd numbers that we add together, the result will always be an odd number.
Then is impossible to add 3 odd numbers and get 10 as the result (Again, 10 is an even number).
Then is not possible to have an odd number of lumps in each cup.
Solve the following system of equations using the elimination method.
6x + 7y = 10
–8x – 14y = –4
Question 3 options:
A)
(3,–8)
B)
(4,–2)
C)
(–4,6)
D)
(7,–3)
Answers
Answer:
Step-by-step explanation:
12x + 14y = 20
-8x - 14y = -4
4x = 16
x = 4
6(4) + 7y = 10
24 + 7y = 10
7y = -14
y = -2
(4, -2)
the solution is B