Please help me to prove this!​

Please Help Me To Prove This!

Answers

Answer 1

Answer:  see proof below

Step-by-step explanation:

Given: A + B + C = π              → A + B = π - C

                                              → B + C = π - A

                                              → C + A = π - B

                                              → C = π - (B +  C)

Use Sum to Product Identity:  cos A + cos B = 2 cos [(A + B)/2] · cos [(A - B)/2]

Use the Sum/Difference Identity: cos (A - B) = cos A · cos B + sin A · sin B

Use the Double Angle Identity: sin 2A = 2 sin A · cos A

Use the Cofunction Identity: cos (π/2 - A) = sin A

Proof LHS → Middle:

[tex]\text{LHS:}\qquad \qquad \cos \bigg(\dfrac{A}{2}\bigg)+\cos \bigg(\dfrac{B}{2}\bigg)+\cos \bigg(\dfrac{C}{2}\bigg)[/tex]

[tex]\text{Sum to Product:}\qquad 2\cos \bigg(\dfrac{\frac{A}{2}+\frac{B}{2}}{2}\bigg)\cdot \cos \bigg(\dfrac{\frac{A}{2}-\frac{B}{2}}{2}\bigg)+\cos \bigg(\dfrac{C}{2}\bigg)\\\\\\.\qquad \qquad \qquad \qquad =2\cos \bigg(\dfrac{A+B}{4}\bigg)\cdot \cos \bigg(\dfrac{A-B}{4}\bigg)+\cos \bigg(\dfrac{C}{2}\bigg)[/tex]

[tex]\text{Given:}\qquad \quad =2\cos \bigg(\dfrac{A+B}{4}\bigg)\cdot \cos \bigg(\dfrac{A-B}{4}\bigg)+\cos \bigg(\dfrac{\pi -(A+B)}{2}\bigg)[/tex]

[tex]\text{Sum/Difference:}\quad =2\cos \bigg(\dfrac{A+B}{4}\bigg)\cdot \cos \bigg(\dfrac{A-B}{4}\bigg)+\sin \bigg(\dfrac{A+B}{2}\bigg)[/tex]

[tex]\text{Double Angle:}\quad =2\cos \bigg(\dfrac{A+B}{4}\bigg)\cdot \cos \bigg(\dfrac{A-B}{4}\bigg)+\sin \bigg(\dfrac{2(A+B)}{2(2)}\bigg)\\\\\\.\qquad \qquad \qquad =2\cos \bigg(\dfrac{A+B}{4}\bigg)\cdot \cos \bigg(\dfrac{A-B}{4}\bigg)+2\sin \bigg(\dfrac{A+B}{4}\bigg)\cdot \cos \bigg(\dfrac{A+B}{4}\bigg)[/tex]

[tex]\text{Factor:}\quad =2\cos \bigg(\dfrac{A+B}{4}\bigg)\bigg[ \cos \bigg(\dfrac{A-B}{4}\bigg)+\sin \bigg(\dfrac{A+B}{4}\bigg)\bigg][/tex]

[tex]\text{Cofunction:}\quad =2\cos \bigg(\dfrac{A+B}{4}\bigg)\bigg[ \cos \bigg(\dfrac{A-B}{4}\bigg)+\cos \bigg(\dfrac{\pi}{2}-\dfrac{A+B}{4}\bigg)\bigg]\\\\\\.\qquad \qquad \qquad =2\cos \bigg(\dfrac{A+B}{4}\bigg)\cdot \cos \bigg(\dfrac{A-B}{4}\bigg)+\cos \bigg(\dfrac{2\pi-(A+B)}{4}\bigg)\bigg][/tex]

[tex]\text{Sum to Product:}\ 2\cos \bigg(\dfrac{A+B}{4}\bigg)\bigg[2 \cos \bigg(\dfrac{2\pi-2B}{2\cdot 4}\bigg)\cdot \cos \bigg(\dfrac{2A-2\pi}{2\cdot 4}\bigg)\bigg]\\\\\\.\qquad \qquad \qquad =4\cos \bigg(\dfrac{A+B}{4}\bigg)\cdot \cos \bigg(\dfrac{\pi-B}{4}\bigg)\cdot \cos \bigg(\dfrac{\pi -A}{4}\bigg)[/tex]

[tex]\text{Given:}\qquad \qquad 4\cos \bigg(\dfrac{\pi -C}{4}\bigg)\cdot \cos \bigg(\dfrac{\pi-B}{4}\bigg)\cdot \cos \bigg(\dfrac{\pi -A}{4}\bigg)\\\\\\.\qquad \qquad \qquad =4\cos \bigg(\dfrac{\pi -A}{4}\bigg)\cdot \cos \bigg(\dfrac{\pi-B}{4}\bigg)\cdot \cos \bigg(\dfrac{\pi -C}{4}\bigg)[/tex]

LHS = Middle [tex]\checkmark[/tex]

Proof Middle → RHS:

[tex]\text{Middle:}\qquad 4\cos \bigg(\dfrac{\pi -A}{4}\bigg)\cdot \cos \bigg(\dfrac{\pi-B}{4}\bigg)\cdot \cos \bigg(\dfrac{\pi -C}{4}\bigg)\\\\\\\text{Given:}\qquad \qquad 4\cos \bigg(\dfrac{B+C}{4}\bigg)\cdot \cos \bigg(\dfrac{C+A}{4}\bigg)\cdot \cos \bigg(\dfrac{A+B}{4}\bigg)\\\\\\.\qquad \qquad \qquad =4\cos \bigg(\dfrac{A+B}{4}\bigg)\cdot \cos \bigg(\dfrac{B+C}{4}\bigg)\cdot \cos \bigg(\dfrac{C+A}{4}\bigg)[/tex]

Middle = RHS [tex]\checkmark[/tex]

Please Help Me To Prove This!
Please Help Me To Prove This!
Please Help Me To Prove This!

Related Questions

6x+2y=12 ( solve for y )
How do I show my work on a piece of paper would help a lot :)

Answers

Answer:

y=6-3x

Step-by-step explanation:

factor out common terms 2(3x+y)=12divide both sides by 2

3x+y=[tex]\frac{12}{2}[/tex] simplify 3x+y=6 subtract 3 from both sides

y=6-3x

HELP PLS THIS IS IMMEDIATE

Answers

Hello the answer is on the paper

You want to buy a new hat that costs $9.20 on the price tag. How much
will you pay at the register if there is an 8.5% tax? Round to the nearest
cent. *

Answers

Answer:

$9.98 :)

Step-by-step explanation:

just multiply the tax (0.085) by the price ($9.20) then add the initial price ($9.20) to that.

9.98 because it is and the thing is like

given a right-angled triangle with perimeter 18. The sum of the squares of the three side lengths is 128. What is the area of the triangle ? ​

Answers

Answer:

9units²

Step-by-step explanation:

A triangle has 3 sides. Let the length of the three sides be a, b, c.

If the perimeter of the triangle is 18, then;

P = a+b+c

a+b+c = 18... 1

If the sum of the squares of the three side lengths is 128, this means;

a²+b²+c² = 128... 2

Since the triangle is also right angled, according to Pythagoras theorem;

c² = a²+b²... 3 (taking c as the hypotenuse)

Substitute equation 3 into 2;

(a²+b²)+c² = 128

c²+c² = 128

2c² = 128

c² = 128/2

c² = 64

c = √64

c = 8

Substituting c = 8 into 1

a+b+c = 18

a+b+8=18

a+b = 10 ...... 3

Substituting c = 8 into 2

a²+b²+c² = 128

a²+b²+8² = 128

a²+b²+64 = 128

a²+b²= 64..... 4

Solve equation 3 and 4 simultaneously;

a+b = 10

a²+b²= 64

From 4; a = 10-b

Substitute a = 10-b into 4;

(10-b)²+b² = 64

100-20b+b²+b² = 64

2b²-20b+100-64 = 0

2b²-20b+36 = 0

Divide through by 2:

b²-10b+18 = 0

b = 10±√10²-4(18)/2

b = 10±√100-72/2

b =( 10±√28)/2

b =(10±√4×7)/2

b = (10±2√7)/2

b = 5±√7

Hence b = 5+√7

If a+ b = 10

a+(5+√7 ) = 10

a = 10-(5+√7)

a = 10-5-√7

a = 5-√7

Area of the triangle A = 1/2 ab

A = 1/2 × (5+√7) × (5-√7)

A = 1/2 × (25-5√7+5√7-7)

A = 1/2 × (25-7)

A = 1/2 × 18

A = 9

Hence the are of the triangle is 9units²

The number of fungi on a culture has increased from 800 spores to 1,600 spores in 1 hour. What is the growth factor for
this fungus?

0.1
1
2
10

Answers

800 x 2 is 1600 so I’d say the answer is 2

Lynn and Max are divers. Lynn's depth gauge says she is 43.6 feet deep. Max's depth gauge says that he is 54.2 feet deep. Who is closer to the surface? Write two statements of inequality to describe the situation.​

Answers

Answer: Lynn is closer to the surface.

Step-by-step explanation:

Lynn is 43.6 deep meaning -46.3. Max is 54.2 feet meaning that he is -54.2 feet below sea level. so the smaller the number the close it is to 0 (which in this case is the surface) hope this helped

What is f(-2) of the function given below? *
f(x) = 3x2 - 4x + 1

Answers

Answer:

f(-2) = 21

Step-by-step explanation:

Step 1: Define

f(x) = 3x² - 4x + 1

f(-2) is x = -2

Step 2: Substitute and Evaluate

f(-2) = 3(-2)² - 4(-2) + 1

f(-2) = 3(4) + 8 + 1

f(-2) = 12 + 9

f(-2) = 21

X3-6x-16=0 solve by factoring

Answers

Answer:

X  =  3  √ 2  ( 3 x + 8 )

Step-by-step explanation:

Evaluate the expression for p = –4.
3p2 − 18(–2 − p) =

Answers

Answer:

12

Step-by-step explanation:

p = -4

3p² - 18(-2 - p)

plug in p

3(-4)² - 18(-2 - (-4))

solve the exponent

3(16) - 18(-2 - (-4))

solve within the parentheses

3(16) - 18(-2 + 4)

3(16) - 18(2)

multiply each expression

48 - 36

subtract

12

Given: KM bisects JKL
Prove: m MKL= 1/2m JKL

please explain

Answers

Start with out given statement:

Line KM bisects <JKL and the reason for that is because it's given.

For statement 2, we know that m<JKM = m<MKL

because of the definition of an angle bisector.

For statement 3, we can use the angle addition

postulate to get m<JKM + m<MKL = m<JKL.

For statement 4, we use substitution from

the info in statement 2 to get m<MKL + m<MKL = m<JKL.

For statement 5, we simplify to get 2(m < KML) = m<JKL.

For statement 5, we use the division property of equality.

All the reasons are underlined.

Beach Bicycles sells two-seater bicycles. Its markup rate is 68%. Which equation represents the new price for a bicycle, y, given an original price, x?

Answers

Answer:

y = 1.68x

Step-by-step explanation:

The markup rate of a product or service refers to how higher the selling price is than the total production cost. In this case, this company doesn't produce the bikes, instead it purchases them from a wholesaler, distributor or even the factory, and then resells them in order to make a profit.

E.g. Beach Bicycles purchases two-seater bikes at $200 each, and then it will resell them to their customers at 1.68 x $200 = $336. The total markup in this example is $136, which represents $136/$200 = 68% of the purchase price.

Find the product -1(-15) . Oh and also -1(-24) pls help

Answers

Answer:

15 and 24

Step-by-step explanation:

distribute the -1 to the -15 amd it will cancel out same for the other

D
Question 1
(02.03)
Read the following statement: Line segment CD is congruent to line segment XY.
Which of the following is an equivalent statement? (4 points)

Answers

???????????????????????

Which of the following is an equation of the line with slope 2 that passes through the point (-1,-4)​

Show complete working format

Answers

Answer:

y = 2x - 2

Step-by-step explanation:

Slope-Intercept Form: y = mx + b

Step 1: Define

Slope m = 2

Random point (-1, -4)

y = 2x + b

Step 2: Find b

-4 = 2(-1) + b

-4 = -2 + b

-2 = b

Step 3: Rewrite linear equation

y = 2x - 2

Answer should be Y=2x-2

which point is the center of a circle

Answers

Answer:

isnt it called the focus thats what i think

Step-by-step explanation:

If she does not make any deposits or withdrawals, what will be the balance in her
account after 2 days?

Answers

Answer:

the same as its been for 2 days

Step-by-step explanation:

Your school band is selling T-shirts. The group purchased the tees wholesale for $3.00 each. If the team has a 310% markup, what price are they selling the t-shirts for?

Answers

Answer:

$9.10

Step-by-step explanation:

hope this helps buddy

Cans of peas are 3 for $1.29. How much would one can of peas and a bag of chips priced at $2.19 cost all together?

Answers

Answer:

It would be 2.62

Step-by-step explanation:

helphelphelphelphelp

Answers

Answer:

Point A

Step-by-step explanation:

You have taken up gardening for relaxation and have decided to fence in your new
rectangular shaped masterpiece. The length of the garden is 4 meters and 34 meters
of fencing is required to completely enclose it. What is the width of the garden?

Answers

Answer:

  13 meters

Step-by-step explanation:

Adjacent sides form half the perimeter, so the sum of the unknown side and the given side is 34/2 = 17 meters.

  4 meters + width = 17 meters

  width = 13 meters . . . . . subtract 4 meters

Simplify the expression 8+10p-4-7p

Answers

Answer: the answer is 4+3p

Evaluate the following expression. cos315°

Answers

Answer:

[tex]\sqrt{2}/2[/tex]

Step-by-step explanation:

Making a unit circle will be very helpful if you don't already have one. You will use it for years to come.

Steve had 48 chocolates but decided to give 8 chocolates to each of his f coworkers. How many chocolates does Steve have left?

Answers

8•(f)=48 f=6
So if Steve divided 48 chocolates by how many he would give to each co worker he would have 6 workers and 0 chocolates left.

Hope this helped!! :))

Answer: 40 chocolates

Step-by-step explanation:

At the fair, Terrell has 47 ride tickets. Each ride on the Ferris wheel costs 6 tickets. After
riding the Ferris wheel as many times as possible, how many tickets will Terrell have left?

Answers

Answer:

Terrell has 5 tickets left.

Step-by-step explanation:

a rectangular piece of metal is 15in longer then it is wide. squares with sides 3in long are cut from the four corners and the flaps are folded upward to form an open box. if the volume of the box is 1092in cubed what were the original dimensions of the piece of metal?

Answers

Step-by-step explanation:

Let x equal the width of the box and x + 15 equal the length of the box.  Subtract 6 (the amount cut out) from both of the these to get the following terms: (x+9) and (x-6).  These are the length and width of the box respectively.  We also know that the height of the box is 3.  Plugging these into the formula for the volume of a box gives you:

3(x+9)(x-6)=1092

Expanding this out gives:

3x^2+9x-162=1092

Subtract 1092 from both sides to make the quadratic equation equal zero.  Plug the a, b and c coefficients into the quadratic formula (or factor) to find x.

3x^2 +9x-1254 = 0

a = 3       b = 9        c = -1254

I hope this helps :)

The volume of a pyramid, V, is calculated using the formula V = 1/3 AH, where A is the area of the base and H is the height of the pyramid.

Rearrange the formula to solve for the base area (A)

Answers

Answer:

A= 3V/H

Step-by-step explanation:

V=1/3 AH

multiply 3 on both sides

3V=AH

divide both sides by H

3V/H=A

Describe the 6 ways to transform a function.

Answers

Translate it up, translate it down, translate it to the right, translate it to the left, stretch it vertically, stretch it horizontally

Find a general term a, for the sequence whose first four terms are given.
1 /1,1/4,1/9,1/16

Answers

Answer:

  an = 1/n^2

Step-by-step explanation:

We notice that the denominators are the squares of successive integers. The general term will be ...

  [tex]\boxed{a_n=\dfrac{1}{n^2}}[/tex]

-1 7/8 + -6 Help, please.

Answers

-7 7/8 is the answer.

Alfred and his 4 friends bought 13 1/4 pounds of gummy bears at $1.20 pound . If the cost is divided between alfred and his four friends what would each person pay

Answers

$3.18

to find the total cost, just multiply 13.25 by $1.20 to get $15.90. to find the cost per person, just divide $15.90 by 5, which results in each person paying $3.18
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