MATHEMATICS HIGH SCHOOL

If A, B,C are the angles of a triangle then prove: (the following in picture)Please help me to prove this.

Answers

Answer 1

Answer:

Answer: see proof below

Step-by-step explanation:

Given: A + B + C = π → A + B = π - C

→ C = π - (A + B)

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

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

Use the Double Angle Identity: cos 2A = 1 - 2 sin² A

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

Proof LHS → RHS:

LHS: cos A + cos B + cos C

= (cos A + cos B) + cos C

$\text{Sum to Product:}\qquad 2\cos \bigg((A+B)/(2)\bigg)\cdot \cos \bigg((A-B)/(2)\bigg)+\cos C$

$\text{Given:}\qquad 2\cos \bigg((\pi -C)/(2)\bigg)\cdot \cos \bigg((A-B)/(2)\bigg)+\cos C\n\n\n.\qquad \qquad =2\cos \bigg((\pi)/(2) -(C)/(2)\bigg)\cdot \cos \bigg((A-B)/(2)\bigg)+\cos C$

$\text{Cofunction:}\qquad 2\sin \bigg((C)/(2)\bigg)\cdot \cos \bigg((A-B)/(2)\bigg)+\cos C$

$\text{Double Angle:}\qquad 2\sin \bigg((C)/(2)\bigg)\cdot \cos \bigg((A-B)/(2)\bigg)+\cos\bigg(2\cdot (C)/(2)\bigg)\n\n\n.\qquad \qquad \qquad =2\sin \bigg((C)/(2)\bigg)\cdot \cos \bigg((A-B)/(2)\bigg)+1-2\sin^2 \bigg((C)/(2)\bigg)\n\n\n.\qquad \qquad \qquad =1+2\sin \bigg((C)/(2)\bigg)\cdot \cos \bigg((A-B)/(2)\bigg)-2\sin^2\bigg((C)/(2)\bigg)$

$\text{Factor:}\qquad 1+2\sin \bigg((C)/(2)\bigg)\bigg[\cos \bigg((A-B)/(2)\bigg)-\sin\bigg((C)/(2)\bigg)\bigg]$

$\text{Given:}\qquad 1+2\sin \bigg((C)/(2)\bigg)\bigg[\cos \bigg((A-B)/(2)\bigg)-\sin\bigg((\pi-(A+B))/(2)\bigg)\bigg]\n\n\n.\qquad \qquad 1+2\sin \bigg((C)/(2)\bigg)\bigg[\cos \bigg((A-B)/(2)\bigg)-\sin\bigg((\pi)/(2)-(A+B)/(2)\bigg)\bigg]$

$\text{Cofunction:}\qquad 1+2\sin \bigg((C)/(2)\bigg)\bigg[\cos \bigg((A-B)/(2)\bigg)-\cos\bigg((A+B)/(2)\bigg)\bigg]$

$\text{Product to Sum:}\qquad 1+2\sin \bigg((C)/(2)\bigg)\bigg[2\sin \bigg((A)/(2)\bigg)\cdot \sin\bigg((B)/(2)\bigg)\bigg]\n\n\n.\qquad \qquad \qquad \qquad =1+4\sin \bigg((C)/(2)\bigg)\bigg[\sin \bigg((A)/(2)\bigg)\cdot \sin\bigg((B)/(2)\bigg)\bigg]\n\n\n.\qquad \qquad \qquad \qquad =1+4\sin \bigg((A)/(2)\bigg)\sin \bigg((B)/(2)\bigg) \sin\bigg((C)/(2)\bigg)$

$\text{LHS = RHS:}\ 1+4\sin \bigg((A)/(2)\bigg)\sin \bigg((B)/(2)\bigg) \sin\bigg((C)/(2)\bigg)=1+4\sin \bigg((A)/(2)\bigg)\sin \bigg((B)/(2)\bigg) \sin\bigg((C)/(2)\bigg)\quad \checkmark$

Answer 2

Answer:

The proof for this is simple. Let's say that A + B + C = π. From here on we require several trigonometric identities that must be applied.

$\cos \left(A\right)+\cos \left(B\right)+\cos \left(C\right) \n= 2 * cos((A + B) / 2) * cos((A - B) / 2) + \cos C \n= 2 * cos((\pi /2) - (C/2)) * cos((A - B) / 2) +\cos C \n= 2 * sin(C/2) * cos((A - B) / 2) + (1 - 2 * sin^2 (C/2)) \n= 1 + 2 sin (C/2) * cos((A - B) / 2) - sin (C/2) \n= 1 + 2 sin (C/2) * cos((A - B) / 2) - sin((\pi /2) - (A + B)/2 ))\n= 1 + 2 sin (C/2) * cos((A - B) / 2) - cos((A + B)/ 2)\n= 1 + 2 sin (C/2) * 2 sin (A/2) * sin(B/2) \n= 1 + 4 sin(A/2) sin(B/2) sin(C/2)$

Hope that helps!

Related Questions

Solve the problem.If a lawn sprinkler sprays 15 feet in every direction, what is the area of the yard that gets watered?
There is a discrete relationship between the number of workers and the number of bricks layed on a wall. Use this information to answer problems 9-11. What is the domain for the function that models the relationship between workers and bricks laid on the wall?
What is an equation in point-slope form of the line that passes through the point (4, −1) and has slope 6?
Veronica earned $150 at work this past week in her paycheck. She wants to buy some necklaces which costs $6 each. She writes a function to model the amount of money she will have left from her paycheck after purchasing a certain number of necklaces. She writes the function, f(x)= 15 - 6x. Determine what x and f(x) stand for in the function. (1) x=weeks; f(x)= dollars left (2) x=dollars left; f(x)= weeks(3) x=necklaces; f(x)= dollars left(4) x=dollars left; f(x)= necklaces
12.75x+250=18.25x74 can someone help with this?

You make an initial deposit of $2500 in an account that offer 6% simple interest per year how much will be in the account after four years?

Answers

I think its 625, deeply sorry if it is not.

Answer:

$600

Step-by-step explanation:

One gallon of fuel mixture contains antifreeze in the ratio of 5 parts fuel to one part antifreeze. To this is added half a gallon of mixture which is one third antifreeze and two thirds fuel. What is the ratio of fuel to antifreeze in the final mixture?

Answers

1 gal (initial) = (5/6) parts fuel & (1/6) parts antifreeze= total of 6 parts
0.5 gal (add'l) = (2/3) parts fuel & (1/3) parts antifreeze = total of 3 parts

adding the two
Fuel: 1gal*(5/6 parts) + 0.5gal*(2/3 parts) = 7/6 parts in 1.5 total gallons
Antifreeze: 1gal*(1/6 parts) + 0.5gal*(1/3 parts) = 2/6 parts in 1.5 total gallons

thus with a common denominator provides a ratio of 7:2 (fuel:antifreeze)

its cost 50 to make a bracelet and 1$ to make a necklace to make a profit the total cost for bracelets and necklaces must be less than 10$ the jeweler can make no more than 14 pieces of jewelry each day write a system of inequalities to model the number of bracelets and necklaces to be made each day

Answers

b + n ≤ 14 and .50b + 1n < 10 This is because the total number of bracelets added to the total number of necklaces made each day must be less than or equal to 14. Also, the price of each bracelet (.50b) added to the price of each necklace (1n) must be less than $10.

A candy company has orders for chocolate bars feom 5 different stores. Each order dontains 45 chocolate bars. Use he distrubutive property to digure out how many chocolate bars the candy company needs to make

Answers

Answer:

CANDY diabeatise sgdigduejsdid

i’m sorry but i genuinely don’t know the answer

Don’t answer just for points i actually need the help

Answers

Answer:

y int is -4. slope is negative. slope is rise/run. in the graph is rises 2 for every 3 it goes to the left. this means your slope is 2/3

put all that together

-2/3x -4 is your equation

How do you find the system of equations for a graph?

Answers

Answer:

The solution of such a system is the ordered pair that is a solution to both equations. To solve a system of linear equations graphically we graph both equations in the same coordinate system. The solution to the system will be in the point where the two lines intersect.

Step-by-step explanation:

Answer:

Step-by-step explanation:

If A, B,C are the angles of a triangle then prove: (the following in picture)Please help me to prove this. ​

Answers

Related Questions

Answers

Answers

Answers

Answers

Answers

Answers

If A, B,C are the angles of a triangle then prove: (the following in picture)Please help me to prove this.