Answers
Answer: see proof below
Step-by-step explanation:
Given: A + B + C = π → A = π - (B + C)
→ B = π - (A + C)
→ C = π - (A + B)
Use Sum to Product Identity: sin A - sin B = 2 cos [(A + B)/2] · sin [(A - B)/2]
Use the following Cofunction Identity: cos (π/2 - A) = sin A
Proof LHS → RHS:
LHS: sin A - sin B + sin C
= (sin A - sin B) + sin C
[tex]\text{Sum to Product:}\quad 2\cos \bigg(\dfrac{A+B}{2}\bigg)\cdot \sin \bigg(\dfrac{A-B}{2}\bigg)+2\cos \bigg(\dfrac{C}2{}\bigg)\cdot \sin \bigg(\dfrac{C}{2}\bigg)[/tex]
[tex]\text{Given:}\qquad 2\cos \bigg(\dfrac{\pi -(B+C)}{2}+\dfrac{B}{2}}\bigg)\cdot \sin \bigg(\dfrac{A-B}{2}\bigg)+2\cos \bigg(\dfrac{C}2{}\bigg)\cdot \sin \bigg(\dfrac{C}{2}\bigg)\\\\\\.\qquad \qquad =2\cos \bigg(\dfrac{\pi -C}{2}\bigg)\cdot \sin \bigg(\dfrac{A-B}{2}\bigg)+2\cos \bigg(\dfrac{C}2{}\bigg)\cdot \sin \bigg(\dfrac{C}{2}\bigg)[/tex]
[tex].\qquad \qquad =2\cos \bigg(\dfrac{\pi}{2} -\dfrac{C}{2}\bigg)\cdot \sin \bigg(\dfrac{A-B}{2}\bigg)+2\cos \bigg(\dfrac{C}2{}\bigg)\cdot \sin \bigg(\dfrac{C}{2}\bigg)[/tex]
[tex]\text{Cofunction:} \qquad 2\sin \bigg(\dfrac{C}{2}\bigg)\cdot \sin \bigg(\dfrac{A-B}{2}\bigg)+2\cos \bigg(\dfrac{C}2{}\bigg)\cdot \sin \bigg(\dfrac{C}{2}\bigg)[/tex]
[tex]\text{Factor:}\qquad 2\sin \bigg(\dfrac{C}{2}\bigg)\bigg[ \sin \bigg(\dfrac{A-B}{2}\bigg)+\cos \bigg(\dfrac{C}{2}\bigg)\bigg][/tex]
[tex]\text{Given:}\qquad 2\sin \bigg(\dfrac{C}{2}\bigg)\bigg[ \sin \bigg(\dfrac{A-B}{2}\bigg)+\cos \bigg(\dfrac{\pi -(A+B)}{2}\bigg)\bigg]\\\\\\.\qquad \qquad =2\sin \bigg(\dfrac{C}{2}\bigg)\bigg[ \sin \bigg(\dfrac{A-B}{2}\bigg)+\cos \bigg(\dfrac{\pi}{2} -\dfrac{(A+B)}{2}\bigg)\bigg][/tex]
[tex]\text{Cofunction:}\qquad 2\sin \bigg(\dfrac{C}{2}\bigg)\bigg[ \sin \bigg(\dfrac{A-B}{2}\bigg)+\sin \bigg(\dfrac{A+B}{2}\bigg)\bigg][/tex]
[tex]\text{Sum to Product:}\qquad 2\sin \bigg(\dfrac{C}{2}\bigg)\bigg[ 2\sin \bigg(\dfrac{A}{2}\bigg)\cdot \cos \bigg(\dfrac{B}{2}\bigg)\bigg]\\\\\\.\qquad \qquad \qquad \qquad =4\sin \bigg(\dfrac{A}{2}\bigg)\cdot \cos \bigg(\dfrac{B}{2}\bigg)\cdot \sin \bigg(\dfrac{C}{2}\bigg)[/tex]
[tex]\text{LHS = RHS:}\quad 4\sin \bigg(\dfrac{A}{2}\bigg)\cdot \cos \bigg(\dfrac{B}{2}\bigg)\cdot \sin \bigg(\dfrac{C}{2}\bigg)=4\sin \bigg(\dfrac{A}{2}\bigg)\cdot \cos \bigg(\dfrac{B}{2}\bigg)\cdot \sin \bigg(\dfrac{C}{2}\bigg)\quad \checkmark[/tex]
Related Questions
Rupert buys a bicycle with a marked price of £ 12500.He gets a rebate of 10% on it. After getting
the rebate ,tax is charged at the rebate of 8%. Find the amount he will have to pay for the bicycle.
Answers
Answer:
£ 12,150
Step-by-step explanation:
Marked price of bicycle = £ 12500
Rebate = 10%
Amount of rebate = 12500*10/100 = £1250
Rebated price = 12500 - 1250 = £ 11,250
Tax = 8%
Amount of tax = 11250*8/100 = £ 900
Total amount paid for bicycle = Rebated price + amount of tax
= £ 11,250 + £ 900
= £ 12,150
Is geometric sequence a linear function? SOMEONE HELP ME
Answers
Answer:
No
Geometric sequences are exponential functions, not linear functions. The answer is no.
select the correct answer
Answers
Answer:
-4x^2+5x-28
Step-by-step explanation:
3x^2 +5x - 7(x^2 +4)
Distribute the -7
3x^2+5x-7x^2-28
Combine like terms
-4x^2+5x-28
A town's pharmacy and hospital are 3 miles apart. If a map of the town has a
scale, where 1 inch=0.25 mile, what is the distance between the pharmacy and
the hospital on the map, in inches?
Answers
because we use simple rule of three
2 inches ---------------- 5 miles
20inch -------------- xmiles
x = 100/2
x = 50
PLEASE HELP!!!!!!!!! A silo is a building shaped like a cylinder used to store grain. The diameter of a particular silo is 6.5 meters, and the height of the silo is 12 meters. Which equation can be used to find V, the volume of this silo in cubic meters? >> Please answer ASAP, I really appreciate your help :) I will give the first answer Brainliest if I can!
Answers
Answer:
311.41m3
Step-by-step explanation:
A=2πrh+2πr2
find the pattern for the following sequence, then use the pattern to show the next two terms. 1, 1, 2, 3, 5, 8, 13...
Answers
Answer I believe the next two terms are 21 and 34!
Step-by-step explanation:
the sequence is 1,1,2,3,5,8,13...
So for the first number, 1 + 0 is 1, for the next sequence 1 + 1 is 2, 1 + 2 is 3.
Add the number with the number behind it to get your next number!
The formula for the volume of a pyramid is V=⅓Bh, where B is the area of the base of the pyramid and h is the height of the pyramid.
What is the volume of this pyramid in cubic inches?
Answers
Answer:
80
Step-by-step explanation:
v=lwh/3= 5*6*8/3 = 80
The volume of this pyramid in cubic inches is 80 inch³.
What is Volume?Volume is a mathematical term used to describe how much three-dimensional space an item or closed surface occupies.
The area that any three-dimensional solid occupies is known as its volume. These solids can take the form of a cube, cuboid, cone, cylinder, or sphere.
We have,
V = (1/3) (b) (h)
So, the Volume of Pyramid is
= 1/3 x base aera x height
= 1/3 x 5 x 6 x 8
= 5 x 2 x 8
= 80 inch³
Learn more about Volume here:
https://brainly.com/question/1578538
#SPJ7
Daniel works at Funplex. Last week, he worked for 12 hours and made $93.84. How much does Daniel make per hour?
Answers
Answer
$7.82
Step-by-step explanation:
12 hrs=$93.84
1 hr= 93.84/12
1 hr=$7.82
ITS DUE IN 10 MINUTES PLEASE HELP find -5(10)(-4)
Answers
Answer:
200
Step-by-step explanation:
-5*10 = -50
-50*-4 = 200
Answer:
200
Step-by-step explanation:
-5 * 10 is equal to -50.
-50 * -4 = 200 ----- the two negatives cancel out, leaving a positive answer.
I have an exam tomorrow, wish me luck.
Answers
Answer:
good luck
Step-by-step explanation:
Answer:
Good luck on your exam!
The diagram represents an enlargement of a rectangle by a scale factor of 4,
4 in
2 in
[Not drawn to scale]
Help ASAP
Answers
Answer:D 16inches by 8 inches
Step-by-step explanation:
Answer:
16 in by 8 in
Step-by-step explanation:
HELPPPPP!!!! tHIS IS REALLY HARD!!!!!
Answers
Answer: J.
Step-by-step explanation: -tanx i believe
Answer:
H: 2
Step-by-step explanation:
[tex]\frac{\sqrt{1-\cos ^2\left(x\right)}}{\sin \left(x\right)}+\frac{\sqrt{1-\sin ^2\left(x\right)}}{\cos \left(x\right)}[/tex]
LCM is [tex]\cos \left(x\right)\sin \left(x\right)[/tex]
Adjust fractions:
[tex]\frac{\sqrt{1-\cos ^2\left(x\right)}\cos \left(x\right)}{\sin \left(x\right)\cos \left(x\right)}+\frac{\sqrt{1-\sin ^2\left(x\right)}\sin \left(x\right)}{\cos \left(x\right)\sin \left(x\right)}[/tex]
Combine:
[tex]\frac{\sqrt{1-\cos ^2\left(x\right)}\cos \left(x\right)+\sqrt{1-\sin ^2\left(x\right)}\sin \left(x\right)}{\sin \left(x\right)\cos \left(x\right)}[/tex]
--------------------------------------------------------
We will take a look at [tex]\sqrt{1-\cos ^2\left(x\right)}\cos \left(x\right)[/tex].
Using the identity that [tex]\cos ^2\left(x\right)+\sin ^2\left(x\right)=1[/tex], we know that [tex]1-\cos ^2\left(x\right)=\sin ^2\left(x\right)[/tex].
Therefore, [tex]\sqrt{1-\cos ^2\left(x\right)}\cos \left(x\right)=\sqrt{\sin ^2\left(x\right)}\cos \left(x\right)=\sin \left(x\right)\cos \left(x\right)[/tex]
-----------------------------------------------------------------------------------------------
Similarly, [tex]1-\sin ^2\left(x\right)=\cos ^2\left(x\right)[/tex] so
[tex]\sqrt{1-\sin ^2\left(x\right)}\sin \left(x\right)=\sqrt{\cos ^2\left(x\right)}\sin \left(x\right)=\cos \left(x\right)\sin \left(x\right)[/tex]
-------------------------------------------------------------------------------
We will finally end up with
[tex]\frac{\sin \left(x\right)\cos \left(x\right)+\cos \left(x\right)\sin \left(x\right)}{\sin \left(x\right)\cos \left(x\right)}[/tex]
Add:
[tex]\frac{2\sin \left(x\right)\cos \left(x\right)}{\sin \left(x\right)\cos \left(x\right)}[/tex]
Cancel out sin(x):
[tex]\frac{2\cos \left(x\right)}{\cos \left(x\right)}[/tex]
Cancel out cos(x):
[tex]2[/tex]
Which number is rational?
exploring real numbers
Answers
Answer:
you didnt give us numbers to choose from
Step-by-step explanation:
Answer:
any number that caan be written as a fraction.
Step-by-step explanation:
3 is rational 3/1
Round 374,462,894 to the nearest thousand. Explain each step of the process.
Answers
Answer:
374,463,000
Step-by-step explanation:
374,462,894
Identify the thousands spot and locate the digit next to it.
374,462,894
If the number is greater than or equal to 5, we round up. If the number is less than or equal to 4, we round down.
8 is greater than or equal to 5. We round up.
374,462,894 ≈ 374,463,000
Hope this helps.
Answer:
[See Below]
Step-by-step explanation:
✦ The number in the thousandth place is [tex]2[/tex].
✧ To round, look at the number to the right, in this case it's [tex]8[/tex].
○ If the number is greater than or equal to [tex]5[/tex], then round up.
○ If the number is less than [tex]5[/tex], round down.
✦ So then we would change the [tex]2[/tex] to a [tex]3[/tex].
✦ Then rewrite the number:
✧ [tex]374463000[/tex]
~Hope this helps Mate. If you need anything feel free to message me.
What is 9.2568 x 10-3 in standard notation?
Answers
Answer:
0.0092568
Step-by-step explanation:
10^-3 means to move the decimal 3 places to the left.
0.0092568
How can the tiles be sorted? (I need it quick please)
Answers
Answer:
by fraction and decimals
Step-by-step explication:
5/2x - 12 = 18 - 1/3x can yall help me im tryna get my grade up so im not a "failure" to my mom
Answers
Answer:
x= 180/17
Step-by-step explanation:
You will need to simplify both sides of the equation, which isolates the variable.
A computer has a list price of $1,200. It is on sale for 30% off list. What is
the sale price of the computer?
A $840
B $800
0
$720
$360
Answers
Answer:
A. $840
Step-by-step explanation:
multiply 1200 x .70 = 840
Solve each equation for the indicated variable 22. 12(m+3x)=18(x-3m);m
Answers
Answer:
m
Step-by-step explanation:
Consider the following information.
U = {all table games}
C = {card games} = {Hearts, Euchre, Go Fish, Uno, Snap, Set}
S = {card games using a standard deck of cards} = {Hearts, Euchre, Go Fish, Snap}
O = {card games using other decks of cards} = {Uno, Set}
B = {board games} = {chess, checkers, backgammon}
The disjoint sets are:
Answers
Answer:
2
Step-by-step explanation:
Two sets A and B are called disjoint if A and B have no elements in common. Another equivalent definition of disjoint sets would be that the intersection of the two sets actually equals the empty set.
11 m
Find the area of the shaded region. All angles are right angles.
4 m
7 m
2 m
2 m
The area of the shaded region is m2.
(Simplify your answer.)
Answers
Answer: 82m2
Step-by-step explanation:
An entry level designer's annual pay is $49,868 based on 52 weeks per year. Due to the economy, his company is having to cut back on the number of weeks that it employs its designers. If the firm cuts the work year to 50 weeks but keeps the same rate of pay, how much should the designer expect his annual pay to decrease? Round your answer to the nearest dollar.
Answers
Answer: his pay will decrease by $1,918
Step-by-step explanation:
49868/52 = 959
959 × 50 = 47950
49868 - 47950 = 1918
One week, Wyatt earned $70.40 at his job when he worked for 4 hours. If he is paid the same hourly wage, how many hours would he have to work the next week to earn $281.60?
Answers
Answer: 8 hours
Step-by-step explanation: he gets 70.40$ to work for 4 hours so you'll have to multiply 4 by 70.40 to represent it's doubling to 8 and you'll get 281.60
70.40x4=281.60
so add this 4 to the other 4 it makes 8 because that's how much it would take to get there 4 steps up to really =8
i'm sooo dificult
Represent the quantity by an integer.
A company posted a net income loss of $14,870 million for the fiscal year 2005.
A net income loss of $14,870 million can be represented by
million dollars.
Answers
HELP ASAP!!!!!!
Which ratio is equivalent to this one?
9/10
16:20
18:20
Answers
Answer:
18/20
Step-by-step explanation:
You just have to multiply by 2
Answer:
18:20
Step-by-step explanation:
all you have to do is 9/10 times two and you see which one it is and then you get your answer
Can i be made brainliest
what steps could be used to solve the equation 1 + 3x = -x + 4.
Answers
Answer:
Combing like terms
Step-by-step explanation:
Answer: x = 3/4
Step-by-step explanation:
1+3x = -x+4
or, 1-4 = -x-3x
or, -3 = -4x
or, 4x = 3
or, x = 3/4
so, x = 3/4
the AP in which 4th term is - 15 and 9th term is - 30 find the sum of the first 10 number
Answers
Answer:
5
Step-by-step explanation:
Enter the fraction as a decimal. Use a repeating decimal if necessary.
14
11
Answers
Answer:
the answer is 14/11
Step-by-step explanation:
8. One pound of fettuccine makes 5 cups cooked. How many cups will 3 pounds?
Answers
Step-by-step explanation:
1 pound = 5 cups
3 pounds = 5 ×3
= 15 cups
For x > 3, values of the function f(x) = –(x – 3)^2(x + 2) are negative. On this same interval, which statement correctly describes the values of the additive and multiplicative inverses? Options: A) Both the additive inverse and the multiplicative inverse are positive. B) Both the additive inverse and the multiplicative inverse are negative. C) The additive inverse is positive, while the multiplicative inverse is negative. D) The additive inverse is negative, while the multiplicative inverse is positive.
Answers
For x > 3, values of the function f(x) = –(x – 3)2(x + 2) are negative. On this same interval, which statement correctly describes the values of the additive and multiplicative inverses?
Answer:
C) The additive inverse is positive, while the multiplicative inverse is negative.
Step-by-step explanation:
The answer is C on Edge. (VERIFIED Edge Answer)
Make sure to click "thanks" and rate this answer to let others know this is the correct answer for this question.
Edge Tip: Answers do not mix or change.
