Answers
Answer: see proof below
Step-by-step explanation:
Given: A + B + C = π → A + B = π - C
→ A + C = π - B
Use the Cofunction Identities: sin (π - A) = sin A
sin (A - π/2) = cos A
cos (π/2 - A) = - sin A
Use the Power Reducing Identity: sin ² A - sin² B = sin (A + B) · sin (A - B)
Use the Sum to Product Identity:
sin A - sin B = 2 [cos (A + B)/2] · [sin (A - B)/2]
Proof LHS → RHS:
LHS: sin² A - sin² B - sin² C
= (sin² A - sin² B) - sin² C
Power Reducing: sin (A + B) · sin (A - B) - sin² C
Given: sin (π - C) · sin (A - B) - sin² C
Cofunction: sin C · sin (A - B) - sin² C
Factor: sin C [sin (A - B) - sin C]
[tex]\text{Sum to Product:}\qquad \sin C\bigg[2\cos \bigg(\dfrac{(A-B)+C}{2}\bigg)\cdot \sin \bigg(\dfrac{(A-B)-C}{2}\bigg)\bigg]\\\\\\.\qquad \qquad \qquad \qquad =2\sin C\bigg[\cos \bigg(\dfrac{(A+C)-B}{2}\bigg)\cdot \sin \bigg(\dfrac{A-(B+C)}{2}\bigg)\bigg][/tex]
[tex]\text{Given:}\qquad \qquad 2\sin C\bigg[\cos \bigg(\dfrac{(\pi - B)-B}{2}\bigg)\cdot \sin \bigg(\dfrac{A-(\pi - A)}{2}\bigg)\bigg]\\\\\\.\qquad \qquad \qquad =2\sin C\bigg[\cos \bigg(\dfrac{\pi -2B}{2}\bigg)\cdot \sin \bigg(\dfrac{2A-\pi}{2}\bigg)\bigg]\\\\\\.\qquad \qquad \qquad =2\sin C\bigg[\cos \bigg(\dfrac{\pi}{2} -B}\bigg)\cdot \sin \bigg(A-\dfrac{\pi}{2}\bigg)\bigg][/tex]
Cofunction: 2 sin C (-sin B) · (cos A)
= -2 sin C · sin B · cos A
= -2 cos A · sin B · sin C
LHS = RHS: -2 cos A · sin B · sin C = -2 cos A · sin B · sin C [tex]\checkmark[/tex]
Related Questions
Kane is training for a marathone. He started by running 3 miles during every training session. Each week plans to increase the distance of his run by 1/4 mile
Answers
Complete question :
Kane is training for a marathon. He starts by running 3 miles during every training session each week plans to increase the distance of his runs by 1/4 mile. Let w be the number of weeks. Write and expression to show the diatance kane runs in a training session after w weeks
Answer:
(11 + w ) / 4
Step-by-step explanation:
Given the following :
Initial miles kane plans to cover per session = 3 miles
Increase in miles covered per session = 1/4 mile per week
Hence, distance covered after w weeks :
Using the sum of an arithmetic progression formula:
Tn = a + (n - 1) d
a = first term = Initial / starting distance = 3 miles
n = nth term = number of weeks = w
d = common difference = constant increase in distance per week = 1/4
Hence, after w weeks :
= 3 + (w - 1) × 1/4
= 3 + (w/4 - 1/4)
= 3 + (w - 1) / 4
Taking L. C. M
(12 + w - 1) / 4
(11 + w ) / 4
The expression that shows the distance Kane runs in a training session after w weeks will be; (11 + w ) / 4
What is an expression?Expression in maths is defined as the collection of numbers variables and functions by using signs like addition, subtraction, multiplication, and division.
Given that the Initial miles kane plans to cover per session = 3 miles
The Increase in miles covered per session = 1/4 mile per week
Using the sum of an arithmetic progression formula to calculate the distance covered after w weeks
Tn = a + (n - 1) d
a = first term = 3 miles
n = nth term = number of weeks = w
d = common difference = the constant increase in distance per week = 1/4
Hence, after w weeks,
= 3 + (w - 1) × 1/4
= 3 + (w/4 - 1/4)
= 3 + (w - 1) / 4
= (12 + w - 1) / 4
= (11 + w ) / 4
To know more about an expression follow;
brainly.com/question/19876186
#SPJ2
Complete question :
Kane is training for a marathon. He starts by running 3 miles during every training session each week plans to increase the distance of his runs by 1/4 mile. Let w be the number of weeks. Write and expression to show the distance kane runs in a training session after w weeks.
A board 91 cm long is to be cut into two parts so that the longer part is 10 cm more than twice as long as the shorter part. Find the length of each part.
Answers
Answer:
The parts are 27 cm and 64 cm long.
Step-by-step explanation:
Let the shorter part be x.
"The longer part is 10 cm more than twice as long as the shorter part."
shorter part: x
longer part: 2x + 10
total length: x + 2x + 10
total length: 91
x + 2x + 10 = 91
3x + 10 = 91
3x = 81
x = 27
2x + 10 = 2(27) + 10 = 54 + 10 = 64
Answer: The parts are 27 cm and 64 cm long.
55,555 times x = 965,777 teacher gave this weird question and i do not under stand help me please. also which pic is prettier be honest
Answers
Answer:
[tex]\frac{965777}{55555}[/tex]
Step-by-step explanation:
The answer is quite simple, just divide both sides and rewrite the equation to get it. Maybe focus more on the question rather than yourself next time :)
Andrew is taking a multiple choice test with a total of 80 points available. Each
question is worth exactly 2 points. What would be Andrew's test score (out of 80) if
he got 8 questions wrong? What would be his score if he got x questions wrong?
Answers
Answer:
Andrew's test score if he got 8 questions wrong = 64 points
Andrew's test score if he got x questions wrong = 80 - 2x
Step-by-step explanation:
Total points of the multiple choice test = 80
Each question = 2 points
Total questions = Total points of the multiple choice test / 2 points
= 80/2
= 40
Total questions = 40
A. What would be Andrew's test score (out of 80) if he got 8 questions wrong
If he got 8 questions wrong out of 40 questions
= Total question points - wrong question points
= 40*2 - 8*2
= 80 - 16
= 64 points
Andrew's test score if he got 8 questions wrong = 64 points
B. What would be his score if he got x questions wrong?
Test points if he got x questions wrong = Total question points - wrong question points
= (40 * 2) - (x * 2)
= 80 - 2x
Andrew's test score if he got x questions wrong = 80 - 2x
A yard has a perimeter of 400 feet. If eight times the length of the yard equals seventeen times the width, what are its dimensions?
Answers
Answer:
Length of the rectangular yard = 136 feet
Width of the rectangular yard = 64 feet
Step-by-step explanation:
Let
Width = 17 × width
Length = 8 × length
8l = 17w
l = 17w/8
Perimeter of a rectangular yard = 2(length + width)
400 = 2(17w/8 + w)
400 = 34w/8 + 2w
400 = 17/4w + 2w
400 = 17w+8w/4
400 = 25w/4
Cross product
1600 = 25w
Divide both sides by 25
w = 1600/25
= 64 feet
Recall,
l = 17w/8
= 17*64 / 8
= 1088/8
= 136 feet
Length of the rectangular yard = 136 feet
Width of the rectangular yard = 64 feet
what’s 2 miles divided by 3
Answers
answer: when you take 5280 feet times two (2 miles) and then divide by three you get 3,520 feet
Write the following standard forms into exponential form 3 x 3 x 3 x 3 and 7 x 7.
Answers
Answer: 3^4, 7^2
Step-by-step explanation:
3 times itself 4 times! Seven times itself 2 times
7*7= 7 with power of 2
Use the distributive property to find the product
Answers
Answer:
1st one: -24a-6b-36c
2nd one: 14mn-12m
Step-by-step explanation:
1st one: Multiply '-6' w/ '4a', 'b', and '6c'
2nd one: Multiply '2m' w/ '7n' and '-6'
What is the domain of the relation? {(−4.5, 0), (−3, 9), (0.9, 0), (4, −7.8), (4, 0), (17, 3)}
{0.9, 4, 17}
{−7.8, 0, 3, 9}
{−4.5, −3, 0.9, 4, 17}
{−4.5, −3}
Answers
Answer:
Domain of the relation is Option (c) : {−4.5, −3, 0.9, 4, 17}
Which of the following are coplanar??
Answers
Answer:
Points T, R and Q are coplanar
Step-by-step explanation:
They are all lying on plane P
If you count to 100 by 2, how many numbers will you say?
Answers
Answer:
If you start at 0, 51. If you start with 2, 50.
Step-by-step explanation:
The amount of numbers that will be said are simply 100/2 which is 50 since counting by two is the same as the amount of even numbers up to 100. If you start at 0, simply add an additional count, making 51.
Cheers.
Kyle has $25 and gets a job walking Chihuahuas for $2 per hour. Sophia has $4 and walks Rottweilers for $9 per hour. How many hours will they both have to work to have the same amount of money?
Answers
Answer:
3 hours
Step-by-step explanation:
Let k represent Kyle's balance after h hours. Let s represent Sophia's balance after h hours. Then we have ...
k = 25 +2h
s = 4 +9h
We want to find h so that k=s:
25 +2h = 4 +9h
21 = 7h . . . . . subtract 4+2h
3 = h . . . . . . . divide by 7
If both work 3 hours, they will have the same amount of money.
I tried using a protractor but i kept getting it wrong? can you help me anyone?
Answers
what is an equation of the line that passes through the points (4,-6) and (8,-4)
Answers
Answer:
[tex]y = \frac{1}{2} x - 8[/tex]
Step-by-step explanation:
The equation of a line is usually written in the form of y=mx+c, where m is the gradient (also known as slope) and c is the y-intercept (the point through which the line cuts through the y-axis).
[tex]\boxed{gradient = \frac{y1 - y2}{x1 - x2} }[/tex]
Using the gradient formula above,
[tex]m = \frac{ -4 - ( - 6)}{8 - 4} \\ m = \frac{ - 4 + 6}{4} \\ m = \frac{2}{4} \\ m = \frac{1}{2} [/tex]
Substitute the value of m into the equation:
y= ½x +c
To find the value of c, substitute a pair of coordinates.
When x=4, y= -6,
-6= ½(4) +c
-6= 2 +c
c= -6 -2 (-2 on both sides)
c= -8 (simplify)
Thus, the equation of the line is y= ½x -8.
Point A (-8, 7) and Point B (6,-9) form a line segment on a coordinate plane. What is the midpoint of line segment AB
Answers
Answer:
-1,-1
Step-by-step explanation:
it is the midpint (-1,-1)
Answer:
( -1 , -1 )
Step-by-step explanation:
( -8 + 6 / 2 ) , ( 7 + -9 / 2)
( -2 / 2) , ( -2 / 2)
( -1 , -1 )
- If there are 12 inches in a foot, what is 4,096 cubic inches converted to cubic feet? a. 28.44 ft3 c. 2.370 ft b. 341.33 ft3 d. 4.096 ft3
Answers
Step-by-step explanation:
2.370.
One cubic foot equals 1728 cubic inches. So, to convert from cubic inches to cubic feet you should divide your cubic inches figure by 1728.
I hope it helps!
Simplify the expression: 4 + 10x + 5 - 9x
Answers
[tex]\text {Hello! Let's Solve this Equation!}[/tex]
[tex]\text {In order to Simplify this Equation you must subtract}[/tex] [tex]\text { it with the same match as it. Let me show what I mean.}[/tex]
[tex]\text {\underline {Subtract}}[/tex]
[tex]\text {10x-9x=1x=x}[/tex]
[tex]\text {\underline {Add}}[/tex]
[tex]\text {4+5=9}[/tex]
[tex]\text {Your Answer would be:}[/tex]
[tex]\fbox{x+9}[/tex]
[tex]\text {Best of Luck!}[/tex]
Hey there!☺
[tex]Answer:\boxed{x+9}[/tex]
[tex]Explanation:[/tex]
[tex]4+10x+5-9x[/tex]
Let's simplify.
[tex]4+10x+5+-9x[/tex]
Now let's combine like terms.
[tex](10x+-9x)+(4+5)[/tex]
[tex]x+9[/tex]
Hope this helps!
Explain how you can use grouping to make multiplying three factors easier.
Answers
Answer:
no To multiply three factors, we first multiply two of the factors together. Then we multiply the product we get by the third factor. Since multiplication is commutative, we may multiply numbers in any order.
Jane builds a ramp made of a triangular prism and a rectangular prism. What is the volume
of the ramp?
7.5 m3
11.5 m
1 m
4 m
28 m
2 m
3.5 m
15 m
Answers
Answer:11.5
Step-by-step explanation:I’ve done this in a test before and that is the correct AnSwEr
What I am really in need of help. If anyone answers this I will award 60 points to anyone!
Answers
What is the slope of the line in the graph?
-4/3
-3/4
3/4
4/3
Answers
Answer: -3/4
Step-by-step explanation: I'M SMART
Find the value of x in each case:
Answers
Answer:
x=15
Step-by-step explanation:
5x+(180-9x)+(180-8x)=180
total of a triangle angles is 180
Answer:
15°
Step-by-step explanation:
Step 1:
180° - 8x + 5x + 180° - 9x = 180° Sum of Triangle
Step 2:
360° - 12x = 180° Algerbra
Step 3:
- 12x = 180° Algerbra
Step 4:
12x = 180° Algerbra
Answer:
x = 15°
Hope This Helps :)
A gopher begins at 7 inches below the surface of a garden and digs
another 9 inches. Which expression represents this problem? *
7-9
9-7
-7-9
-7-(-9)
Answers
yesterday jack drove 33 1/2 miles. He used 1 1/4 gallons of gasloine
Answers
what is 02.15cm in meters
Answers
Answer:3200
Step-by-step explanation:
When Bryan was 11 years old he was 150.25 centimeters tall. For the next 4 years, he grew 65 centimeters a year. How tall was Bryan when he was 15 years old? Help PLEASE
Answers
Step-by-step explanation:
He grew 65cm a year??? Omg
I think it is
65cm x 4(years)= 260cm
150.25 + 260= 410.25cm
So, when he was 15 years old he was 410.25cm tall
Your car's gas tank is 3/4 full before you leave for your trip. When you stop at the gas station, the tank is only 1/12 full. Show the amount of change as a rational number.
Answers
Answer:
2/3
Step-by-step explanation:
State if the given functions are inverses. f(x) = 1/x + 3 g(x) = 4/x + 2
Answers
Answer:
they are not inverses
Step-by-step explanation:
f(1) = 1/1 +3 = 1 +3 = 4
g(4) = 4/4 +2 = 1 +2 = 3 ≠ 1
If the functions were inverses, we would have g(f(1)) = 1, which we do not.
The given functions are not inverses.
_____
The same would be true if your functions were
f(x) = 1/(x+3) and g(x) = 4/(x+2)
f(1) = 1/(1+3) = 1/4
g(1/4) = 4/(1/4 +2) = 4/(9/4) = 16/9 ≠ 1
A 24 foot long tree casts a shadow that is 9 feet long at the same time a nearby building casts a shadow of 45 feet long how tall is the building
Answers
Answer:
i think the answer you're looking for is 60
Step-by-step explanation:
24-9=15
45+15=60
A new zoo is coming to your city and they want to give away prizes. Every 4th customer will receive a panda hat, every 10th customer will receive a giraffe beach towel and every 20th customer will receive a stuffed gorilla. Out of the first 100 people to enter the zoo how many will get all three items?
Answers
Answer:
5 people
Step-by-step explanation:
Let's find LCM of 4, 10 and 20:
4 = 2*210 = 2*520 = 2*2*5LCM(4,10,20) =2*2*5 = 20
There are 100/20 = 5 twenties in 100
So every 20th person will get all 3 prizes, in total 5 people