Answers
Answer: see proof below
Step-by-step explanation:
Use the following Sum to Product Identities:
[tex]\sin x+\sin y=2\sin \bigg(\dfrac{x+y}{2}\bigg)\cos \bigg(\dfrac{x-y}{2}\bigg)\\\\\\\sin x-\sin y=2\cos \bigg(\dfrac{x+y}{2}\bigg)\sin \bigg(\dfrac{x-y}{2}\bigg)\\\\\\\cos x+\cos y=2\cos \bigg(\dfrac{x+y}{2}\bigg)\cos \bigg(\dfrac{x-y}{2}\bigg)\\\\\\\cos x+\cos y=-2\sin \bigg(\dfrac{x+y}{2}\bigg)\sin \bigg(\dfrac{x-y}{2}\bigg)[/tex]
Proof LHS → RHS
[tex]\text{LHS:}\qquad \qquad \qquad \dfrac{\sin 5-\sin 15+\sin 25 - \sin 35}{\cos 5-\cos 15- \cos 25 + \cos 35}[/tex]
[tex]\text{Reqroup:}\qquad \qquad \qquad \dfrac{(\sin 25+\sin 5)-(\sin 35 + \sin 15)}{(\cos 35+\cos 5)-(\cos 25 + \cos 15)}[/tex]
[tex]\text{Sum to Product:}\quad \dfrac{2\sin \bigg(\dfrac{25+5}{2}\bigg)\cos \bigg(\dfrac{25-5}{2}\bigg)-2\sin \bigg(\dfrac{35+15}{2}\bigg)\cos \bigg(\dfrac{35-15}{2}\bigg)}{2\cos \bigg(\dfrac{25+15}{2}\bigg)\cos \bigg(\dfrac{25-15}{2}\bigg)-2\cos \bigg(\dfrac{35+5}{2}\bigg)\cos \bigg(\dfrac{35-5}{2}\bigg)}[/tex][tex]\text{Simplify:}\qquad \qquad \dfrac{2\sin 15\cos 10-2\sin 25\cos 10}{2\cos 20\cos 15-2\cos 20\cos 5}[/tex]
[tex]\text{Factor:}\qquad \qquad \dfrac{2\cos 10(\sin 15-\sin 25)}{2\cos 20(\cos 15-\cos 5)}[/tex]
[tex]\text{Sum to Product:}\qquad \dfrac{\cos 10\bigg[2\cos \bigg(\dfrac{15+25}{2}\bigg)\sin \bigg(\dfrac{15-25}{2}\bigg)\bigg]}{\cos 20\bigg[-2\sin \bigg(\dfrac{15+5}{2}\bigg)\sin \bigg(\dfrac{15-5}{2}\bigg)\bigg]}[/tex]
[tex]\text{Simplify:}\qquad \qquad \dfrac{\cos 10[2\cos 20\sin (-5)]}{\cos 20[-2\sin 10\sin 5]}\\\\\\.\qquad \qquad \qquad =\dfrac{-2\cos10 \cos 20 \sin 5}{-2\sin 10 \cos 20 \sin 5}\\\\\\.\qquad \qquad \qquad =\dfrac{\cos 10}{\sin 10}\\\\\\.\qquad \qquad \qquad =\cot 10[/tex]
LHS = RHS: cot 10 = cot 10 [tex]\checkmark[/tex]
Related Questions
Jacques is standing 1200 meters from a cliff he is planning to climb. The angle of elevation from the ground to the top of the cliff is 35º. To the nearest meter, how tall is the cliff?
Answers
Answer:
840 m
Step-by-step explanation:
Please refer to the attached photo.
In the attached photo, h is the height of the cliff.
The following data were obtained from the attached photo:
Opposite = h
Adjacent = 1200
θ = 35 °
The height, h of the cliff can be obtained by using the Tan ratio as follow:
Opposite = h
Adjacent = 1200
θ = 35 °
Tan θ = Opposite /Adjacent
Tan 35 = h/1200
Cross multiply
h = 1200 × Tan 35
h = 1200 × 0.7
h = 840 m
Therefore, the cliff is 840 m tall.
Yvonne's car with an 11-gallon gas tank gets an average of 32 miles per gallon. She travels 12 hours at an average speed of 64 miles per hour. Gas costs $3.25 per gallon. How much does the gasoline cost for this trip?
Answers
Answer:
$78
Step-by-step explanation:
Here we have to find total distance first then find amount of gasoline used based on distance traveled by car in one gallon.
_____________________________________________
Given
time = 12 hours
speed = 64 miles per hour
we know distance = speed * time
distance = 12*64 = 768 miles.
Thus, car ran 768 miles.
given that car gets an average of 32 miles per gallon.
It means in one gallon gasoline car travels 32 miles.
let the gallons used be x
in 1 gallon gasoline car travels 32 miles.
in 1*x gallon gasoline car travels 32*x miles = 32x
Thus, 32x should be equal to 768 miles as car traveled that much only
32x = 768 miles
x = 768/32 = 24
Thus, gallons of gasoline used is 24.
Cost of 1 gallon gasoline = $3.25
since, gallons found is 24, we multiply lHS and RHS by 24
Cost of 1 *24= 24 gallon gasoline = $3.25*24 = $78
Thus,
cost of 24 gallon is $78 which will be the gasoline cost for the trip.
express -8 as a sum of a positive and a negative integer.
Answers
Answer:
-10 + 2
-88 + 80
These are Literal Equations. Solve for g: aeg=10.
Answers
Answer:
g = 10/(ae)
Step-by-step explanation:
When variables are shown like that, in reality they are multiplied:
aeg = 10 is the same as a * e * g = 10
Isolate the variable, g. Note the equal sign, what you do to one side, you do to the other. Simply divide variables a & e from both sides:
(aeg)/ae = (10)/ae
g = 10/ae
g = 10/(ae) is your answer.
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A contractor bought 34 loads of soil each weighing 12 tonnes at $13 per tonne. What was the total cost?
Answers
Answer:
$5304
Step-by-step explanation:
One load of soil is 12 tons.
Multiply 34 with 12: 34 x 12 = 408 tonnes in all.
Next, multiply the cost per tonne with the amount of tonnes:
408 x 13 = 5304
$5304 is the total cost.
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Give two more examples of irrational numbers between 10 and 11.
Answers
Using the midpoint formula, determine the midpoint between the given coordinates. a. (-4, 4) and (-2, 2) M=( -3, ) b. (9, -10) and (-1, 26) M=( , )
Answers
Answer:
Step-by-step explanation:
a. (-4 - 2)/2 = -6/2= -3
(4 + 2)/2= 6/2 = 3
(-3, 3)
b. (9-1)/2= 8/2 = 4
(-10+26)/2= 16/2= 8
(4,8)
What is the distance between -7 and 21 on a number line?
Answers
Answer:
The distance is 28
Add 7 to 21
2. The seventh grade class is putting on a variety show to raise
money. It cost $700 to rent the banquet hall that they are going to
use. If they charge $15 for each ticket, how many tickets do they
need to sell in order to raise at least $1000?
Answers
Answer:
67
Step-by-step explanation:
15 x 10 = 150
150 x 6 = 900
15 x 7 = 105
900 + 105 = 1005
closest to 1000 without decimals
with decimals 66.6666666667
Please mark brainliest
$1000 ÷ 15 = 66.67 ~ 67 tickets
What fraction accurately represents the decimal 2.3
Answers
Answer:
23/10. or 2 3/10
Step-by-step explanation:
NOTE: Angles not necessarily drawn to scale.
Answers
The answer is 47 degrees , using Linear Pair Axiom
plz mark Brainliest
What is the scale if: 50 km in the actual distance correspond to 4 cm on the map?
Answers
The scale is 1 cm: 12.5 km
To get this answer, you start off with the ratio 4 cm: 50 km which is given to us. It means that 4 cm corresponds to 50 km.
Divide both parts of the ratio by 4
4/4 = 1
50/4 = 12.5
The ratio 4 cm: 50 km simplifies to 1 cm: 12.5 km
1 cm on paper represents 12.5 km in real life.
The scale is 1cm on map represents 12.5 km in actual distance.
How to calculate the scale factor?Suppose the initial measurement of a figure was x units.
And let the figure is scaled and new measurement is of y units.
Since the scaling is done by multiplication of some constant, that constant is called scale factor. Let that constant be 's'.
Then we have:
[tex]s \times x = y\\s = \dfrac{y}{x}[/tex]
Thus, scale factor is the ratio of the new measurement to the old measurement.
The ratio 4 cm: 50 km which is given to us means that 4 cm corresponds to 50 km.
Divide both parts of the ratio by 4;
4/4 = 1
50/4 = 12.5
Thus, The ratio 4 cm: 50 km simplifies to 1 cm: 12.5 km
Therefore, 1 cm on paper represents 12.5 km in real life.
Learn more about scale factors here :
https://brainly.com/question/8765466
#SPJ2
What is the value of x that makes the given equation true?
4x−3=−12x+13
a) x=−3
b) x = 0
c) x = 1
d) x = 16
Answers
A very large book is numbered to 1 to 11521 How many digits are used to print all these page numbers
Answers
Answer:
46,499 digits
Step-by-step explanation:
Total pages of book = 11,521
9 pages numbered 1 - 9 that need 1 digit each = 9 × 1
= 9 digits
90 pages numbered 10 - 99 that need 2 digits each = 90 × 2
= 180 digits
900 pages numbered 100 - 999 that need 3 digits each = 900 × 3
= 2,700 digits
9000 pages numbered 1000 - 9999 that need 4 digits each = 9000 × 4
= 36,000 digits
1522 pages numbered 1000 - 11521 that need 5 digits each = 1522 × 5
= 7,610 digits
Total digits needed = 9 digits + 180 digits + 2,700 digits + 36,000 digits + 7,610 digits
= 46,499 digits
A quarterback is sacked for a loss of 4 yards. On the next play, his team loses 10 yards. Then the team gains 12 yards on the third play. Write an addition expression to represent the situation. Then find the sum and explain its meaning.
Answers
Answer:
Kindly check explanation
Step-by-step explanation:
Given the question:
Play 1: Quarterback was sacked for a loss of 4 yards
Play 2 : Another loss of 10 yards
Play 3: a gain of 12 yards
Write an addition expression to represent the situation. Then find the sum and explain its meaning.
A loss of 4 yards = - 4
Loss of 10 yards = - 10
Gain of 12 yards = + 12
Addition expression for the scenario :
(-4) + (-10) + 12
Result :
-4 + (-10) + 12
= - 4 - 10 + 12
= - 14 + 12
= - 2
Hence, after two consecutive losses of 4 and 10 yards, the team wins 12 yards on the third play and the aggregate yards lost is reduced to 2 yards.
The graph will show the height of the red car as the time increases, Predict what this graph will look like before pressing start to run the Ferris wheel. Height (meters) Which statements about the graph are true? Check all that apply Each input generates an output value. As time increases, the height always increases. The input of height generates an output value of time. 80 40 After 60 seconds, the graph will display a second rotation, which looks the same. The graph shows the relationship of time to height.. 30 sec 60 sec Start Intro Done
Answers
Answer:1,4,5
Step-by-step explanation:
I just did it
Sam wants to buy some new shoes. The shoes normally cost $79.99, but with her employee discount they only cost $51.99. How much was her employee discount? (to the nearest percent) _______ % off
Answers
Answer:
35%
Step-by-step explanation:
79.99-51.99=29/79.99-.35 rounded
What is the area of a triangle with a base of 120 m and a height of 90 m
Answers
Answer:
Hey there!
Area=1/2bh
Area=1/2(120)(90)
Area=60(90)
Area=5400
Let me know if this helps :)
3;7;11;15... are the first four terms of an arithmetic sequence.
Given the last term is 671 determine the sum of the terms that are not divisible by 3
Answers
dAnswer:
BAN ME
Step-by-step explanation:
\fdsa
How much would a school save by using Company A instead of Company C to serve 800 students?
Answers
Answer:
500
Step-by-step explanation:
company a looks like it costs 3000 to serve 800 students and company c looks like it costs 3500 to serve 800
Answer:
600
Step-by-step explanation:
Find the value of X on the equation below. Please explain step by step and I will mark you brainliest.
g=x-c+y
Answers
g = x - c + y
First, let's add c to both sides.
g + c = x + y
Now, let's subtract y from both sides.
g + c - y = x
Please answer my question i will mark brainlyest.
Answers
Step-by-step explanation:
2 * 6 - 15 / 3 * 8
= 12 - 5 * 8
= 12 - 40
= -26.
8x + 20 = 36
8x = 16
x = 2.
#3 The question says the sun of two consecutive integers is 345. What are the integers?
#5 The sum of two consecutive even integers is 446. What are the two.integers?
Need help please
Answers
Step-by-step explanation:
#3 let the two consecutive integers be a, a+1.
a+a+1=345
2a+1=345
2a=345-1
2a=344
a=344/2=172...
the integers are 172, 173
#5 let the consecutive even integers be a, a+2
a+a+2=446
2a=446-2
2a=444
a=444/2=222..so the even integers are 222, 224
what is the surface area of this cylinder ?q
Answers
Answer:
130 pi or 408.41
Step-by-step explanation:
formula for sa of a cylinder is 2pir^2+2pirh
you just fill in the r and h
Answer:
[tex]\approx 408.2[/tex] in²
Step-by-step explanation:
The surface area of a cylinder can be represented as [tex]2\pi rh+2\pi r2[/tex]. Let's represent pi as 3.14 and r is 5 and h is 8.
[tex](2\cdot 3.14\cdot 5\cdot8) + (2\cdot 3.14 \cdot 5^2)\\\\251.2 + 157 = 408.2[/tex]
This means the surface area of this cylinder is 408.2 in².
Hope this helped!
APPLYING EXPONENTS RULES 2.0 PLEASE helppppp
Answers
Answer:
gotta chill. hi
Step-by-step explanation:
Answer:
Hi, how are you?
Please help me!!! A bag is filled with small and large marbles. The ratio of small to large marbles is 1:3. If there are eight more large marbles than small marbles, how many total marbles are in the bag?
Answers
Answer:
10 Marbles in total
Step-by-step explanation:
A ratio is pretty much a diversity between two things.
Examples:
Apples, green vs red
1 green apple : 3 red apples
Your Questions Explanation:
Marbles, small vs large
1 small marble : 3 large marbles
BUT if there are 8 MORE large marbles than small marbles
what that looks like as a ratio...
1:9
Then in total there are 10 marbles in the bag.
<3 Enjoy,
Dea
can three lines intersect at one point?
Answers
Answer:
yes
Step-by-step explanation:
use this picture as an example
-2 (2x + 5) SIMPLIFY
I get one answer but my calculator gives me another pls help!!
Answers
Answer:
−2⋅(2x+5)
Step-by-step explanation:
Please help ill give u a ( b r i l l i a n t)
Answers
Answer:
1.
[tex] \frac{91}{2} [/tex]
2.
[tex] \frac{121}{4} [/tex]
3.
[tex] \frac{226}{3} [/tex]
Step-by-step explanation:
1.
[tex] \frac{4}{1} \times \frac{5}{1} \\ 4 \times 5 \\ 20[/tex]
2.
[tex] \frac{3}{4} \times \frac{7}{6} \\ 18 \times 28 \\ 504[/tex]
Given the following funtions: f (x) = x^2 g (x) = x-3 find the composition of the two functions and show your process: f (g(x))
Answers
Answer:
Step-by-step explanation:
g(x)= x - 3
f(x - 3) = (x - 3)^2 = x^2 - 6x + 9