Answers
Answer: see proof below
Step-by-step explanation:
Use the following Sum to Product Identities:
sin (A) - sin (B) = 2 cos (A+B)/2 · sin (A-B)/2
sin (A) + sin (B) = 2 sin (A+B)/2 · cos (A-B)/2
Given: sin Ф = k sin β --> (sin Ф)/(sin β) = k
Proof RHS → LHS
[tex]\text{RHS:}\qquad \qquad \qquad \dfrac{k-1}{k+1}\tan\dfrac{\theta+\beta}{2}[/tex]
[tex]\text{Given:}\qquad \qquad \dfrac{\frac{\sin \theta}{\sin \beta}-1}{\frac{\sin \theta}{\sin \beta}+1}}\cdot \tan\dfrac{\theta +\beta}{2}[/tex]
[tex]\text{Simplify:}\qquad \qquad \dfrac{\frac{\sin \theta}{\sin \beta}-\frac{\sin \beta}{\sin \beta}}{\frac{\sin \theta}{\sin \beta}+\frac{\sin \beta}{\sin \beta}}}\cdot \tan\dfrac{\theta +\beta}{2}\\\\\\.\qquad \qquad \qquad = \dfrac{\sin \theta -\sin \beta}{\sin \theta +\sin \beta}\cdot \tan\dfrac{\theta +\beta}{2}[/tex]
[tex]\text{Product to Sum:}\qquad \quad \dfrac{2\cos \frac{\theta+\beta}{2}\cdot \sin \frac{\theta-\beta}{2}}{2\sin \frac{\theta+\beta}{2}\cdot \cos \frac{\theta-\beta}{2}}\cdot \tan\dfrac{\theta +\beta}{2}[/tex]
[tex]\text{Expand:}\qquad \qquad \dfrac{2\cos \frac{\theta+\beta}{2}\cdot \sin \frac{\theta-\beta}{2}}{2\sin \frac{\theta+\beta}{2}\cdot \cos \frac{\theta-\beta}{2}}\cdot \dfrac{\sin\frac{\theta +\beta}{2}}{\cos \frac{\theta +\beta}{2}}[/tex]
[tex]\text{Simplify:}\qquad \qquad \quad \quad \dfrac{\sin \frac{\theta +\beta}{2}}{\cos \frac{\theta+ \beta}{2}}\\\\\\.\qquad \qquad \qquad \quad =\tan\dfrac{\theta +\beta}{2}[/tex]
[tex]\text{LHS = RHS:}\quad \tan\dfrac{\theta +\beta}{2}=\tan\dfrac{\theta +\beta}{2}\quad \checkmark[/tex]
Related Questions
What is the greatest four-digit integer that meets the following three restrictions? 1- All of the digits are different. 2- The greatest digit is the sum of the other three digits. 3- The product of the four digits is divisible by 10 and not equal to zero.
Answers
What is 350 divided by 55 as a decimal? A) 6.6 B) 6.36 C) 6.363 D) 6.363
Answers
350 divided by 55 = 6.36
The result of 350/55 is a non-terminating, repeating decimal. The repeating pattern above, 36, is called repetend, and denoted overlined with a vinculum. The answer is C
6.363 is the correct answer as it represents the repeating decimal correctly.
So, the correct answer is D) 6.363
To find the decimal equivalent of the division 350 divided by 55, you simply divide 350 by 55 using long division. Here's the step-by-step calculation:
6.363636...
_______________
55 | 350.000000
- 330
_______
200
- 165
______
350
- 330
______
200
- 165
______
350
- 330
______
200
- 165
______
...
The division continues indefinitely with the decimal repeating the sequence 36.
Therefore, the decimal equivalent of 350 divided by 55 is 6.363636...
Based on the options provided:
A) 6.6 is not correct as it rounds up the decimal prematurely.
B) 6.36 is not correct as it does not account for the repeating decimal.
C) 6.363 is not correct as it rounds down the decimal prematurely.
D) 6.363 is the correct answer as it represents the repeating decimal correctly.
So, the correct answer is D) 6.363.
Learn more about repeating decimal click;
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9. This piecewise function represents the Social Security taxes for 2009. How much did Rachel pay in Social
Security tax if she earned $102,000 in 2009?
(0.062x
when 0< x < 106,800
($6,621.60
when x > 106,800
Nax)
a. $6,824.60
b. $6,355.10
c. $6,324.00
d. $6,220.00
Answers
Answer:
Option c.
Step-by-step explanation:
It is given that the piecewise function represents the Social Security taxes for 2009.
[tex]Tax=\begin{cases}0.062x & \text{ when } 0< x < 106,800 \\ \$6,621.60 & \text{ when } x > 106,800 \end{cases}[/tex]
We need to find Rachel's security tax if she earned $102,000 in 2009.
Since 102,000 lies in [tex]0<x<106,800[/tex], therefore
[tex]Tax=0.062x[/tex]
Put x=102000.
[tex]Tax=0.062(102000)[/tex]
[tex]Tax=6324[/tex]
So, the Social Security tax is $6324.
Therefore, the correct option is c.
How do I solve this problem?
Answers
Answer:
Step-by-step explanation:
The first 2 pieces of this function agree at pi/4, so we will set them equal to each other and sub in pi/4 for x to solve for k:
[tex]ksin(\frac{\pi}{4})+(\frac{\pi}{4})^2=\frac{\pi}{4}+2[/tex] and
[tex]k(\frac{\sqrt{2} }{2})+\frac{\pi^2}{16}=\frac{\pi}{4}+2[/tex] and we'll multiply everything by 16 to get rid of the fractions. Doing that gives us:
[tex]8k\sqrt{2}=4\pi+32-\pi^2[/tex] and
[tex]k=\frac{4\pi+32-\pi^2}{8\sqrt{2} }[/tex] so
k = 3.067
The next one is a bit tricky if you're not up on your natural log equations.
Again, we set these equal to each other because they meet at the value of e. We sub in e for x and solve for m:
[tex]x +2=mlne^x[/tex]
We'll sub in e for x. ln of e to the x power is equal to x, so
e + 2 = me and
[tex]m=\frac{e+2}{e}[/tex] so
m = 1.736
Hope that helps!
Point E is on line segment DF. Given DE = 2x, EF = 2x – 6, and
DF = 3x + 5, determine the numerical length of EF.
Answer: EF =
Answers
Answer:
EF is 16.
Step-by-step explanation:
So we know that Point E is somewhere on Line DF. In other words, DE plus EF must equal DF. In an equation, this is:
[tex]DF=DE+EF[/tex]
We are given that DE is 2x, EF is 2x-6, and DF is 3x+5. So, substitute:
[tex](3x+5)=(2x)+(2x-6)[/tex]
Now, solve for x.
On the right, combine like terms:
[tex]3x+5=4x-6[/tex]
Subtract 4x from both sides:
[tex]-x+5=-6[/tex]
Subtract 5 from both sides:
[tex]-x=-11[/tex]
Divide both sides by -1:
[tex]x=11[/tex]
Now that we know x is 11, substitute this back into the equation for EF to find EF.
[tex]EF=2x-6[/tex]
Substitute 11 for x:
[tex]EF=2(11)-6[/tex]
Multiply:
[tex]EF=22-6[/tex]
Subtract:
[tex]EF=16[/tex]
And we're done!
10
Write the numbers in order from least to greatest (**HINT: You need to
make them all decimals or all fractions***): 5/1000, 0.003, 9/1000
Answers
Answer:
0.003, 5/1000, 9/1000
Step-by-step explanation:
0.003 is thousandths.
5/1000 is thousandths.
9/1000 is thousandths.
I converted them to fractions.
5/1000 is already a fraction
Same for 9/1000
0.003=3/1000
So 3/1000, 5/1000, 9/1000
889×27+50 is blank than 889×27
Answers
Answer:
50
Step-by-step explanation:
889*27=24003
889*27+50=24053
24053
-24003
00050
I NEED HELP ASAP!
7x-3y= 6; х= 3
Answers
Answer:
If you are looking for y it's 5
Step-by-step explanation:
Plug in 3 for x
7(3) -3y = 6
21 - 3y = 6
subtract 21 from both sides
-3y = - 15
divide by three from both sides
y = 5
Is 22 + 32 = 42 a true statement?
Answers
False because 22 + 32 =54
Answer:
No. 22+32=54
Step-by-step explanation:
1.5y-9.82=-9.37-10.1y-14.37 solve for y
Answers
Step-by-step explanation:
1.5y - 9.82 = -9.37 - 10.1y - 14.37
1.5y + 10.1y = 9.82 -9.37 - 14.37
11.6y = −13.92
y = (-13.92)/11.6
y = -1.2
Point Q'Q ′ Q, prime is the image of Q(0,6)Q(0,6)Q, left parenthesis, 0, comma, 6, right parenthesis under the translation (x,y)\to(x+7,y-5)(x,y)→(x+7,y−5)left parenthesis, x, comma, y, right parenthesis, \to, left parenthesis, x, plus, 7, comma, y, minus, 5, right parenthesis. What are the coordinates of Q'Q
Answers
Q is located at (0,6)
The translation rule is [tex](x,y) \to (x+7,y-5)[/tex] which says to add 7 to the x coordinate and subtract 5 from the y coordinate. Doing that to (0,6) moves it to (7,1) which is where point Q' is located.
In other words, if you shift point Q(0,6) seven units to the right and five units down, then it arrives at Q ' (7,1)
Answer: (7, 1)The coordinates of Q' are [tex]Q'(x,y) = (7, 1)[/tex].
Note - There are typing mistakes in the question. Correct statement is presented below:
Point Q' is the image of [tex]Q(0,6)[/tex] under the translation [tex](x,y) \to (x+7, y-5)[/tex]. What are the coordinates of Q'.
In this question we should use the point and the operation described in statement. Vectorially speaking, a translation is defined by the following operation:
[tex]P'(x,y) = P(x,y) + T(x,y)[/tex] (1)
Where:
[tex]P(x,y)[/tex] - Original point.[tex]P'(x,y)[/tex] - Translated point. [tex]T(x,y)[/tex] - Translation vector.If we know that [tex]Q(x,y) = (0,6)[/tex], then the coordinates of [tex]Q'(x,y)[/tex] are:
[tex]Q'(x,y) = Q(x,y) + (7,-5)[/tex]
[tex]Q'(x,y) = (0,6) + (7,-5)[/tex]
[tex]Q'(x,y) = (7, 1)[/tex]
The coordinates of Q' are [tex]Q'(x,y) = (7, 1)[/tex].
We kindly invite to see this problem on rigid transformations: https://brainly.com/question/18613109
solve for x: 4/x=5/10
Answers
1. 4/x = 1/2
multiply both sides by x
2. 4= 1/2 x
simply 1/2x to x/2
3. 4= x/2
multiply both sides by 2
4. 4x2= x
simplify 4 x 2 to 8
5. 8= x
switch sides
x= 8
ANSWER; x= 8
Answer:
x = 8
Step-by-step explanation:
4/x = 5/10
=> 4/x = 1/2
=> 4 x 2 = x
=> 8 = x
=> x = 8
what is the horonzontial distance between the two points(3,4) and (8,4)
Answers
Answer:
Distance = 5.099
Step-by-step explanation:
Given h(x)= 4x^2, find h(3)
Answers
Explanation:
h(x) = 4x^2
Find h(3):
Simply plug x = 3 in the function:
h(3) = 4(3)^2
h(3) = 4(9)
h(3) = 36
write an expression that represents the total amount of fencing needed to enclose the two fields. -how would you do this?-
Answers
Answer:
8x + 60
Step-by-step explanation:
4(x + 5) + 2(x - 10) + 2(x + 30)
= 4x + 20 + 2x - 20 + 2x + 60 (combine like terms)
= 8x + 60
What does x equal?
2(3x - 2) = 20
Answers
Answer:
4
Step-by-step explanation:
2(3x-2)=20
3x-2=10
3x=12
x=4
Answer:
x=4
Step-by-step explanation:
- 3(4r - 8) = 36 - 2r
Answers
Answer:
r= -1.2
Step-by-step explanation:
first use distributive property
- 3(4r - 8) = 36 - 2r
-12r +24 = 36-2r
+2r. +2r
-10r +24= 36
-24. -24
-10r = 12
/-10. /-10
r= -1.2
Answer:
Step-by-step explanation:
Equation: -3(4r - 8) = 36 - 2r
Step 1:
-12r + 24 = 36 - 2r Multiply (4r - 8) by -3 using the Distributive Property.
Step 2:
-10r + 24 = 36 Add -12r and -2r by 2r.
Step 3:
-10r = 12 Subtract 24 and 36 by 24.
Step 4:
r = -12/10 Divide both sides by -10
Step 5:
r = -6/5 Simplify the fraction.
Convert the given amount to the given unit.
27 yd; ft
27 yd = ft
Answers
Answer:
81 ft
Step-by-step explanation:
1 yard=3ft
27 x 3=81
So the answer is 81 ft
Which statement is true? 2.289 < 2.297 2.297 < 2.289 3.01 < 2.297 2.289 > 3.01
Answers
Answer:
The first one, 2.2289<2.297
Step-by-step explanation:
Answer:
The answer is 2.2289<2.297 because if you look in the hundredths place there is a 9 in 2.297 but in 2.2289 there is an 2 and 9>2 so that is your answer.
Solve the inequation (2x+3x)4 using distributive method.
Answers
Answer:
First, distribute the 4.
2x times 4 = 8x
3x times 4 = 12x
then add/ combine like terms.
8x + 12x = 20x
Step-by-step explanation:
Answer:
20x
Step-by-step explanation:
distribute the 4 to the 2x and 3x: 4 * 2x= 8x and 4 * 3x= 12x
then add the x's together: 8x+12x=20x
so the answer is 20x
x2 + y2 = 13
x² - y2 = -16
Solve
Answers
x + 2y = 4 , x - y = -3
x - y = 9 , x - y = 6
There
8(4√7 - √18) I breed the answer fast please
Answers
Answer:
32[tex]\sqrt{7}[/tex] - 24[tex]\sqrt{2}[/tex]
Alternative form:
≈50.72292
Step-by-step explanation:
Step one,
Simplify the radical expression:
8(4√7 - √18)
Still Stuck? *( √18 Simplified is 3√2)*
Your answer should look like this: 8(4√7 - 3√2)
Step two,
Distribute 8 through the parentheses:
8(4√7 - 3√2)
Still Stuck? *( Use distribuitave property to distribute 8 to (4) and (-3). You should get (32) and (-24)*
Your answer should look like this: 32√7 - 24√2
Solution,
32[tex]\sqrt{7}[/tex] - 24[tex]\sqrt{2}[/tex]
Alternative form:
≈50.72292
*For alternative form, plug the final answer into your calculator. Round it if necessary*
What is the value of x in the equation -x = 6 - 3x + 8?
Answers
Answer:
x=7
Step-by-step explanation:
Collect like terms:
-x=6-3x+8
-x+3x=6+8
Simplify
2x=14
Simplify
x=7
O points
Mr. Freeman has a square cornfield. Which of the following could be the
area of the cornfield if the sides are measured in whole numbers?
150 square acres
189 square acres
200 square acres
225 square acres
Answers
Step-by-step explanation:
Notice that 15 squared = (3 * 5)^2 = 9 * 25 = 225.
So the area of the cornfield can be 225 square acres.
PLEASE HELP ASAP Write the following equation in general form.
-x - 5y = -19
–x - 5y + 19 = 0
x + 5y - 19 = 0
x + 5y + 19 = 0
Answers
Answer: -x - 5y + 19 = 0, choice A
All we're doing is adding 19 to both sides to get everything on the same side.
General form is Ax+By+C = 0. Some books use Ax+By = C which is just a variation of the same idea.
Solve the following:
2x-7\4 = x+3/3
Hope u get this!
Answers
Answer:
hello it is so
Step-by-step explanation:
simple simply do the division if don't I can
Daniel growing corn in a square field. The length of one side is 100 yards he needs to purchase fertilizer for the corn field. Each bag covers 50 square yards .How many bags do he need?
Answers
Answer:
200 bags are needed
Step-by-step explanation:
First let us determine the area of the corn field:
Shape of cornfield = square
Area of a square = Length × Length
where: Length = 100 yards
∴ Area = 100 × 100 = 10,000 square yards
1 bag of fertilizer is needed to cover 50 square yards:
50 square yards = 1 bag of fertilizer
[tex]\therefore 1 \ square\ yard\ = \frac{1}{50}\ bag\\10,000\ square\ yards\ =\ \frac{1}{50} \times 10,000 = \frac{10l000}{50} \\ = 200 bags[/tex]
Hence, 200 bags are needed.
Answer:
200 bags are needed
Step-by-step explanation:
I did the test and got it correct good luck everyone !!
Consider this expression.
(x + 9)(x - 2) - 9x - 6
What is equivalent?
Answers
Answer:
We can find an expression equivalent to the one given by expanding the expression given,
[tex](x + 9)(x - 2) - 9x - 6\\x^2-2x+9x-18-9x-6\\x^2-2x-24[/tex]
So [tex]x^2-2x-24[/tex] is a possible expression.
The required expression equivalent to (x + 9)(x - 2) - 9x - 6 is (x + 4)(x - 6).
Given that,
(x + 9)(x - 2) - 9x - 6, an equivalent expression is to be determined.
Functions are the relationship between sets of values. e g y=f(x), for every value of x there is its exists in a set of y. x is the independent variable while Y is the dependent variable.
= (x + 9)(x - 2) - 9x - 6
= x² + 9x -2x -18 - 9x - 6
= x² - 2x - 24
= x² -6x + 4x - 24
= x (x - 6) + 4 (x - 6)
= (x + 4)(x - 6)
Thus, the required expression equivalent to (x + 9)(x - 2) - 9x - 6 is (x + 4)(x - 6).
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What is the ratio for 2 red paper clips to 6 blue paper clips? Write it as a fraction.
Answers
Answer:
2 over 6 as a fraction simplifies as 1 over three.
Step-by-step explanation:
If you are writing as a fraction, you put 2 over 6 and simplify. Your answer is 1 over 3
Answer:
2/6 2/6 2/6 2/6 2/6 2/6 2/6 2/6 2/6 2/6 2/6 2/6 2/6 2/6 2/6 2/6 2/6 2/6 2/6 2/6 2/6 2/6 2/6 2/6 2/6 2/6 2/6 2/6 2/6 2/6 2/6 2/6 2/6 2/6 2/6 2/6 2/6 2/6 2/6 2/6 2/6 2/6 2/6 2/6 2/6 2/6 2/6 2/6 2/6 2/6 2/6 2/6 2/6 2/6 2/6 2/6 2/6 2/6 2/6 2/6 2/6 2/6 2/6 2/6 2/6 2/6 2/6 2/6 2/6 2/6 2/6 2/6 2/6 2/6 2/6 2/6 2/6 2/6 2/6 2/6 2/6 2/6 2/6 2/6 2/6 2/6 2/6 2/6 2/6 2/6 2/6 2/6 2/6 2/6 2/6 2/6 2/6 2/6 2/6 2/6 2/6 2/6
Step-by-step explanation:
its just 2/6
This question is confusing. Please help me!
Answers
Answer:
C. 490.
Step-by-step explanation:
25% = 0.25.
If the number of emails that Ryan and Taylor each received is x emails then Sara received 2x + 0.25 * 2x = 2.5 emails.
So 2x + 2.5x = 882
x = 882/4.5
x = 196.
So Sara received 2.5 * 196
= 490 emails.