Answers
Answer:
Step-by-step explanation:
These are really difficult, but one thing I noticed pretty much across the board is that we state our trig ratios in terms of sin and cos using the Pythagorean identity:
[tex]sin^2x+cos^2x=1[/tex], even this one where tangents are involved. Let's begin:
If tan⁻1(x) = u (this is just a substitution to make things easier down the road), then tan(u) = x
If tan⁻1(y) = v, then tan(v) = y.
If that be the case, what we have as a rewrite from our original problem is
sin(u + v) which, according to the identity is:
[tex]sin(u-v)=sin(u)cos(v)-sin(v)cos(u)[/tex] which also tells us that we need those tangents in terms of sines and cosines, as stated earlier. We will take the Pythagorean identity and solve it for both sinx and cosx. Sinx first:
If [tex]cos^2x+sin^2x=1[/tex] then solving for sin:
[tex]\frac{cos^2x}{sin^2x}+\frac{sin^2x}{sin^2x}=\frac{1}{sin^2x}[/tex] (as you can see, I just divided everything by sin-squared):
[tex]cot^2x+1=\frac{1}{sin^2x}[/tex] and
[tex]\frac{1}{tan^2x}+1=\frac{1}{sin^2x}[/tex] and work on eliminating the denominators by multiplying through by sin-squared times tan-squared:
[tex]sin^2x*tan^2x(\frac{1}{tan^2x}+1=\frac{1}{sin^2x})[/tex] which simplifies to
[tex]sin^2x+sin^2x*tan^2x=tan^2x[/tex] and factor out on the left:
[tex]sin^2x(1+tan^2x)=tan^2x[/tex] and solve for sin-squared:
[tex]sin^2x=\frac{tan^2x}{1+tan^2x}[/tex] and take the square root of both sides:
[tex]sinx=\frac{tanx}{\sqrt{1+tan^2x} }[/tex] . Do the same to solve for cos(x) to get:
[tex]cosx=\frac{1}{\sqrt{1+tan^2x} }[/tex] and now we're ready to fill in our formula for sin(u + v):
[tex]\frac{tanu}{\sqrt{1+tan^2u} }*\frac{1}{\sqrt{1+tan^2v} }[/tex] - [tex]\frac{tanv}{\sqrt{1+tan^2} }*\frac{1}{\sqrt{1+tan^2u} }[/tex] and...
since tanu = x and tanv = y:
[tex]\frac{x}{\sqrt{1+x^2}}*\frac{1}{\sqrt{1+y^2}}-\frac{y}{\sqrt{1+y^2}}*\frac{1}{\sqrt{1+x^2} }[/tex] which, in the end, simplifies to
[tex]\frac{x-y}{\sqrt{1+x^2}*\sqrt{1+y^2} }[/tex] and you're done! Yikes to that!!!!
Related Questions
what’s the scale reading
Answers
Answer:
0.026
Step-by-step explanation:
6 out of 10 steps between 0.02 and 0.03.
The square pyramid pictured below has a surface area of
9 m
6 m
6 m
Answers
What is the slope of the line that contains the points (–2, 7) and (2, 3)?
A. 4
B. 1
C. –4
D. –1
Answers
Answer:
D. -1
Step-by-step explanation:
Use rise over run, (y2 - y1) / (x2 - x1)
Plug in the points:
(y2 - y1) / (x2 - x1)
(3 - 7) / (2 + 2)
-4 / 4
= -1
So, the slope is -1.
The correct answer is D. -1
Tricia starts school at 7:00 AM and has lunch at 12:00 PM. She wants to make sure she has something to eat in between. Determine what time she should eat her snack if she is to eat at exactly a time between starting school and eating lunch. (Hint: Set up a horizontal number line as a timeline.) A. A. 9:30 AM
B. 10:00 AM
C. 9:00 AM
D. 10:30 AM
Answers
Answer:
D) 9:00 am
Step-by-step explanation:
Because 9:00 is the midpoint of 7 and 12
Which is the mode for the data set? −1, 0, 3, 4, −3, −1
0.3 −1 2 −0.5
Answers
Answer:
-1
Step-by-step explanation:
Mode is the number that appears most often
−1, 0, 3, 4, −3, −1
-1 appears twice which is more than any other number
-1 is the mode
Pls help ———————————-
Answers
Area of square 1 = 84
Step-by-step explanation:
a^2 + b^2 = c^2
√48^2 + 6^2 = c^2
48 + 36 = 84
area of square 1 = 84
The side of square 2 is 48/4 = 12 so the area = 12x22 = 144
Using Pythagoras the area of square 3 = 144 + 81 = 225 sq.units
You are not asked for the side of square 3 but it is 15 units
what is the equation of the line that passes through the point (3,-7) and has a undefined slope
Answers
Answers:-
y = mx+b where m is the slope and b is the y intercept.
The line must pass through (3,-7) therefore
-7 =3 m +b
Now solve for b
b=-7 - 3m
Now substitute b into the definition of a line
y = mx -7 -3m
y=m*(x-3) -7
Now check the solution to see if for every m it passes through (3,-7)
substituting x in to the solution.
y=m(3–3)-7
y=7 hence the solution is correct. for any m
Plz follow ma );
Answer:
y = 7
Step-by-step explanation:
y = mx+b where m is the slope and b is the y intercept.
The line must pass through (3,-7) therefore ---> -7 =3 m +b
Now solve for b ---> b=-7 - 3m
Now substitute b into the definition of a line, y = mx -7 -3m ---> y=m*(x-3) -7
Now check the solution to see if for every m it passes through (3,-7)
Substitute x in to the solution ---> y=m(3–3)-7
y=7 therefore the solution is correct.
What is the area of the polygon below
Answers
What I do it cut the polygon into something easier like a square and rectangle.
Rectangle: 6•12=72
Square: 4•3=12
Add them together and you have 84
The cross-section of a searchlight mirror is shaped like a parabola. The light bulb is located 3 centimeters from the base along the axis of symmetry. If the mirror is 20 centimeters across at the opening, find its depth in centimeters. (Round your answer to the nearest tenth if necessary.)
Answers
The depth of the mirror of the cross-section of a searchlight would be 8.33 cm if the light bub has a vertex at 3 cm and the mirror is 20 centimeters across at the origin.
A cross-section is perpendicular to the axis of the symmetry goes through the vertex of the parabola. The cross-sectional shape of the mirrored section of most searchlights or spotlights is parabolic.
It helps in maximizing the output of light in one direction.The equation of the cross-section of the parabola is - [tex]y^{2} = 4ax[/tex], where a is the focus and x is the depth of the mirror from its origin.Given:
a = 3
y = [tex]\frac{20}{2}[/tex] cm = 10 cm
Solution:
from the equation [tex]y^{2} = 4ax[/tex]
[tex]y^{2} = 4*3*x\\ y^{2} = 12x[/tex]
putting x, 10 cm in the equation
[tex]x=\frac{10^{2} }{12} \\\\x= \frac{100}{12} \\\\x= 8.33 cm[/tex]
thus, the depth of the mirror would be - 8.33 cm
Learn more about other problems of the parabola:
https://brainly.com/question/12793264
Which of the following equations has exactly one solution?
A. 4x – 4 + 2x = 6x - 4
B. 2(4x + 5) = 8x + 10
C. 4x - 8 = 4(x - 4)
D. 3x + 5 = 2x - 6
Answers
A. 0X = 0 Every number
B. 8X+10 = 8X+ 10
B. 8X -8X = 10-10
B. 0X = 0. Every number
C. 4x -8 = 4X -16
C. 4x-4x = -16+ 8
C. 0x = -8 >>no solution
D. 3X + 5 = 2x -6
D. 3x-2X = -6-5
D. x = -11 one solution
Answer D
The equation with exactly one solution is 3x + 5 = 2x - 6. Option D is correct.
To determine which equation has exactly one solution, we need to solve each equation and see if we end up with a unique value for 'x'.
4x – 4 + 2x = 6x - 4
Combine the x terms on the left side: 6x - 4 = 6x - 4
This equation is an identity and holds true for all values of x. It doesn't have a unique solution.
B. 2(4x + 5) = 8x + 10
Distribute the 2 on the left side: 8x + 10 = 8x + 10
Similar to the previous equation, this equation is also an identity and holds true for all values of x.
It doesn't have a unique solution.
4x - 8 = 4(x - 4)
Distribute the 4 on the right side: 4x - 8 = 4x - 16
Subtract 4x from both sides: -8 = -16
This equation is contradictory and has no solution.
3x + 5 = 2x - 6
Subtract 2x from both sides: x + 5 = -6
Subtract 5 from both sides: x = -11
This equation has a unique solution, x = -11.
So, the equation with exactly one solution is 3x + 5 = 2x - 6. Option D is correct.
To learn more on Equation click here:
https://brainly.com/question/10413253
#SPJ3
3&4&5 pls please …………..
Answers
Answer:
Step-by-step explanation:
Looks hard
i  genuinely dont know help
Answers
Answer:
-13 1/2 < -6/8
Step-by-step explanation:
*Any negative in either the numerator or denominator will make the entire fraction negative.
1/2 = .5
-13 1/2 = -13.5
-6/8 = -3/4
-3/4 = -0.75
-6/8 = -0.75
-13 1/2 = -13.5
Which means:
-13 1/2 < -6/8
Hope this helped.
Answer:
[tex] < [/tex]
Step-by-step explanation:
[tex] \frac{6}{8} [/tex]
Is close to 0 (to positive) you can use a time line to determine which is a correct answer
Solve for b.
106 + 18 = 28
Answers
Answer:
106+18≠28
106+18=124
in case the b was misplaced with the 6
10b=28-18
10b=10
10b/10=10/10
b=1
hope this is helpful
106+18=28
124/10=6.8...
Gabriel's family is investing $5,000 into two different bonds for college. One bond gave
an interest of 5% per year, and the other 3%. If the combined interest for the year was
$220, how much money was invested in each bond?
Answers
The amount invested in the bond that yields the 5% interest is $3500 while the amount invested in the bond that yields the 3% interest is $1500
In order to determine the amount invested in each bond, simultaneous equation would be used
From the question, two sets of equations can be derived
0.05x + 0.03y = 220 equation 1
x + y = 5000 equation 2
Where :
x = amount invested in the bond that gives 5% interest
y = amount invested in the bond that gives 3% interest
Multiply equation 2 by 0.03
0.03x + 0.03y = 150 equation 3
Subtract equation 1 from 3
0.02x = 70
Divide both sides of the equation by 0.02
x = 3500
Substitute for x in equation 2
3500 + y = 5000
Subtract y from both sides of the equation
y = 5000 - 3500
y = 1500
To learn more about simultaneous equation, please check : https://brainly.com/question/15165507?referrer=searchResults
Solve x^3 = 64/27
Picture has answer choices
Answers
Answer:
x = 4/3
Step-by-step explanation:
x^3 = 64/27
Take the cube root of each side
(x^3)^1/3 =( 64/27)^1/3
x = 4/3
Answer:
4th option
Step-by-step explanation:
Given
x³ = [tex]\frac{64}{27}[/tex] ( take the cube root of both sides )
[tex]\sqrt[3]{x^3}[/tex] = [tex]\sqrt[3]{\frac{64}{27} }[/tex]
x = [tex]\frac{\sqrt[3]{64} }{\sqrt[3]{27} }[/tex] = [tex]\frac{4}{3}[/tex]
If two functions, f(x) and g(x), are inverse of each other, then the:
(select all that apply)
- range of f(x) is the same as the domain of g(x)
-domain of f(x) is the same as the domain of g(x)
-domain of f(x) is the same as the range of g(x)
- range of f(x) is the same as the range of g(x)
Answers
Answer: assuming g(x) is the inverse of f(x)
domain of g(x) = range of f(x)
range of g(x) = domain of f(x)
PLS HELP!!! WILL GIVE 100 points
Answers
$125.00
Step-by-step explanation:
$185.50 x 2 + $4.00 =
$371 + $4.00 =
$375 =
so $375 ÷ 3 = $125.00
So Rayne sell each desk for $125.00
Answer:
It's $125.00
(B btw)
Step-by-step explanation:
I got it it right lol
maths work summer vacations homework
Answers
Answer:
Step-by-step explanation:
line is made up of points and its endless
eg:........................................................................................it goes on forever even though it stopped here
a line segment is not endless it starts a one point and stops at another
eg: :-----------------------: see it stops its like starting a sentence and finshing it .
a ray is diffrent . i dont know how to explain so heres a pic
those orange things are called rays.
the set is points indicated by 12.is the ___ of the angle
Answers
Answer:
Vertex
Step-by-step explanation:
The vertex is the point were two rays meet to form an angle.
If my answer is incorrect, pls correct me!
If you like my answer and explanation, mark me as brainliest!
-Chetan K
URGENT, WILL GIVE BRAINLIEST IF CORRECT.
8 and 13
11.5 and 23.6
22 and 15
Answers
Answer:
5 < x < 21
Step-by-step explanation:
I did this bfore.
can uh help me in this
Answers
Answer:
Step-by-step explanation:
b) a² - 1 + b(b - 2a) = a² -1 + b*b - b*2a {distributive property}
=a² - 1 + b² - 2ab
= (a² - 2ab + b² ) - 1
= (a - b)² - 1² {use identity x²-y²= (x+ y)(x -y)}
= (a - b + 1)(a -b +1)
b) 15 +8y + y² - 2x - x² = (y² + 8y + 15) - x² - 2x
= (y² + 5y + 3y + 15) - x(x + 2)
= [ y(y + 5) + 3(y +5)] - x(x + 2)
= [(y + 5)(y+3) ] - x(x +2)
What is the radius of a sphere with a surface area of 100πcm2?
Answers
Answer:
5cm
Step-by-step explanation:
A=4πr2 = 100πcm2
r = 5
solve for the measure of c pt 2 (18pts)
Answers
Answer:
d: 20
Step-by-step explanation:
alternative angle
Since this triangle starts at the exact centre of the circle, both of the “arms” are radii of the circle (and therefore the same length), making this an isosceles triangle (two equal sides). In isosceles triangles the angles UNDER those two sides are also identical (therefore 20 degrees in this case). The final angle must add with those two 20’s to a total of 180 degrees. Two 20 degree angles = 40, so c must equal 140 degrees. B is 40. E and d are 20.
a cake recipe calls for 4 1/2 cups of sugar. a caterer has 50 1/2 cups of sugar on hand. how many cakes can he make?
Answers
Answer:
11 cakes. he can make 11 cakes
Answer:
11
Step-by-step explanation:
You have to do 4 x the amount you know you can such as 10.
Then you have to multiply the 1/2 by that same number
So we'd get 4x10=40 and 1/2x10=5 to get 45
Then we add 45+4 1/2 = 49 1/2
They can make 11 cakes
2^3*2^4=2^x
Solve for X PLZ HELP
Answers
Answer:
7
Step-by-step explanation:
if the bases of the exponents are the same then we can add the powers. so 2^3 * 2^4 = 2^7 = 128.
Answer:
7
Step-by-step explanation:
2^3⋅2^4= 2^7
2^7=2^x
since it had the same base you wouldn't need to worry about it you can just focus on the exponent.
7=x
swap it over
x=7
What is the value of the expression below when y = 5? Show your work and write your answer in a complete sentence.
Y exponent 2-3y+6
Answers
Answer:
y=1
Step-by-step explanation:
5=2-3y+6
3y=2-5+6
3y= -3+6
3y= 3
3y÷3=3÷3
y=1
Answer:
16
Step-by-step explanation:
Substitute y = 5 into the expression
y² - 3y + 6
= 5² - 3(5) + 6
= 25 - 15 + 6
= 16
On two investments totaling $9,500, Peter lost 3% on one and earned 7% on the other. If his net annual receipts were $169, how much was each investment?
Answers
Answer:
23$ was each investment
Step-by-step explanation:
[tex]\sqrt{x} x^{2}[/tex] +3
√2+1
b) Rationalize the denominator:
√2+1
Answers
Answer:
there is not a complete question here ...
[tex]\sqrt{2} +1[/tex] is not in a denominator to rationalize
if it were you would multiply the rational expression (the fraction)
by [tex]\frac{\sqrt{2} -1}{\sqrt{2} -1}[/tex] the result (in the denominator would end up being "2 -1"
which is 1
Step-by-step explanation:
calculate using the distributive property
7×19
Answers
Answer:
133
Step-by-step explanation:
7 * 19
Rewrite 19 as 20 -1
7 * (20-1)
Distribute
7*20 - 7*1
140 - 7
133
arrange in order smallest to largest 0.38, 0.31 , 0.4
Answers
Answer:
0.4 , 0.31 , 0.38 is the answer
