Answer:
[tex]\displaystyle x = \frac{\pi}{4} + k\, \pi[/tex] for integer [tex]k[/tex] (including negative numbers.)
Step-by-step explanation:
Pythagorean Identity: [tex]\sin^{2}(x) + \cos^{2}(x) = 1[/tex]. Equivalently, [tex]\cos^{2}(x) = 1 - \sin^{2}(x)[/tex].
Rewrite the original equation and apply this substitution to eliminate [tex]\cos(x)[/tex]:
[tex]\displaystyle \sin^{6}(x) + \cos^{6}(x) = \frac{1}{4}[/tex].
[tex]\displaystyle (\sin^{2}(x))^{3} + (\cos^{2}(x))^{3} = \frac{1}{4}[/tex].
[tex]\displaystyle (\sin^{2}(x))^{3} + (1 - \sin^{2}(x))^{3} = \frac{1}{4}[/tex].
Let [tex]y = \sin(x)[/tex] ([tex]-1 \le y \le 1[/tex].) The original equation is equivalent to the following equation about [tex]y[/tex]:
[tex]\displaystyle y^{6} + (1 - y^{2})^{3} = \frac{1}{4}[/tex].
Expand the cubic binomial in the equation:
[tex]\displaystyle y^{6} + 1 - 3\, y^{2} + 3\, (y^{2})^{2} - (y^{2})^{3} = \frac{1}{4}[/tex].
[tex]\displaystyle y^{6} + 1 - 3\, y^{2} + 3\, y^{4} - y^{6} = \frac{1}{4}[/tex].
Simplify to obtain:
[tex]\displaystyle 1 - 3\, y^{2} + 3\, y^{4} = \frac{1}{4}[/tex].
Rearrange and simplify:
[tex]12\, y^{4} - 12\, y^{2} + 3 = 0[/tex].
[tex]3\, (2\, y^{2} - 1)^{2} = 0[/tex].
[tex]2\, y^{2} - 1 = 0[/tex].
[tex]\displaystyle y^{2} - \frac{1}{2} = 0[/tex].
Solve for [tex]y[/tex]:
Either [tex]\displaystyle y = \frac{1}{\sqrt{2}}[/tex] or [tex]\displaystyle y = -\frac{1}{\sqrt{2}}[/tex].
If [tex]\displaystyle \sin(x) = y = \frac{1}{\sqrt{2}}[/tex], then [tex]\displaystyle x = \frac{\pi}{4} + 2\, k\,\pi[/tex] for all [tex]k\in \mathbb{Z}[/tex].
On the other hand, if [tex]\displaystyle \sin(x) = y = \frac{1}{\sqrt{2}}[/tex], then [tex]\displaystyle x = \frac{3\, \pi}{4} + 2\, k\,\pi = \frac{\pi}{4} + (2\, k + 1) \, \pi[/tex] for all [tex]k\in \mathbb{Z}[/tex].
Combine both situations to obtain:
[tex]\displaystyle x = \frac{\pi}{4} + 2\, k\, \pi[/tex] for all [tex]k \in \mathbb{Z}[/tex].
A group has more than $120 to spend. The inequality 5x + 20 > 120 represents the cost to rent a hall for a conference, where x represents the number of hours the group can rent the hall. Which of the following is the solution for this situation?
Answer:
x>20
Step-by-step explanation:
You want to try and get the x alone so
5x + 20 > 120
-20 -20 Subtract 20 from each side
5x > 100
5x/5 > 100/5 Divide 5 from each side
x > 20
b) Tìm ba cách viết số hữu tỉ -11/25 dưới dạng tổng của hai số hữu tỉ dương.
Can anyone help pls :)? Thank you
Answer:
It's D:5.3
Step-by-step explanation:
√28 =5.29
Round off therefore is 5.3
f The difference of 8 and 3 is multiplied by one-quarter of the sum of 7 and 5 37 vedanta Excel in Mathematics
Answer:
15
Step-by-step explanation:
first find f which is (8-3) × ( 1/4 × 7 + 5)
simplify
5 × (1/4 ×12)
5 × 3
15 is thw answer
A fair spinner has 10 equal sections: 3 red, 3 blue and 4 green.
It is spun twice.
What is the probability of getting 2 different colours?
Answer:
11/30
Step-by-step explanation:
Fifteen years from now, Atli’s age will be 4 times his current age. What is his current age?
Answer:
he is three i think
Step-by-step explanation:
Select the correct answer.
Which equation matches the function shown in the graph?
Answer:
D. y = cos(4x)
Step-by-step explanation:
The graph passes through the points y = 0 and x = 3π/8
So you solve x = 3π/8 into 4 answers, which is equal to 0 is the correct answer.
Triangle QRP is congruent to triangle YXZ. What is the perimeter of triangle YXZ?
What is the perimeter of triangle YXZ?
Answer:
Perimeter of XYZ = Perimeter of QRP
Step-by-step explanation:
Since congruent then
P of XYZ = P of QRP
Multiply: (2x+y) (n2-3xy+y2)
Answer:
[tex]{ \tt{(2x + y)( {n}^{2} - 3xy + {y}^{2} )}} \\ = { \tt{(2x {n}^{2} - 3 {x}^{2}y + 2x {y}^{2} + {n}^{2}y - 3x {y}^{2} + {y}^{3} )}} \\ = { \tt{ {y}^{3} - xy(y + 3x) + {n}^{2} y }}[/tex]
Hello!
(2x+y) (n2-3xy+y2)
2x* n²= 2xn²
2x* -3xy = -6x²y
2x* y² = 2xy²
y*n² = yn²
y*-3xy = -3xy²
y* y² = y³
=>
2xn²- 6x²y + 2xy² +yn²- 3xy² +y³
Please help, the question is in the picture
Answer:
per = 18a 7a+2a+7a+2a
area = 14 [tex]a^{2}[/tex] 7a*2a
Step-by-step explanation:
Answer:
(i) 18a
(ii) 14[tex]a^2[/tex]
Step-by-step explanation:
i: 7a+7a+2a+2a
14a+4a
18a
ii: 7a*2a
14a2
I hope this helps!
Given the function g(x) = x^2 + 5x + 14, determine the average rate of change
of the function over the interval 1 < x < 7.
Answer:
13
Step-by-step explanation:
The average rate of change is
f(7) - f(1)
---------------
7-1
f(7) = 7^2 + 5*7 +14 = 49 +35 +14=98
f(1) = 1^2 +5(1)+14 = 1+5+14=20
98-20
--------
7-1
78/6
13
Which answer shows 0.00897 written in scientific notation?
From what I heard it's not D
Answer:
B
Step-by-step explanation:
The decimal number 0.00897 written in scientific notation is 8.97 ×10^3 and it has significant figures
Answer:
8.97 * 10⁻³
Step-by-step explanation:
0.00897 = [tex]\frac{8.97}{1000}=8.97 * 10^{-3}[/tex]
Place the decimal point after the first nonzero number.
Write the power of 10,
(i) If we move the decimal to the right, the power of 10 decrease by number of moves.
(ii) If we move the decimal to the left, the power of 10 increase by the number of moves
. Doanh nghiệp dự tính đầu tư góp vốn liên doanh với số tiền ban đầu là 500 trđ, trong vòng 4 năm, lãi suất dự kiến là 12%/năm. Tính tổng số lãi của hoạt động đầu tư liên doanh thu được sau khi kết thúc liên doanh ?
Write the expression 4^4(4^-7)(4) using a single
exponent.
4^-28
4^-4
4^-3
4^-2
Answer:
4^(-2)
Step-by-step explanation:
4^4(4^-7)(4)
We know that a^b * a^c = a^(b+c)
4^4(4^-7)(4^1)
4^(4+-7+1)
4^(-2)
HELPPP MEE PLEASE!!!
Consider the function shown on the graph.
Answer:
Domain x ≥ - 5
Range y ≥ - 2
Step-by-step explanation:
x ≥ - 5
y ≥ - 2
Help help help help help
Answer: 0 and 4
Step-by-step explanation:
someone please help me with this question
Answer:
Step-by-step explanation:
The diagram is given to you below. I could not change <C so that it was theta. So C = theta.
So C is the reference angle. Call it theta when you think of it. The diagram shows you how cos(C) must be set up.
The adjacent side is 7
The hypotenuse is 18
theta = C = cos-1(7/18) = 67.11
To the nearest degree 67.11 = 67
If 55% people in a city like Cricket, 30% like Football and the remaining like nother games, then what per cent of the people like other games? If the total people is 60 lakh, find the exact number who like each type of games Answer it fast, no spam please
write in notebook
[tex]\large{\dag\:{\underline{\underline{\frak{\pmb{\red{Answer:-}}}}}}}[/tex]
Cricket = 55%Football = 30%Other games = Remaining (?)So, the percent of people who like other games equals:
= 100 – (55 + 30)
= 100 – 85
= 15%
If the total no. of. people is 60,00,000:
★ Cricket
= 60,00,000 × 55/100
= 33,00,000
★ Football
= 60,00,000 × 30/100
= 18,00,000
★ Other sports
= 60,00,000 × 15/100
= 9,00,000
help me please and thank youu
Answer:
-3
Step-by-step explanation:
Sally spent half of her allowance going to the movies. She washed the family
car and earned 6 dollars. What is her weekly allowance if she ended with
17 dollars ?
does it give the answer choice or is it a blank space?
Please help!
This is a part 2 of a question, I found the first one and couldn’t find the second one. These are from edginuity. If you guys need the first part lmk.
Answer:
The vertical line test is basically a way to determine whether or not a graph is a function. All three of the options are accurate, so I would select all three.
Let me know if you have any other questions!
Х
49°
X =
degrees
What do I do
Answer:
x = 139
Step-by-step explanation:
The exterior angle is equal to the sum of the opposite interior angle
x = 49 +90
x = 139
An exterior angle of a triangle is equal to the sum of its interior opposite angles.
[tex] \bf \large \implies \: x \: = \: 49 \degree \: + \: 90 \degree[/tex]
[tex] \bf \large \implies \: x \: = \: 139 \degree [/tex]
The mean age of 5 women in an office is 35 years old.
The mean age of 5 men in an office is 24 years old.
What is the mean age (nearest year) of all the people in the office?
Answer:
29.5
Step-by-step explanation:
(5×35)+(5×24)/10
175+120/10
295/10
=29.5
find the value of x.
Answer: x=35
Step-by-step explanation:
To find the value of x, we can use the Pythegorean Theorem because this is a right triangle.
Pythagorean Theorem: a²+b²=c²
[tex]12^2+x^2=(x+2)^2[/tex] [exponent and distribute]
[tex]144+x^2=x^2+4x+4[/tex] [subtract both sides by 144]
[tex]x^2=x^2+4x-140[/tex] [subtract both sides by x²+4x]
[tex]-4x=-140[/tex] [divide both sides by -4]
[tex]x=35[/tex]
Now, we know that x=35.
Click the photo to solve the photo
Answer:
A=2
B=4
C=6
D=5
E=7
F=8
G=3
H=1
Step-by-step explanation:
explanation is in the picture!
nick was scuba diving at -32 1/2 feet if he descends another 8 3/5 feet what is his location ?
Answer:
-41 1/10
Step-by-step explanation:
subtract the two numbers may i get brainliest plz
This is a really simple fraction problem. All we have to do is realize that, because he is descending, the [tex]8\frac{3}{5}[/tex] is actually negative, so we get [tex]-32\frac{1}{2}-8\frac{3}{5}=-41\frac{1}{10}[/tex] which can also equal -41.1
Find the vertical asymptotes of the function (x-1)(x-3)^2(x+1)^2/(x-2)(x+2)(x-1)(x+3)
Answer:
x=2, x=-2, x=-3
Step-by-step explanation:
-check if anything simplifies
(x-1)(x-3)²(x+1)² / (x-2)(x+2)(x-1)(x+3), simplify (x-1)
(x-3)²(x+1)² / (x-2)(x+2)(x+3)
-make the denominator 0 to find the asymptotes
(x-2)(x+2)(x+3) = 0
(x-2) =0 gives x = 2 asymptote
(x+2) =0 gives x= -2 asymptote
(x+3) = 0 gives x=- 3 asymptote
A kindergarten class had a total of 15 boys 9 girls , what is the ratio of girls to boys in that class ?
Answer:
9:15
girls : boys
Step-by-step explanation:
Answer:
9:15
Or 3:5 when reduced
Step-by-step explanation:
what is the solution to Y=-2x²+4x+8
Y=-4x+16
Hi there!
[tex]\large\boxed{(2, 8)}[/tex]
Solve by setting the two equations equal to each other:
-2x² + 4x + 8 = -4x + 16
Move all terms to one side:
-2x² + 4x + 4x + 8 - 16 = 0
Simplify:
-2x² + 8x - 8 = 0
Factor out a negative 2 from each term:
-2(x² - 4x + 4) = 0
Factor:
-2(x - 2)² = 0
Set the factor equal to 0 to solve:
(x - 2)² = 0
x - 2 = 0
x = 2
Substitute this value of x into an equation to find the shared y value:
y = -4(2) + 16 = 8
Find the shortest side of a triangle whose perimeter is 64 if the ratio of two of its sides is 4:3 and the third side is 20 less than the sum if the other two
Answer:
The shortest side of the triangle is 18
Step-by-step explanation:
Let the sides the triangle be x, y and z.
From the question, the perimeter of the rectangle is 64, that is
x + y + z = 64 ...... (1)
Also, the ratio of two of its sides is 4:3, that is x:y = 4:3, then we can write that x/y = 4/3 ⇒ 3x = 4y ...... (2)
The third side, z, is 20 less than the sum of the other two, that is
z + 20 = x + y ...... (3)
Substitute equation (3) into (1)
Then,
z + 20 + z = 64
2z +20 = 64
2z = 64 - 20
2z = 44
z = 44/2 k
z = 22
From equation (3)
z + 20 = x + y
Then, k
22 + 20 = x +y
42 = x + y
x = 42 - y ...... (4)
Substitute this into equation 2
3x = 4y
3(42-y) = 4y
126 - 3y = 4y
4y + 3y = 126
7y = 126
y = 126/7
y = 18
Substitute this into equation (4)
x = 42 - y
x = 42 - 18
x = 24
∴ x = 24, y = 18 and z = 22
Hence, the shortest side of the triangle is 18.
