sin^6x + cos^6x = 1/4
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].
Find a (Round to the nearest tenth). PLS HURRY!!
Answer:
a = 56.3°
Step-by-step explanation:
tan(a) = 9/6 = 1.5
atan(1.5) = 56.31°
Answer:
[tex]\boxed {\boxed {\sf 56.3 \textdegree}}[/tex]
Step-by-step explanation:
We are asked to find the measure of angle a.
This triangle is a right triangle because of the small triangle in the corner representing a 90 degree or right angle. Therefore, we can use trigonometric functions. The three main functions are:
sinθ=opposite/hypotenuse cosθ= adjacent/hypotenuse tanθ= opposite/adjacentThe side measuring 6 is adjacent or next to angle a. The side measuring 9 is opposite angle a. Therefore, we will use the tangent function.
[tex]tan \theta= \frac{ opposite}{adjacent}[/tex]
[tex]tan \ a = \frac{ 9}{6}[/tex]
Since we are solving for an angle measure, we use the inverse trigonometric function.
[tex]tan ^{-1} * tan \ a = tan ^{-1} * \frac{9}6}[/tex]
[tex]a= tan ^{-1} * \frac{9}6}[/tex]
[tex]a= 56.30993247[/tex]
Round to the nearest tenth. The 0 in the hundredth place tells us to leave the 3 in the tenth place.
[tex]a \approx 56.3[/tex]
The measure of angle a is approximately 56.3 degrees.
what is the solution to the equation below? sqrt x-7 = 5
A. 144
B. 12
C. 2
D. 4
Answer:
I think the answer is B.12
If this not correct, Sorry.
Can someone help me on this
Answer:
The choose (C)
F(x)=x/ (x+1)(x-2)
Determine which equations have the same solution set as 2/3 -x +1/6 = 6x by recognizing properties, rather than solving. Check all that apply.
Answer:
The answer is "0.1190".
Step-by-step explanation:
Given:
[tex]\to \frac{2}{3} -x +\frac{1}{6} = 6x\\\\\to \frac{2}{3} +\frac{1}{6} = 6x+x\\\\\to \frac{4+1}{6} = 7x\\\\\to \frac{5}{6} = 7x\\\\\to x=\frac{5}{6\times 7} \\\\\to x=\frac{5}{42}\\ \\\to x= 0.1190[/tex]
Answer:
A.) 4 - 6x + 1 = 36x
B.) 5/6 - x = 6x
F.) 5 = 42x
Step-by-step explanation:
edge.
Find the approximate side length of a square game board with an area of 145 in 2 Plz help!
Answer:
Side length ≈ 12.04
Step-by-step explanation:
145 = x²
144 is the closest square, with the root 12
The square root of 145 is approximately 12.04
If my answer is incorrect, pls correct me!
If you like my answer and explanation, mark me as brainliest!
-Chetan K
Answer:
The approximate side length is 12.0 in
Step-by-step explanation:
The area of a square is given by
A = s^2 where s is the side length
145 = s^2
Taking the square root of each side
sqrt(145) = sqrt(s^2)
12.04159458 = s
The approximate side length is 12.0 in
The equation y=2(x-1)^2-5y=2(x−1)
2
−5y, equals, 2, left parenthesis, x, minus, 1, right parenthesis, squared, minus, 5 is graphed in the xyxyx, y-plane. Which of the following statements about the graph is true?
Answer:
(b) It is symmetrical about [tex]x = 1[/tex]
Step-by-step explanation:
Given
[tex]y = 2(x - 1)^2 - 5[/tex]
See attachment for options
Required
True statement about the graph
First, we check the line of symmetric
[tex]y = 2(x - 1)^2 - 5[/tex]
Expand
[tex]y = 2(x^2 - 2x + 1) - 5[/tex]
Open bracket
[tex]y = 2x^2 - 4x + 2 - 5[/tex]
[tex]y = 2x^2 - 4x -3[/tex]
A quadratic equation [tex]y = ax^2 + bx + c[/tex] has the following line of symmetry
[tex]x = -\frac{b}{2a}[/tex]
By comparison, the equation becomes:
[tex]x = -\frac{-4}{2*2}[/tex]
[tex]x = \frac{4}{4}[/tex]
[tex]x = 1[/tex]
Hence, the line of symmetry is at: [tex]x = 1[/tex]
(b) is true.
Answer: It is symmetrical about x=1
Step-by-step explanation:
Find the Value of y. 70 60 65 40
Answer:
125
Step-by-step explanation:
360-70-60-65-40
= 125
Answered by GAUTHMATH
please helpp!!!!!!!!
Step-by-step explanation:
the answer is in picture
Hassan drove 78 miles in 1ị hours. If he drove at a constant rate, how far did he
travel in one hour? Enter your answer as a whole number, proper fraction, or mixed
number in simplest form.
Answer:
1 1/3 hours
Step-by-step explanation:
I took the same test
Is {3,…} a defined set
Answer:
no it's not a defined state it's undefined
Classify the polynomial 5x3 + 4x - 2 by degree.
Answer:
3
Step-by-step explanation:
3 would be the degree of the polynomial since it has the highest degree.
( 2 + 3 ) ^-1 x ( 2 ^-1 + 2^-1 )
Answer:
Step-by-step explanation:
[tex]\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\la\la\la\la\ddddddddddddddddddddddddddddddddcleverdddddd\ffffffffffffffffffffffffffffffffffffffff\pppppppppppppppppppppppppppppppppppp\ddddddddddddddddddd\displaystyle\ \Large \boxed{ \boxed{\boldsymbol{Rule : a^{-1}=\frac{1}{a} }}} \\\\\\\\ (2+3)^{-1} \times (2^{-1}+2^{-1}) = \\\\1)\ (2+3)^{-1}=5^{-1}=\frac{1}{5} \\\\2)\ 2^{-1}+2^{-1}=\frac{1}{2} +\frac{1}{2} } =1 \\\\3)\ \frac{1}{5} \cdot 1=\boxed{\frac{1}{5} }[/tex]
Given the functions below, find f(x) + g(x)
f(x) = 3x - 1
g(x) = x2 + 4
Answer:
x^2+3x+3
Step-by-step explanation:
f(x) = 3x - 1
g(x) = x^2 + 4
f(x) + g(x) = 3x-1+ x^2 +4
Combine like terms
= x^2+3x+3
Simplify: 0.9(2b-1)-0.5b+1
Answer:
Step-by-step explanation:
0.9*2b = 1.8b
0.9*-1 = -0.9
So far we have 1.8b-0.9. It can't be simplified further.
Then, we add the 2nd part, -0.5b+1.
We have:
1.8b-0.9-0.5b+1. Next we combine like terms.
1.8b-0.5b = 1.3b.
-0.9+1 = 0.1
Then we put it together.
1.3b+0.1 is our answer.
Hope this helped! Have a nice day :D
Hi there!
»»————- ★ ————-««
I believe your answer is:
1.3b + 0.1
»»————- ★ ————-««
Here’s why:
⸻⸻⸻⸻
[tex]\boxed{\text{Simplifying...}}\\\\0.9(2b-1)-0.5b+1\\--------------\\\rightarrow 1.8b - 0.9 - 0.5b + 1\\\\\rightarrow 1.8b - 0.5b - 0.9 + 1\\\\\rightarrow \boxed{1.3b +0.1}[/tex]
⸻⸻⸻⸻
»»————- ★ ————-««
Hope this helps you. I apologize if it’s incorrect.
Solve the following.
2x^2-7x-4/6x^2+7x+2<0
Can someone explain how to solve this.
Answer:
Which one have same variable gather or decrease up
Step-by-step explanation:
2x^2-4.6x^2=1.4x^2
-7x+7x=0
1.4x^2+2<0
x<0
HELP ASAP
The graph of f(x)= |x| is chosen below. Write the equation for the stretched graph, g(x).
Answer:
y = |3x|
Step-by-step explanation:
The option which is not a solution of the equation 2x + 3y = 6 is:
(A) (0, 2)
(B) (1, 1)
(C) (-3, 4)
(D) (3, 0).
Answer:
Step-by-step explanation:
B OR TRUE
Express 12 000 iin standard form?
Answer:
the answer will be
1.2x10⁴
hope it helps
Answer:
We have been provided the number, 3430000. Therefore, the standard form is, 3430000=3.43×106, here, we have moved 6 places to the left. Hence, the standard form of 3430000 is 3.43×106. Note: It is important to note that the standard form of representing numbers is also called scientific form or standard index form.
A company determines that its weekly online sales, Upper S (t ), in hundreds of dollars, t weeks after online sales began can be estimated by the equation below. Find the average weekly sales for the first 3 weeks after online sales began. Upper S (t )equals2 e Superscript t
Answer:
$1272.36 ( average weekly sales for first 3 weeks )
Step-by-step explanation:
weekly online sales = S(t)
Determine average weekly sales for first 3 weeks
S(t) = 2e^t
total sales = ∫ S(t) dt
∴ Average weekly sales for first 3 weeks ( note : S(t) = 2e^t )
= [tex]\int\limits^3_0 {2e^t \, dt / ( 3 - 0 )[/tex]
= 2 [ e^t ] ³₀ / 3 = 2 ( e^3 - e^0 ) / 3 = 2 ( 6.3618 ) = 12.7236 hundreds
= $1272.36 ( average weekly sales for first 3 weeks )
There is a sequence of rigid transformations that takes A to A', B to B', and C to "
same sequence takes D to D'. Draw and label D':
Answer:
I think it's D
Step-by-step explanation:
Answer: I think its D
Step-by-step explanation:
Rigid transformation moves a shape without changing the size of the shape.
See attachment for the diagram that shows the position of D'
PT= 3x+4 and TQ=5x-8
Answer:
So if PT=TQ and TQ=7x-9
PT=5x+3=TQ=7x-9
5x+3=7x-9
minus 5x both sides
3=2x-9
add 9 both sides
12=2x
divide 2
6=x
PT=5x+3
PT=5(6)+3
PT=30+3
PT=33
PT=QT=33
x=6
Hoped I helped you.please help me with that
Answer:
[tex]\frac{16}{81}[/tex]
Step-by-step explanation:
[tex](\frac{27}{8} )^{-\frac{4}{3} }[/tex]
[tex]=((\frac{3}{2} )^3)^{-4/3}[/tex]
[tex]=(\frac{3}{2} )^{-4}[/tex]
[tex]=(\frac{2}{3} )^{4}[/tex]
[tex]=\frac{16}{81}[/tex]
Answer:
16/81
Step-by-step explanation:
a negative exponent means 1/...
the number in the numerator means "to the power of".
the number in the denominator means take the root of that power.
so, we have to take the third root of the expression, or this then to the power of 4, and finally build 1/... if the whole result.
and the sequence is not making a difference.
the third root of of 27/8 = 3/2
this to the power of 4 = 81/16
this 1/... = 16/81
What is the perimeter, rounded to the nearest tenth?
The area of the regular hexagon is 169.74 ft2.
A regular hexagon has an apothem with length 7 feet and an area of 169.74 feet squared.
What is the perimeter, rounded to the nearest tenth?
24.2 ft
28.3 ft
48.5 ft
56.8 ft
Answer:
48.5 ft
Step-by-step explanation:
The average of four different positive integers is 9. What is the greatest value for one of the integers?
Answer:
30Step-by-step explanation:
Sum of those 4 integers is:
4*9 = 36If the smallest ones are 1, 2 and 3, then the greatest possible integer is:
36 - (1 + 2 + 3) = 30Please help explanation if possible
Answer:
17.6
Step-by-step explanation:
Answer:
80 pennies ; 160 nickels
Step-by-step explanation:
Given the 2 equations
0.01p + 0.05n = 8.80 → (1)
n = 2p → (2)
Substitute n = 2p into (1)
0.01p + 0.05(2p) = 8.80
0.01p + 0.1p = 8.80
0.11p = 8.80 ( divide both sides by 0.11 )
p = 80
Substitute p = 80 into (2)
n = 2 × 80 = 160
There are 80 pennies ; 160 nickels
If the curved surface area of a cylinder with height 15cm is 1320cm², find total surface area
Answer:
2552cm^2
Step-by-step explanation:
C.S.A=1320cm^2 ;r=?
h=15cm.
[C.S.A. = 2πrh]
(r=1320×7/660=14cm)
Now,
TSA of cylinder = 2πr (h + r) sq
TSA=2×22/7(15+14)=2552cm^2
Let be the density function for the shelf life of a brand of banana which lasts up to weeks. Time, , is measured in weeks and . Incorrect answer icon Your answer is incorrect. Find the mean shelf life of a banana using . Round your answer to one decimal place. Mean
The question is incomplete. The complete question is :
Let [tex]p(t) = -0.0375t^2 + 0.225t[/tex] be the density function for the shelf life of a brand of banana which lasts up to 4 weeks. Time, t, is measured in weeks and [tex]$0 \leq t \leq 4$[/tex]. Incorrect answer icon Your answer is incorrect. Find the mean shelf life of a banana using . Round your answer to one decimal place.
Answer:
2.4
Step-by-step explanation:
Given :
[tex]p(t) = -0.0375t^2 + 0.225t[/tex]
Mean :
[tex]$=\int_0^4 tp (t) \ dt$[/tex]
[tex]$=\int_0^4 t (0.0375 t^2 + 0.225t) \ dt$[/tex]
[tex]$=-0.0375 \int_0^4 t^3 \ dt + 0.225 \int_0^4 t^2 \ dt$[/tex]
[tex]$=-0.0375 \left[ \frac{t^4}{4} \right]^4_0 + 0.225 \left[ \frac{t^3}{3} \right]^4_0$[/tex]
[tex]$=-0.0375 (64) + 0.225 \left( \frac{64}{3} \right)$[/tex]
[tex]$=-2.5 + 4.8$[/tex]
= 2.4
Therefore, the mean is 2.4
A pair of shoes is on sale for 30% off. The original price is p. Which expression can be used to find the price of the shoes after the discount?
a) 0.30p
b) 0.70p
c) 1.30p
d) 30p
Answer:
Step-by-step explanation:
the answer is b
Graph the relation shown in the table. Is the relation a function? Why or why not?
Answer:
what can i help u with
Step-by-step explanation:
No; the relation passes the vertical-line test. Yes; only one range value exists for each domain value
Yes; two domain values exist for range
yes; only one range value exists for each domain.