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 room of size 16m ×6m is to be carpeted. whats the breadth of the carpet required if the given length is 60 m?
Data obtained from the question
Dimension of room = 16 m × 6 m
Length (L) = 60 m
Breadth (B) =?Step 2:Determination of the area of the room.
Dimension of room = 16 m × 6 m
Area of the room = 16 × 6
Area of the room = 96 m²Step 3:Determination of the breadth
Area of the room (A) = 96 m²
Length (L) = 60 m
Breadth (B) =?[tex]A = L * B\\96 = 60 * B[/tex]
Divide both side by 60
[tex]B = \frac{96}{60}[/tex]
B = 1.6 mTherefore, the breadth of the carpet required is 1.6 m.
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BRAINLIEST
HOW CAN POSTULATES AND THEOREMS RELATING TO SIMILAR AND CONGRUENT ANGLES BE USED TO WRITE A PROOF?
Answer:
How can postulates and theorems relating to similar and congruent triangles be used to write a proof? If the three sets of corresponding sides of two triangles are in proportion, the triangles are similar. ... If the three sets of corresponding sides of two triangles are in proportion, the triangles are similar.
Step-by-step explanation:
Hope this helps :D good luck!!
One thousandth of the base unit in length :
One millionth of the base unit in mass :
One thousandth of the base unite in time :
Step-by-step explanation:
[tex]1 \div 100th[/tex]
The temperature on a winter was -23 °F. The temerature rise by 5 °F when the sun came up. When the sun set again, the temperature dropped by. 7°F. Write and evaluate an exspression to find the temperature after the sun set.
Answer:
-25
Step-by-step explanation:
First, add 5 to -23 since the temperature is getting hotter.
so -23 +5= -18
Second, minus the answer by 7 since the temperature is now falling down after the sunset.
so -18 -7 = -25
or...
this step can be simplified as:
-23 +5 -7 =. -25
Pls pls help me! What is it
Answer:
H
Step-by-step explanation:
Jackie
[tex] \sqrt{9} + \sqrt{25} = 3 + 5 = 8[/tex]
Jesse
[tex] \sqrt{9 + 25} = \sqrt{34} [/tex]
Taking the difference, we get the answer to be
[tex]8 - \sqrt{34} [/tex]
How to slove this one?
Answer:
turning it I think.........
Help me pls
Find the missing segment in the image below
Answer:
The answer is 12
Step-by-step explanation:
You see its because.
Answer:
?=12
Step-by-step explanation:
Using the Thalès's theorem:
[tex]\dfrac{6}{6+4}=\dfrac{?}{20} \\\\?*10=6*20\ (cross\ products)\\\\?=\dfrac{6*20}{10} \ dividing\ by\ 10\\\\?=12[/tex]
For the function y = (x – 4)(2x + 3)
the 2 x-intercepts are
Answer:
x=4 -3/2 =x
Step-by-step explanation:
y = (x – 4)(2x + 3)
To find the x intercepts set y=0 and solve for x
0 = (x – 4)(2x + 3)
Using the zero product property
0= x-4 0= 2x+3
x=4 -3 = 2x
x=4 -3/2 =x
Answer:
We can find the x intercepts by setting this equation equal to 0.
So we have:
0 = (x-4)(2x+3)
So we get:
x-4 = 0, x = 4
2x+3 = 0, x = -3/2
The x intercepts are 4 and -3/2
Hope this helps!
Christopher walks 5km south then walks on a bearing of 036º until he is due east of his starting point. How far is he from his starting point, to 1 decimal place?
Christopher's distance from his starting point is 3.6 km
Since Christopher initially walks South 5 km and then walks on a bearing of of 036º until he is due east of his starting point.
His distance South, his distance from his starting point and his distance from his 036º bearing, all form a right-angled triangle.
This right-angled triangle with opposite side to the angle 036º, as his distance from his starting point, x and the adjacent side to the angle 036º, as his distance 5 km south.
Since we have both the opposite and adjacent sides of a right-angled triangle,
From trigonometric ratios,
tanФ = opposite/adjacent
tanФ = x/5 km
Now Ф = 036º
So, tan36º = x/5km
x = 5 km(tan36º)
x = 5 km (0.7265)
x = 3.633 km
x ≅ 3.6 km to 1 decimal place.
So, Christopher's distance from his starting point is 3.6 km.
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Laura brought some fish; salt water ($2 each) and fresh water ($1 each). If she has 15 fish - how many of each did she buy?
Answer:
are you sure there is no further info
Step-by-step explanation:
f(x)=-4+7, what is f(2)
Answer:
3
Step-by-step explanation:
7-4
The following are house prices in Southern California:
$750,000 $680,000 $600,000 $880,000 $1,200,000 $760,000 $480,000
What is the median house price?
Answer:
$750,000
Step-by-step explanation:
$750,000 $680,000 $600,000 $880,000 $1,200,000 $760,000 $480,000
Write the prices in increasing order and choose the middle one.
$480,000 $600,000 $680,000 $750,000 $760,000 $880,000 $1,200,000
Answer: $750,000
What is the answer to 1-x=16-4x and how did u get it?
Answer:
x=5
Step-by-step explanation:
1 - x = 16 - 4x
Rearrange :
-x +4x = 16 - 1
3x = 15
x = 5.
Hence, there is one solution : {5}
Answer:
x=5
Step-by-step explanation:
1-x=16-4x
-1 -1
-x=15-4x
+4x. +4x
3x=15
/3. /3
x=5
Teacher Thomas did a survey of primary 6 students to find out how the students go to school.the result are down. given that 110 student go to school by train, how many students go to school by bus? THE RESULT
BUS 140*
CAR 67.2*
CYCLE 44.8*
walk 20*
train 88*
Answer:
(88°/360°) * total number of students= 110
88x=39600
x=450 students
x is the total number of students
(140/360)*450=175 students going by bus
if u like my answer mark as brainliest
simplify [tex]x^{2} - x - 12[/tex]
Answer:
Step-by-step explanation:
x2 - x - 12
x2 - 4x + 3x - 12
x(x - 4) + 3(x - 4)
(x - 4) (x+ 3)
-4 + 3 will make -1 for -1x
-4 x 3 will make -12
Thus, the factored form will be (x-4)(x+3)
At the laundromat, Carol used to pay $1.50 per load, but the company has increased the price per load by 20%. Estimate the amount Carol will pay to was 6 loads of laundry.
Answer:
$10.80 (about $11)
Brainliest, please!
Step-by-step explanation:
20% = 0.2
+20% = x 1.2
1.5 x 1.2 = 1.8
1.8 x 6 = 10.8
Angle Terminology with Equations
Aug 03, 8:18:32 AM
?
ZA and ZB are complementary angles. If m2A = (2x – 10)° and m
B = (x - 2), then find the measure of ZA.
EN O
Answer:
Submit Answer
attempt 1 out of 2
Pls help
If both are complementary then their sum will be 90°
[tex]\\ \sf\longmapsto m<A+m<B=90[/tex]
[tex]\\ \sf\longmapsto 2x-10+x-2=90[/tex]
[tex]\\ \sf\longmapsto 2x+x-10-2=90[/tex]
[tex]\\ \sf\longmapsto 3x-12=90[/tex]
[tex]\\ \sf\longmapsto 3x=90+12[/tex]
[tex]\\ \sf\longmapsto 3x=102[/tex]
[tex]\\ \sf\longmapsto x=\dfrac{102}{3}[/tex]
[tex]\\ \sf\longmapsto x=34[/tex]
Now
[tex]\\ \sf\longmapsto <A=2x-10=2(34)-10=68-10=58[/tex]
Pls help answer, also explain. I will give brainliest
Answer:
8 lb box
Step-by-step explanation:
Find which laundry detergent has a lower unit price
The unit price is how much something costs for a single unit
In this case, it is how much the detergent costs for 1 pound
7.4/8 lbs = 0.925 dollars per lb
5.38/4 lbs = 1.345 dollars per lb
Since 1.345 > 0.925, the 8 lb box is a better buy
Case 1
Cost of 8 pound detergent = $7.40
So, cost of 1 pound detergent
= $7.40 ÷ 8
= $0.925
Case 2
Cost of 8 pound detergent = $5.38
So, cost of 1 pound detergent
= $5.38 ÷ 4
= $1.345
$0.925 < $1.345
$0.925 < $1.345So, cost per pound in 8 pound box < cost per pound in 4 pound box
So, buying the 8 pound box is better.
solve 6 X + 1 / 3+1=×-3/6
A square garden has area 1764meter find it's length and perimeter
Answer:
[tex]area \: of \: a \: squre = {a}^{2} \\ {a}^{2} = 176 \\ a = 13.26m \\ perimeter = 4a = 4 \times 13.26 = 53.04m^2 \\ thank \: you[/tex]
Answer:
Length = 42 mPerimeter = 168 mConcept:
Here, we need to know how to find the area and perimeter of the square.
Area (square) = s²
s = side lengthPerimeter (square) = 4s
s = side lengthSolve:
PART I: Find the lengthGiven information
Area = 1764 m²
Given formula
A = s²
Substitute values into the formula
1764 = s²
Square root both sides
√s² = √1764
[tex]\boxed{s=42m}[/tex]
--------------------------------------------------------------------------------------------------------
PART II: Find the perimeterGiven information
s = 42 m
Given formula
P = 4s
Substitute values into the formula
P = 4 (42)
Simplify by multiplication
[tex]\boxed{P=168m}[/tex]
Hope this helps!! :)
Please let me know if you have any questions
5. There are 5,280 feet in a mile. What part of a mile, in decimal form, will you drive until you reach the exit?It is 1,000 feet away. I need it quick plz I will GIVE you 50pts!
Answer:
1 mile = 5280 feet, then
1 foot = 1 / 5280 milesFind 1000 feet in miles:
1000 feet = 1000 * 1/5280 miles = 1000/5280 miles = ~0.1894 miles[tex]\\ \sf\longmapsto 1feet= \dfrac{1}{5280}miles[/tex]
Now
[tex]\\ \sf\longmapsto 1000feet[/tex]
[tex]\\ \sf\longmapsto 1000\times \dfrac{1}{5280}[/tex]
[tex]\\ \sf\longmapsto \dfrac{1000\times 1}{5280}[/tex]
[tex]\\ \sf\longmapsto \dfrac{1000}{5280}[/tex]
[tex]\\ \sf\longmapsto 0.18miles[/tex]
Find the sum of square root 5 and 2 square root 16 in simplest form. Also, determine whether the result is rational or irrational and explain your answer.
Answer:
sqrt(5) + 8
The result is irrational because the sqrt(5) is a non-repeating, non terminating decimal that cannot be written as a fraction.
Brainliest, please!
Step-by-step explanation:
sqrt(5) + 2sqrt(16)
sqrt(5) + 2(4)
sqrt(5) + 8
The sum of square root 5 and 2 square root 16 in simplest form is, √5 + 8 and it is irrational number
What is square root?In mathematics, square roots is a concept in which, if we multiply square roots of any number by itself then it will give us an original number.
if 'a' is square root of 'b' then it will represent as, a = √b.
Given that,
Two numbers
square root 5
2 square root 16
sum of square root 5 and 2 square root 16 in simplest form = ?
Sum = √5 + 2√16
= √5 + 2×4
= √5 + 8
the maximum possible simplest form is √5 + 8
As it is clear that, it is a irrational number
Because, √5 can not be written in P/Q form
Hence, the sum is √5 + 8
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Can somone help me pls
Answer:
Step-by-step explanation:
calculate the square of 31
Answer:
square of 31 is 961
Step-by-step explanation:
31^2
31×31
=961
1. How can you find the decimal forms of 3/12 and 2/9 .
Answer:
3/12 as a decimal is 0.25
2/9 as a decimal is 0.2 reapeted
Answer:
Answer written below
Step-by-step explanation:
Divide 3 by 12 and 2 by 9 using long division solve like normal long division if your divisor is bigger than your dividend add a zero on top and add a zero to the dividend.
Decimal form of 3/12:0.25
Decimal form of 2/9:0.22223
For a company picnic, Jacob ordered a box of fresh-baked gingerbread cookies and sugar cookies. The box included a total of 68 cookies, and 75% of them were gingerbread. How many gingerbread cookies did Jacob get?
Answer:
51 gingerbread cookies
Step-by-step explanation:
Find how many gingerbread cookies he got by finding 75% of 68:
68(0.75)
= 51
So, Jacob got 51 gingerbread cookies.
What is the sum of the geometric series below?
3+1+1/3+1/9+1/27
Answer:
The correct answer for your geometric series is 4 13/27 (mixed form) or 121/27.
Step-by-step explanation:
Answer:
Step-by-step explanation:
r =term 1 ÷ term 2 = 1 ÷ 3 = 1/3
a =term 1 = 3
n = 5
[tex]Sum = \frac{a(1-r^{n})}{1-r}[/tex]
[tex]= \frac{3*(1-(\frac{1}{3})^{5}}{1-\frac{1}{3}}\\\\=\frac{3*(1-\frac{1}{243})}{\frac{2}{3}}\\\\=\frac{3*\frac{242}{243}}{\frac{2}{3}}\\\\=3*\frac{242}{243}*\frac{3}{2}\\\\=\frac{121}{27}\\\\=4 \frac{13}{27}[/tex]
What percent of 500 is 125
Answer:
25%
Step-by-step explanation:
125 of 500 can be written as: 125 /500
To find a percentage, we need to find an equivalent fraction with the denominator 100. Multiply both numerator & denominator by 100.
125 /500 × 100 /100
= ( 125 × 100/ 500 ) × 1 /100 = 25 /100
Answer:
25%
Step-by-step explanation:
Of means multiply and is means equals
P *500 = 125
Divide each side by 500
P = 125/500
P = .25
Change to percent form
P = 25%
Hello! I really need some help with this (:
Answer:
y = 3x + 3
Step-by-step explanation:
As x increases, y increases by 3. Therefore, the slope of the function is 3.
Setting x = 0 will give an output of 3. Therefore, the y-intercept is (0, 3).
Overall, the function rule for the given data set is y = 3x + 3.
Write the following expression
as a
simplified polynomial in standard form.
2(x – 5)^2 + 2(x – 5) + 5
Answer:
2x^2 - 18x + 45
Step-by-step explanation:
Standard form is ax^2 + bx + c
Expand the (x-5)^2 = (x-5)(x-5) --
x^2 - 5x - 5x + 25
x^2 - 10x + 25
So we now have
2(x^2 - 10x + 25) + 2(x-5) + 5
From here, distribute the 2 to each term in x^2 - 10x + 25, and distribute the 2 to each term in x - 5.
2x^2 - 20x + 50 + 2x - 10 + 5
Combine like terms
2x^2 - 20x + 2x + 50 - 10 + 5
2x^2 - 18x + 40 + 5
2x^2 - 18x + 45
