Step-by-step explanation:
sin ∅ = -(√3)/2
Note that
cos²∅ + sin²∅ = 1
cos²∅ = 1 - sin²∅
= 1 - (-√3 / 2)²
= 1 - (-√3)²/ 2²
= 1 - 3/4
= 1/4
cos²∅ = 1/4
Taking square root of both sides
cos∅ = 1/2
Note that tan∅ = sin∅/cos∅
therefore, tan∅ = -(√3)/2 ÷ 1/2
= -(√3)/2 × 2/1
= -√3
tan∅ = -√3
Since sin∅ = -√3 /2
Then ∅ = -60⁰
The value of ∅ for the given range (third quadrant) is 240⁰.
NB: sin∅ = sin(180-∅)
Also, since 180⁰ is π radians, then ∅ = 4π/3
ab=6cm ac=12 calculate the length of cd
Answer:
is that the full question?
Answer:
Solution:-
Given,
ab =perpendicular (p)= 6cm
ac =hypotenuse (h)= 12cm
cd =base (b)= ?
using , Pythagoras theorem we have ,
b²=√h²-p²
or,cd²=√ac²-ab²
= √12²-6²
= √144-36
=√108
=√10.8²
=10.8cm
the length of cd is 10.8 cm
hope it is helpful to you
The radius of a sphere is increasing at a rate of 2 mm/s. How fast is the volume increasing (in mm3/s) when the diameter is 60 mm?
Answer:
22608 mm³/s
Step-by-step explanation:
Applying chain rule,
dV/dt = (dV/dr)(dr/dt)............... Equation 1
Where dV/dr = rate at which the volume is increasing
But,
V = 4πr³/3
Therefore,
dV/dr = 4πr²............... Equation 2
Substitute equation 2 into equation 1
dV/dt = 4πr²(dr/dt).............. Equation 3
From the question,
Given: dr/dt = 2 mm/s, r = 60/2 = 30 mm
Consatant: π = 3.14
Substitute these values into equation 3
dV/dt = 4×3.14×30²×2
dV/dt = 22608 mm³/s
1. Find the exact value of sin( a−B), given that sin a=−4/5 and cos B=12/13, with a in quadrant III and B in quadrant IV.
2. Find all real numbers in the interval [0,2pi) that satisfy the equation.
3sec^2 x tan x =4tan x
3. Simplify the following trigonometric expressions, using identities as needed:
sin(x)/1−cos(x) + 1−cos(x)/sin(x)
(1) Recall that
sin(x - y) = sin(x) cos(y) - cos(x) sin(y)
sin²(x) + cos²(x) = 1
Given that α lies in the third quadrant, and β lies in the fourth quadrant, we expect to have
• sin(α) < 0 and cos(α) < 0
• sin(β) < 0 and cos(β) > 0
Solve for cos(α) and sin(β) :
cos(α) = -√(1 - sin²(α)) = -3/5
sin(β) = -√(1 - cos²(β)) = -5/13
Then
sin(α - β) = sin(α) cos(β) - cos(α) sin(β) = (-4/5) (12/13) - (-3/5) (-5/13)
==> sin(α - β) = -63/65
(2) In the second identity listed above, multiplying through both sides by 1/cos²(x) gives another identity,
sin²(x)/cos²(x) + cos²(x)/cos²(x) = 1/cos²(x)
==> tan²(x) + 1 = sec²(x)
Rewrite the equation as
3 sec²(x) tan(x) = 4 tan(x)
3 (tan²(x) + 1) tan(x) = 4 tan(x)
3 tan³(x) + 3 tan(x) = 4 tan(x)
3 tan³(x) - tan(x) = 0
tan(x) (3 tan²(x) - 1) = 0
Solve for x :
tan(x) = 0 or 3 tan²(x) - 1 = 0
tan(x) = 0 or tan²(x) = 1/3
tan(x) = 0 or tan(x) = ±√(1/3)
x = arctan(0) + nπ or x = arctan(1/√3) + nπ or x = arctan(-1/√3) + nπ
x = nπ or x = π/6 + nπ or x = -π/6 + nπ
where n is any integer. In the interval [0, 2π), we get the solutions
x = 0, π/6, 5π/6, π, 7π/6, 11π/6
(3) You only need to rewrite the first term:
[tex]\dfrac{\sin(x)}{1-\cos(x)} \times \dfrac{1+\cos(x)}{1+\cos(x)} = \dfrac{\sin(x)(1+\cos(x))}{1-\cos^2(x)} = \dfrac{\sin(x)(1+\cos(x)}{\sin^2(x)} = \dfrac{1+\cos(x)}{\sin(x)}[/tex]
Then
[tex]\dfrac{\sin(x)}{1-\cos(x)}+\dfrac{1-\cos(x)}{\sin(x)} = \dfrac{1+\cos(x)+1-\cos(x)}{\sin(x)}=\dfrac2{\sin(x)}[/tex]
In an examination every student took history or geography or both of 500 candidates 60% took history whiles 72% took geography. How many students took both subjects
Answer:
80 students
Step-by-step explanation:
Answer:
80
Step-by-step explanation:
60% of 500 = 300
72% of 500 = 360
40% of 500 = 200
28% of 500 = 140
300+360 = 660
660 - 2x = 500
660 - 500 = 2x
160 = 2x
2x = 160
x = 80
A statistician calculates that 7% of Americans are vegetarians. If the statistician is correct, what is the probability that the proportion of vegetarians in a sample of 403 Americans would differ from the population proportion by more than 3%
Answer:
0.0182 = 1.82% probability that the proportion of vegetarians in a sample of 403 Americans would differ from the population proportion by more than 3%.
Step-by-step explanation:
To solve this question, we need to understand the normal probability distribution and the central limit theorem.
Normal Probability Distribution
Problems of normal distributions can be solved using the z-score formula.
In a set with mean [tex]\mu[/tex] and standard deviation [tex]\sigma[/tex], the z-score of a measure X is given by:
[tex]Z = \frac{X - \mu}{\sigma}[/tex]
The Z-score measures how many standard deviations the measure is from the mean. After finding the Z-score, we look at the z-score table and find the p-value associated with this z-score. This p-value is the probability that the value of the measure is smaller than X, that is, the percentile of X. Subtracting 1 by the p-value, we get the probability that the value of the measure is greater than X.
Central Limit Theorem
The Central Limit Theorem establishes that, for a normally distributed random variable X, with mean [tex]\mu[/tex] and standard deviation [tex]\sigma[/tex], the sampling distribution of the sample means with size n can be approximated to a normal distribution with mean [tex]\mu[/tex] and standard deviation [tex]s = \frac{\sigma}{\sqrt{n}}[/tex].
For a skewed variable, the Central Limit Theorem can also be applied, as long as n is at least 30.
For a proportion p in a sample of size n, the sampling distribution of the sample proportion will be approximately normal with mean [tex]\mu = p[/tex] and standard deviation [tex]s = \sqrt{\frac{p(1-p)}{n}}[/tex]
A statistician calculates that 7% of Americans are vegetarians.
This means that [tex]p = 0.07[/tex]
Sample of 403 Americans
This means that [tex]n = 403[/tex]
Mean and standard deviation:
[tex]\mu = p = 0.07[/tex]
[tex]s = \sqrt{\frac{p(1-p)}{n}} = \sqrt{\frac{0.07*0.93}{403}} = 0.0127[/tex]
What is the probability that the proportion of vegetarians in a sample of 403 Americans would differ from the population proportion by more than 3%?
Proportion below 7 - 3 = 4% or above 7 + 3 = 10%. Since the normal distribution is symmetric, these probabilities are equal, which means that we find one of them, and multiply by 2.
Probability the proportion is below 4%
p-value of Z when X = 0.04.
[tex]Z = \frac{X - \mu}{\sigma}[/tex]
By the Central Limit Theorem
[tex]Z = \frac{X - \mu}{s}[/tex]
[tex]Z = \frac{0.04 - 0.07}{0.0127}[/tex]
[tex]Z = -2.36[/tex]
[tex]Z = -2.36[/tex] has a p-value of 0.0091
2*0.0091 = 0.0182
0.0182 = 1.82% probability that the proportion of vegetarians in a sample of 403 Americans would differ from the population proportion by more than 3%.
evaluate the expression when b= -6 and c=3
-4c+b
Answer:
-18
Step-by-step explanation:
b = -6
c = 3
-4c + b = ?
Plug in the value of each variable into the equation
-4c + b = ?
= -4(3) + (-6)
= -12 - 6
= -18
What is y=-2(x+3)^2+2
Answer:
y = -2(x + 3)² + 2
y = 2{ -(x + 3)²+ 1}
y = 2{ -(x² + 6x + 9) + 1}
y = 2{ -x² - 6x - 9 + 1}
y = 2{ -x² - 6x - 8 }
y = -2 { x² + 6x + 8}
OR
y = -2{(x + 4)(x + 2)}
Use the parametric equations of an ellipse, x=acosθ, y=bsinθ, 0≤θ≤2π , to find the area that it encloses.
Answer:
Area of ellipse=[tex]\pi ab[/tex]
Step-by-step explanation:
We are given that
[tex]x=acos\theta[/tex]
[tex]y=bsin\theta[/tex]
[tex]0\leq\theta\leq 2\pi[/tex]
We have to find the area enclose by it.
[tex]x/a=cos\theta, y/b=sin\theta[/tex]
[tex]sin^2\theta+cos^2\theta=x^2/a^2+y^2/b^2[/tex]
Using the formula
[tex]sin^2x+cos^2x=1[/tex]
[tex]\frac{x^2}{a^2}+\frac{y^2}{b^2}=1[/tex]
This is the equation of ellipse.
Area of ellipse
=[tex]4\int_{0}^{a}\frac{b}{a}\sqrt{a^2-x^2}dx[/tex]
When x=0,[tex]\theta=\pi/2[/tex]
When x=a, [tex]\theta=0[/tex]
Using the formula
Area of ellipse
=[tex]\frac{4b}{a}\int_{\pi/2}^{0}\sqrt{a^2-a^2cos^2\theta}(-asin\theta)d\theta[/tex]
Area of ellipse=[tex]-4ba\int_{\pi/2}^{0}\sqrt{1-cos^2\theta}(sin\theta)d\theta[/tex]
Area of ellipse=[tex]-4ba\int_{\pi/2}^{0} sin^2\theta d\theta[/tex]
Area of ellipse=[tex]-2ba\int_{\pi/2}^{0}(2sin^2\theta)d\theta[/tex]
Area of ellipse=[tex]-2ba\int_{\pi/2}^{0}(1-cos2\theta)d\theta[/tex]
Using the formula
[tex]1-cos2\theta=2sin^2\theta[/tex]
Area of ellipse=[tex]-2ba[\theta-1/2sin(2\theta)]^{0}_{\pi/2}[/tex]
Area of ellipse[tex]=-2ba(-\pi/2-0)[/tex]
Area of ellipse=[tex]\pi ab[/tex]
What is the coefficient of x2 in the expansion of (x + 2)??
O A.
2
OB.
3
O C.
4
OD.
6
x+2 in expansion of (x+2) ?
A
How would I draw the reflection over the line y=2x+5?
Answer:
Step-by-step explanation:
Solve using the substitution method
16x – 4y = 16
4x - 4 = y
Answer:
y = 4 x − 4
Step-by-step explanation:
What is the volume of a cone with a height of 27 cm
and a radius of 13 cm? Round your answer to the
nearest tenth.
Use the button on your calculator to complete this
problem.
V=
cm3
Answer:4778.3 cm^3
Step-by-step explanation: The formula for volume of a cone is V=1/3h pi r^2. By plugging in the height and the radius we get our answer.
Answer:
4778.4 :)
Step-by-step explanation:
Assume that there is a 8% rate of disk drive failure in a year. a. If all your computer data is stored on a hard disk drive with a copy stored on a second hard disk drive, what is the probability that during a year, you can avoid catastrophe with at least one working drive?
Answer:
0.9936 = 99.36% probability that during a year, you can avoid catastrophe with at least one working drive
Step-by-step explanation:
For each disk, there are only two possible outcomes. Either it works, or it does not. The probability of a disk working is independent of any other disk, which means that the binomial probability distribution is used to solve this question.
Binomial probability distribution
The binomial probability is the probability of exactly x successes on n repeated trials, and X can only have two outcomes.
[tex]P(X = x) = C_{n,x}.p^{x}.(1-p)^{n-x}[/tex]
In which [tex]C_{n,x}[/tex] is the number of different combinations of x objects from a set of n elements, given by the following formula.
[tex]C_{n,x} = \frac{n!}{x!(n-x)!}[/tex]
And p is the probability of X happening.
Assume that there is a 8% rate of disk drive failure in a year.
So 100 - 8 = 92% probability of working, which means that [tex]p = 0.92[/tex]
Two disks are used:
This means that [tex]n = 2[/tex]
What is the probability that during a year, you can avoid catastrophe with at least one working drive?
This is:
[tex]P(X \geq 1) = 1 - P(X = 0)[/tex]
In which
[tex]P(X = x) = C_{n,x}.p^{x}.(1-p)^{n-x}[/tex]
[tex]P(X = 0) = C_{2,0}.(0.92)^{0}.(0.08)^{2} = 0.0064[/tex]
[tex]P(X \geq 1) = 1 - P(X = 0) = 1 - 0.0064 = 0.9936[/tex]
0.9936 = 99.36% probability that during a year, you can avoid catastrophe with at least one working drive
HELP ASAP! I don’t know how to solve this problem nor where to start. Can someone please help me out here?
===============================================
Explanation:
It might help to draw out the picture as shown below. The pool itself (just the water only) is the inner rectangle. The outer rectangle is the pool plus the border of those 1 by 1 tiles.
The pool is a rectangle 90 feet by 80 feet. If we add on the tiles, then we get a larger rectangle that is 90+2 = 92 feet by 80+2 = 82 feet.
We add on 2 since we're adding two copies of "1" on either side of each dimension.
The larger rectangle's area is 92*82 = 7544 square feet
The smaller rectangle's area is 90*80 = 7200 square feet
The difference in areas is 7544-7200 = 344 square feet.
Each 1 by 1 tile is of area 1*1 = 1 sq foot, meaning that 344 tiles will get us the 344 square foot border around the pool.
the average of two number is xy.if one number is x the other i
Answer:
z = (2xy-x)
Step-by-step explanation:
Let the first number be x and the other number is z.
According to question,
The average of two number is xy i.e.
[tex]\dfrac{x+z}{2}=xy\\\\x+z=2xy\\\\z=2xy-x[/tex]
So, the value of z is (2xy-x) i.e. the other number is (2xy-x).
You are dealt one card from a 52-card deck.
a) Find the odds in favor of getting a red king.
b) Find the odds against getting a red king.
Answer:
(a)So, there are 2 kings in red- one of hearts and the other of diamonds. Therefore, the probability of selecting a red king from a deck of cards= 2/52 or 1/26.
(b) There are 6 red face cards in a 52-card deck (so 46 other cards). PROBABILITIES compare the number of favorable outcomes to the total number of possible outcomes: The PROBABILITY of getting a red face card is 6/52 = 3/26.
The odds in favor of getting a red king will be 1/26. And the odds against getting a red king will be 25/26.
What is probability?Its basic premise is that something will almost certainly happen. The percentage of favorable events to the total number of occurrences.
You are dealt one card from a 52-card deck.
Total events = 52
The odds in favor of getting a red king will be
Favorable events = 2
Then the probability will be
P = 2/52
P = 1/26
The odds against getting a red king will be
q = 1 – P
q = 1 – 1/26
q = 25/26
More about the probability link is given below.
https://brainly.com/question/795909
#SPJ2
please answer me as soon as posible
Answer:
yes your answer is right
Answer:
Yes it's Perfectly correct
Let v=-9i+j and w=-i-6j find 8v-6w
Answer:
78i+52j
Step-by-step explanation:
8(9i+2j)-6(-i-6j)
72i+16j+6i+36j
=78i+52j
Which of the following numbers is rational? Assume that the decimal patterns continue.
Answer:
[tex]\sqrt{49}[/tex]
Step-by-step explanation:
Define a rational number by a number able to expressed a fraction where the denominator is not 0 or 1.
Non-terminating (never-ending) decimals cannot be expressed as a fraction and therefore are irrational. However, recall that [tex]\sqrt{49}=7[/tex], which can be expressed as a fraction (e.g. [tex]\frac{14}{2}[/tex], etc). Thus, the answer is [tex]\boxed{\sqrt{49}}[/tex].
The awnser for this question
i need help solving this .
Answer:b
Step-by-step explanation:
Answer:
just be smart trust me u dont need us to give u the answer ur super smart
Step-by-step explanation:
make me brainliest to help people be encourage
1 Find the value of C to that the function probability density function defined as follow, also calculate the man and variance /4
X -1 0 1
f(x) 3c 3c 6c
Answer:
[tex]c = \frac{1}{12}[/tex]
The mean of the distribution is 0.25.
The variance of the distribution is of 0.6875.
Step-by-step explanation:
Probability density function:
For it to be a probability function, the sum of the probabilities must be 1. The probabilities are 3c, 3c and 6c, so:
[tex]3c + 3c + 6c = 1[/tex]
[tex]12c = 1[/tex]
[tex]c = \frac{1}{12}[/tex]
So the probability distribution is:
[tex]P(X = -1) = 3c = 3\frac{1}{12} = \frac{1}{4} = 0.25[/tex]
[tex]P(X = 0) = 3c = 3\frac{1}{12} = \frac{1}{4} = 0.25[/tex]
[tex]P(X = 1) = 6c = 6\frac{1}{12} = \frac{1}{2} = 0.5[/tex]
Mean:
Sum of each outcome multiplied by its probability. So
[tex]E(X) = -1(0.25) + 0(0.25) + 1(0.5) = -0.25 + 0.5 = 0.25[/tex]
The mean of the distribution is 0.25.
Variance:
Sum of the difference squared between each value and the mean, multiplied by its probability. So
[tex]V^2(X) = 0.25(-1-0.25)^2 + 0.25(0 - 0.25)^2 + 0.5(1 - 0.25)^2 = 0.6875[/tex]
The variance of the distribution is of 0.6875.
Choose the system of inequalities that best matches the graph below.
Answer:
"D" is the correct answer
Step-by-step explanation:
In which quadrant do the points have negative x-coordinates and negative y-coordinates?
Hi there!
»»————- ★ ————-««
I believe your answer is:
Quadrant III
»»————- ★ ————-««
Here’s why:
⸻⸻⸻⸻
The plane is split into four quadrants. Quadrant III houses all the points with negative signs for both X and Y values.⸻⸻⸻⸻
See the attached picture for reference.
⸻⸻⸻⸻
»»————- ★ ————-««
Hope this helps you. I apologize if it’s incorrect.
sets A and B have 3 and 6 elements respectively. what can be the minimum number of elements in AUB
Answer:
6
Step-by-step explanation:
n(AUB) = n(A) + n(B) - n(AnB)
n(AUB) can have the minimum number of elements if n(AnB) has the maximum number of elements.
n(AnB) maximum = 3
so n(AUB) = 3+6-3 = 6
I need help with this.
Answer:
A and b Are in quadrant 2. F and D are in quadrant 1. F is in quadrant 3 and C is in quadrant 4
Step-by-step explanation:
each quadrant is in the boxes and the question is asking what is each coordinate is in what quadrant
If y- 1 equals 10 then y
Answer:
11
Step-by-step explanation:
y-1=10
Any figure that crosses equal sign, the operational sign changes.
y=10+1
y= 11
Can someone solve this for me and a couple more questions ?
Answer:
C. -4
Step-by-step explanation:
Answer:
(c) - 4
is your right answer
Identify an equation in point-slope from for the line perpendicular to y=-4x-1 that passes through (-2, 4)
Answer:
y - 4 = ¼(x + 2)
Step-by-step explanation:
Point-slope form equation is given as y - b = m(x - a). Where,
(a, b) = a point on the line = (-2, 4)
m = slope = ¼ (sleep of the line perpendicular to y = -4x - 1 is the negative reciprocal of its slope value, -4 which is ¼)
✔️To write the equation, substitute (a, b) = (-2, 4), and m = ¼ into the point-slope equation, y - b = m(x - a).
y - 4 = ¼(x - (-2))
y - 4 = ¼(x + 2)
None of the options are correct
Simplify. v80
A. 16v5
B. 5v4
C. 4v5
D. 20v4
Hi!
√80 = √(16 • 5) = √(4² • 5) = 4√5
