Let theta be an angle such that cos(theta) = -1/9 and 0 <=theta<=2pi
Assume that cos(theta/2) < 0. Then if
Terminal point of theta/2 = (a1, a2), enter a1 and a2 in that order.
Answers
In this case, the terminal point of theta/2 can be determined using the following steps:
Since cos(theta/2) < 0 and the cosine function has a range of [-1, 1], it follows that theta/2 must be in the second or third quadrant (90 degrees to 180 degrees or 180 degrees to 270 degrees).
To find the terminal point of theta/2, we can use the formula for converting from polar to Cartesian coordinates: x = rcos(theta) and y = rsin(theta), where (x,y) is the Cartesian coordinates of the terminal point, r is the radius, and theta is the angle in radians.
In this case, the radius is 1 (since the terminal point is on the unit circle) and the angle is theta/2. Substituting these values into the formulas above, we get:
x = 1 * cos(theta/2)
y = 1 * sin(theta/2)
To find the values of cos(theta/2) and sin(theta/2), we can use the identity cos(theta/2) = sqrt((1 + cos(theta))/2) and sin(theta/2) = sqrt((1 - cos(theta))/2).
Substituting the given value of cos(theta) (-1/9) and the identities above into the formulas for x and y, we get:
x = sqrt((1 + (-1/9))/2) = -sqrt(2)/3
y = sqrt((1 - (-1/9))/2) = sqrt(8)/3
Thus, the coordinates of the terminal point of theta/2 are (-sqrt(2)/3, sqrt(8)/3).
The terminal point (a₁, a₂) for theta/2 is (-2/3, √(5) / 3).
How to find terminal point?To find the terminal point (a₁, a₂) of theta/2, use the half-angle identity for cosine:
cos(theta/2) = ± √((1 + cos(theta)) / 2)
Given that cos(theta) = -1/9 and cos(theta/2) < 0, the negative sign should be used in the half-angle identity.
Calculate cos(theta/2):
cos(theta/2) = - √((1 + cos(theta)) / 2)
cos(theta/2) = - √((1 - 1/9) / 2)
cos(theta/2) = - √((8/9) / 2)
cos(theta/2) = - √(8/18)
cos(theta/2) = - √(4/9)
cos(theta/2) = - 2/3
Cos(theta/2) = -2/3. To find the terminal point (a₁, a₂), determine the values of a₁ and a₂.
Since cos(theta/2) is negative, a₁ is negative. To find the value of a₂, use the Pythagorean identity:
cos²(theta/2) + sin²(theta/2) = 1
Since cos(theta/2) = -2/3, calculate sin(theta/2):
sin²(theta/2) = 1 - cos²(theta/2)
sin²(theta/2) = 1 - (-2/3)²
sin²(theta/2) = 1 - 4/9
sin²(theta/2) = 5/9
sin(theta/2) = ± √(5/9)
sin(theta/2) = ± √(5) / 3
Since theta is in the second quadrant (cos(theta) = -1/9), sin(theta) is positive in the second quadrant. Therefore, sin(theta/2) = √(5) / 3.
Now, sin(theta/2) = √(5) / 3.
So, the terminal point (a₁, a₂) for theta/2 is (-2/3, √(5) / 3).
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Related Questions
a + 65 = 92
please help
Answers
Answer:
a=27
Step-by-step explanation:
Answer:
a = 27
Step-by-step explanation:
a + 65 = 92
subtract 65 from both sides
a = 27
3. Jada has a coin jar containing n nickels and d dimes worth a total of $3.65. The
equation 0.05n + 0.1d = 3.65 is one way to represent this situation.
Which equation is equivalent to the equation 0.05n + 0.1d = 3.65?
A. 5n + d = 365
B. 0.5n + d = 365
C. 5n + 10d =/365
D. 0.05d + 0.1n = 365
Answers
The equivalent equation for the given equation is 5n+10d=365. Therefore, option A is the correct answer.
The given equation is 0.05n + 0.1d = 3.65.
What is an equivalent equation?Equivalent equations are algebraic equations that have identical solutions or roots. Adding or subtracting the same number or expression to both sides of an equation produces an equivalent equation. Multiplying or dividing both sides of an equation by the same non-zero number produces an equivalent equation.
Now, multiply by 100 on both sides of the given equation.
That is, 100(0.05n + 0.1d) = 3.65×100
⇒ 0.05n×100+0.1d×100=365
⇒5n+10d=365
The equivalent equation for the given equation is 5n+10d=365. Therefore, option A is the correct answer.
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A cable that weighs 8 lb/ft is used to lift 650 lb of coal up a mine shaft 600 ft deep. Find the work done. Show how to approximate the required work by a Riemann sum.
Answers
Answer:
work = 1,830,000 ft·lb
Step-by-step explanation:
You want the work done to lift 650 lb of coal 600 ft up a mine shaft using a cable that weighs 8 lb/ft.
ForceFor some distance x from the bottom of the mine, the weight of the cable is ...
8(600 -x) . . . . pounds
The total weight being lifted is ...
f(x) = 650 +8(600 -x) = 5450 -8x
WorkThe incremental work done to lift the weight ∆x feet is ...
∆w = force × ∆x
∆w = (5450 -8x)∆x
We can use a sum for different values of x to approximate the work. For example, the work to lift the weight the first 50 ft can be approximated by ...
∆w ≈ (5450 -8·0 lb)(50 ft) = 272,500 ft·lb
If we use the force at the end of that 50 ft interval instead, the work is approximately ...
∆w ≈ (5450 -8·50 lb)(50 ft) = 252,500 ft·lb
SumWe can see that the first estimate is higher than the actual amount of work, because the force used is the maximum force over the interval. The second is lower than the actual because we used the minimum of the force over the interval. We expect the actual work to be close to the average of these values.
The attached spreadsheet shows the sums of forces in each of the 50 ft intervals. The "left sum" is the sum of forces at the beginning of each interval. The "right sum" is the sum of forces at the end of each interval. The "estimate" is the average of these sums, multiplied by the interval width of 50 ft.
The required work is approximated by 1,830,000 ft·lb.
__
Additional comment
The actual work done is the integral of the force function over the distance. Since the force function is linear, the approximation of the area under the force curve using trapezoids (as we have done) gives the exact integral. It is the same as using the midpoint value of the force in each interval.
Because the curve is linear, the area can be approximated by the average force over the whole distance, multiplied by the whole distance:
(5450 +650)/2 × 600 = 1,830,000 . . . . ft·lb
Another way to look at this is from consideration of the separate masses. The work to raise the coal is 650·600 = 390,000 ft·lb. The work to raise the cable is 4800·300 = 1,440,000 ft·lb. Then the total work is ...
390,000 +1,440,000 = 1,830,000 . . . ft·lb
(The work raising the cable is the work required to raise its center of mass.)
HELP!!!
The solutions to (x + 3)^2- 4=0 are x = -1 and x = -5
True or false
Answers
Answer:
False
Step-by-step explanation:
We can simplify this equation and then solve for x.
\((x+3)^3-4=0\\\\x^2+6x+9-4=0\\\\x^2+6x+5=0\\\\(x+2)(x+3)=0\\\\x=-3\\x=-2\)
As you can see, the solutions are not x=-1 and x=-5.
Therefore, the answer is false.
Answer:
True
Step-by-step explanation:
Given
(x + 3)² - 4 = 0 ( add 4 to both sides )
(x + 3)² = 4 ( take the square root of both sides )
x + 3 = ± \(\sqrt{4}\) = ± 2 ( subtract 3 from both sides )
x = - 3 ± 2
Thus
x = - 3 - 2 = - 5
x = - 3 + 2 = - 1
(x-2y)^5 using pascal triangle
Answers
Step-by-step explanation:
I edit this on computer, because I have class later
A recipe uses 3 eggs for every 8 cups of flour. What is the ratio of eggs to flour in the recipe?
8 to 3
StartFraction 3 over 8 EndFraction
3 to 11
8:3
Answers
Answer:
3 to 8
Step-by-step explanation: It cannot be further simplified so 3 eggs to 8 cups or 3:8 ratio
HELP 7TH GRADE I WILL GIVE BRAINLIEST!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!
Answers
Answer:
8
Step-by-step explanation:
I believe that is correct
What is the slope for these cordinates
(-8,11) , (17,4)
Answers
Answer:
\(\displaystyle m = \frac{-7}{25}\)
General Formulas and Concepts:
Pre-Algebra
Order of Operations: BPEMDAS
Brackets Parenthesis Exponents Multiplication Division Addition Subtraction Left to RightAlgebra I
Coordinate Planes
Coordinates (x, y)Slope Formula: \(\displaystyle m = \frac{y_2 - y_1}{x_2 - x_1}\)
Step-by-step explanation:
Step 1: Define
Identify.
Point (-8, 11)
Point (17, 4)
Step 2: Find slope m
Simply plug in the 2 coordinates into the slope formula to find slope m.
Substitute in points [Slope Formula]: \(\displaystyle m = \frac{4 - 11}{17 - -8}\)[Order of Operations] Evaluate: \(\displaystyle m = \frac{-7}{25}\)\(\huge{\purple{\underline{\underline{\bf{\pink{ANSWER:-}}}}}}\)
We've been asked to find slope of the following coordinates which are (-8,11) and (17,4).
The standard formula to calculate slope is given by,
\(:\implies\footnotesize\rm{Slope = \frac{y_2 - y_1}{x_2 - x_1} }\)\(:\implies\footnotesize\rm{Slope = \frac{4 - 11}{17 - ( - 8)} }\)\(:\implies\footnotesize\rm{Slope = \frac{ - 7}{17 + 8} }\)\(:\implies\footnotesize\rm{Slope = \frac{ - 7}{25} }\)The slope is -7/25.-Three independent fair coins have been tossed. For i = 1, 2, 3 let
Xi = 1 if the ith coin landed heads and Xi = 0 otherwise. Let Y = X1 + X2 and
Z = X2 + X3.
a. Give Y and Z as functions on the sample space Ω. Determine the joint
probability mass function of (Y, Z).
b. Are Y and Z independent? Why (not)?
Answers
Answer:
a. Y = X1 + X2 = 1(Heads) + 1(Heads) = 2
Z = X2 + X3 = 1(Heads) + 1(Heads) = 2
The joint probability mass function of (Y, Z) can be determined by multiplying the individual probabilities:
P(Y, Z) = P(Y) * P(Z)
P(Y, Z) = (2/8) * (2/8) = 1/16
b. Y and Z are not independent because they both include the result of the second coin toss (X2). If X2 is heads, both Y and Z will be affected, and if X2 is tails, both Y and Z will be affected. Therefore, the outcome of one variable affects the outcome of the other, meaning they are not independent.
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After 4 years, $20,000 deposited in a savings account with simple interest had earned $800 in interest. What was the interest rate?
Answers
The interest rate for the savings account is 5% after 4 years, $20,000 deposited in a savings account with simple interest earned $800 in interest.
We can use the formula for simple interest to solve the problem:
Simple interest = (Principal * Rate * Time) / 100
where Principal is the initial amount deposited, Rate is the interest rate, and Time is the time period for which the interest is calculated.
We know that the Principal is $20,000 and the time period is 4 years. We are also given that the interest earned is $800. So we can plug in these values and solve for the interest rate:
$800 = (20,000 * Rate * 4) / 100
Multiplying both sides by 100 and dividing by 20,000 * 4, we get:
Rate = $800 / (20,000 * 4 / 100) = 0.05 or 5%
Therefore, the interest rate for the savings account is 5%.
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RIDDLE TIMEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEE
What is more useful when it is broken?
Answers
Answer:
a egg
Step-by-step explanation:
there are​ 15,958,866 adults in a region. if a polling organization randomly selects 1235 adults without​ replacement, are the selections independent or​ dependent? if the selections are​ dependent, can they be treated as independent for the purposes of​ calculations?
Answers
The selections in this scenario are dependent, but can be treated as independent for calculations if the sample size is less than 5% of the total population.
This is because once an individual is selected, they are no longer in the pool of potential selections for subsequent choices.
∴ The probability of selecting a certain individual changes with each selection.
However, the selected no.of adults 1235 are less than 5% of the total population (15,958,866), then the selections can be treated as independent for the purposes of calculations.
This is because the impact of the sample on the overall population is negligible, and any changes in probability after each selection are minimal.
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Let f(x)=x2+3x−10. Enter the x-intercepts of the quadratic function in the boxes. x = and x =
Answers
Answer:
X= 2.
X= -5.
Step-by-step explanation:
Hope it was helpful ;)
Are the following triangles Similar? State why or why not. *
Answers
Answer:
the answer is 56
Step-by-step explanation:
because i siad so
Marcela took out a $600 discounted loan with a 4% annual interest rate over a period of 8 months. What is the effective annual interest rate for the loan? Round to two decimal places.
Answers
find the value of the height of the tree.
Answers
Answer:
20 ft
Step-by-step explanation:
First we must find out what the scale factor is.
26 / 6.5 = 4
the scale factor is 4.
Now we must multiply 5 by 4 to find x.
5 * 4 = x
20 = x
x = 20
The height of the tree is 20 ft.
help pls im not sure what to do
Answers
Answer: 2
Step-by-step explanation:
The points (0, 4) and (2, 8) lie on the line. Using the gradient formula,
\(m=\frac{8-4}{2-0}=2\)
Please help I really do need it
Answers
Area=(3.14*1^2)/4=0.785 in^2
Will give u Brainly
Answers
Answer:
Option B
Step-by-step explanation:
Set 1:
med: 76, Range: 50-65= 15IQR: 77-73= 4
find the median, range, and IQRfor each data set.
Set 2:
med: 60, Range: 68-55= 13,IQR: 63-59= 4
76- 60 =16
find the difference of the medians 16 is 4 times 4
so the difference of the medians is 4 times the IQR
----------------------------
hope it helps...
have a great day!!
Which equation represents the line that is parallel to segment RS in the triangle below and passes through point T?
Answers
Answer:
\(y = \frac{1}{2}x +4\)
Step-by-step explanation:
Given:
\(R = (0,0)\)
\(S= (8,4)\)
\(T = (-2,3)\)
First, we have that the line is parallel to RS.
This means that the line has the same slope as RS and the slope of RS is calculated as follows:
\(m = \frac{y_2 - y_1}{x_2 - x_1}\)
\(m = \frac{4-0}{8-0}\)
\(m = \frac{4}{8}\)
\(m = \frac{1}{2}\)
So, the line has a slope of \(m = \frac{1}{2}\)
Next, we have that the line passes through \(T = (-2,3)\).
The equation of the line is then calculated using the following formula
\(y - y_1 = m(x - x_1)\)
\(y - 3 = \frac{1}{2}(x - (-2))\)
\(y - 3 = \frac{1}{2}(x +2)\)
Open bracket
\(y - 3 = \frac{1}{2}x +1\)
Make y the subject
\(y = \frac{1}{2}x +1+3\)
\(y = \frac{1}{2}x +4\)
Y=x. Given A(7,-4), B(6,4), and C(-1,-5), what are the coordinates of the vertices of A’B’C for the reflection
Answers
Answer:
The coordinates of the vertices of A'B'C' is:
A'(-4, 7)B'(4, 6)C'(-5, -1)Step-by-step explanation:
Given the coordinates of the triangle
A(7,-4)B(6,4)C(-1,-5)Reflection in the line y=x
We know that the rule of reflection in the line is y = x is:
(x, y) → (y, x)
Thus,
The coordinates of the vertices of A'B'C' for the reflection in the y = x
(x, y) → (y, x)
A(7, -4) → A'(-4, 7)
B(6, 4) → B'(4, 6)
C(-1, -5) → C'(-5, -1)
Thus, the coordinates of the vertices of A'B'C' is:
A'(-4, 7)B'(4, 6)C'(-5, -1)Below are two parallel line intersecting them
Answers
Answer:
x = 53°
Step-by-step explanation:
alternate exterior angles are equal
Answer:
Step-by-step explanation:
Two lines are parallel.
x = 53
Alternate exterior angles are equal
AC is a diameter of OE, the area of the
circle is 289 units2, and AB = 16 units.
Find BC and mBC.
B
A
C
E. plssss hurry !!
Answers
The measure of arc BC is 720 times the measure of angle BAC.
Given that AC is the diameter of the circle and AB is a chord with a length of 16 units, we need to find BC (the length of the other chord) and mBC (the measure of angle BAC).
To find BC, we can use the property of chords in a circle. If two chords intersect within a circle, the products of their segments are equal. In this case, since AB = BC = 16 units, the product of their segments will be:
AB * BC = AC * CE
16 * BC = 2 * r * CE (AC is the diameter, so its length is twice the radius)
Since the area of the circle is given as 289 square units, we can find the radius (r) using the formula for the area of a circle:
Area = π * r^2
289 = π * r^2
r^2 = 289 / π
r = √(289 / π)
Now, we can substitute the known values into the equation for the product of the segments:
16 * BC = 2 * √(289 / π) * CEBC = (√(289 / π) * CE) / 8
To find mBC, we can use the properties of angles in a circle. The angle subtended by an arc at the center of a circle is double the angle subtended by the same arc at any point on the circumference. Since AC is a diameter, angle BAC is a right angle. Therefore, mBC will be half the measure of the arc BC.
mBC = 0.5 * m(arc BC)
To find the measure of the arc BC, we need to find its length. The length of an arc is determined by the ratio of the arc angle to the total angle of the circle (360 degrees). Since mBC is half the arc angle, we can write:
arc BC = (mBC / 0.5) * 360
arc BC = 720 * mBC
Therefore, the length of the arc BC equals 720 times the length of the angle BAC.
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what is a congruent polygon
Answers
A congruent polygon refers to two or more polygons that have the same shape and size. There must be an equal number of sides between two polygons for them to be congruent.
Congruent polygons have parallel sides of equal length and parallel angles of similar magnitude. When two polygons are congruent, they can be superimposed on one another using translations, rotations, and reflections without affecting their appearance or dimensions. Concluding about the matching sides, shapes, angles, and other geometric properties of congruent polygons allows us to draw conclusions about them.
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Which choices are equivalent to the fraction below? Check all that apply.
18/36
1.6/18
2.8/18
3.1/3
4.1/2
5.3/4
6.6/12
Answers
The population of Nashville, TN is 691,243. The population of Spring Hill, TN is 39,602. How many more people live in Nashville?
Answers
Answer:
651,641 more people live in Nashville.
Step-by-step explanation:
Since more people live in Nashville, we subtract the population of Nashville by the population of Spring Hill to get:
691,243 - 39,602
= 651,641
А
Bag A contains 6 white beads and 3 black beads.
Bag B contains 6 white beads and 4 black beads.
One bead is chosen at random from each bag.
Find the probability that
(1) both beads are black,
(il) at least one of the two beads is white.
The beads are not replaced.
A second bead is chosen at random from each bag.
Find the probability that
(iii) all four beads are white,
Answers
If f(-3) = 7 and f'(x) ≤ 9 for all x, what is the largest possible value of f(4)?
Answers
Answer:
The maximum value f(4) can have is 70
f(4) = 70
Step-by-step explanation:
For the largest possible value, the derivative must be greatest,
so, for our case, since f'(x) ≤ 9,
but for largest value, f'(x) must be greatest, hence it must be,
f'(x) = 9.
With this derivative,
Using the value,
f(-3) = 7,
with each step, we increase by 9 units
so, f(-2) = f(-3) + 9 = 7 + 9 = 16
f(-2) = 16
going till f(4),
f(-1) = 16+9
f(-1) = 25
f(0) = 25 + 9 = 34
f(1) = 34 + 9 = 43
f(2) = 43 = 9 = 52
f(3) = 52 + 9 = 61
f(4) = 70
So,
the maximum value f(4) can have is 70
Evaluate: 2-4 А. =100 В. -8 ОО С. -16 D. 1 16
Answers
Answer:
D. 1/16
Step-by-step explanation:
Evaluate: 2^-4
А. =100
В. -8ОО
С. -16
D. 1/16
Given
2^-4
= 1 / 2⁴
= 1 / (2 * 2 * 2 * 2)
= 1 / 16
Therefore,
2^-4 = 1/16
D. 1/16
Which polygon is a unit tile in this tessellation
Answers
Answer:
Option no. B is a unit tile in that tesselation
i.e rectangular shape
Step-by-step explanation:
Well, it's because we all know that tesselation
only use one shape and it's sum is always 360°
meanwhile sum of rectangle=360°
as each side of rectangle are 90°
90°*4=360°##
I hope it helped you to understand more clearly...