Evaluate sin[2 arccos( -8/17 )].

Answers

Answer 1

Recall that [tex]\sin2x=2\sin x\cos x[/tex], so that

[tex]\sin\left(2\arccos\left(-\dfrac8{17}\right)\right)=2\sin\left(\arccos\left(-\dfrac8{17}\right)\right)\cos\left(\arccos\left(-\dfrac8{17}\right)\right)[/tex]

Dealing with the cosine terms is simple, since [tex]\cos(\arccos x)=x[/tex] (so long as [tex]0\le \arccos x\le\pi[/tex], which is true here). So

[tex]\cos\left(\arccos\left(-\dfrac8{17}\right)\right)=-\dfrac8{17}[/tex]

To find the sine, we can make a substitution of [tex]\theta=\arccos\left(-\dfrac8{17}\right)[/tex]. In other words, we suppose that [tex]\theta[/tex] is some angle that satisfies [tex]\cos\theta=-\dfrac8{17}[/tex].

If this is the case, then we have

[tex]\sin^2\theta=1-\cos^2\theta=\dfrac{225}{289}\implies\sin\theta=\pm\sqrt{\dfrac{225}{289}}=\pm\dfrac{15}{17}[/tex]

But which value do we choose? We know that [tex]\cos\theta[/tex] is negative, which means [tex]\theta[/tex] must be between [tex]\dfrac\pi2[/tex] and [tex]\pi[/tex]. The sine of any angle [tex]\theta[/tex] in this interval will always be positive, so in fact

[tex]\sin\theta=\sin\left(\arccos\left(-\dfrac8{17}\right)\right)=\dfrac{15}{17}[/tex]

Putting everything together, we get

[tex]\sin\left(2\arccos\left(-\dfrac8{17}\right)\right)=2\left(\dfrac{15}{17}\right)\left(-\dfrac8{17}\right)=-\dfrac{240}{289}[/tex]

Related Questions

The volume of a cube is 1,953.125 centimeters cubed. Find the length of the sides.

Answers

It will be [12.5] I think. Tell me if it is right

Rational and Irrational Numbers what is it explain

Answers

A rational number is part of a whole expressed as a fraction, decimal or a percentage. ... It is a numberthat cannot be written as a ratio of two integers (or cannot be expressed as a fraction). For example, the square root of 2 is an irrational number because it cannot be written as a ratio of two integers.

rational numbers can be expressed as fraction decimal or percentage but irrational numbers cannot be written as two integers

A student is selling baked goods in the halls before school as a fundraiser. She sells cinnamon rolls, donuts, and croissants. Students purchase the goods according to the probability distribution:

X(Product): cinnamon roll, donut, croissant
P(x): 0.20, 0.55, 0.25

A): If a cinnamon roll yields $0.25 profit, a donut yields $0.20 profit, and a croissant yields a $0.30 profit, how much profit will be made on each customer?

B):If there are 50 customers, how much can you expect her to make?

Answers

a). Profit made by each customer: (0.25 * 0,2) + (0.20 * 0.55) + (0.30 * 0.25) = 0.05 + 0.11 + 0.07 = $0.235

b). Money made if there are 50 customers: $0.235 * 50 = $11.75

deena has a set of 4 coasters on her coffee table, and each coaster is shaped like a cylinder with a height of 0.5 inches and a radius of 2.5 inches. which is the total surface area of the 4 coasters?

Answers

It’s about 188.5 in squared

Final answer:

The total surface area for the 4 cylindrical coasters, each with a height of 0.5 inches and a radius of 2.5 inches, is approximately 188.4956 square inches.

Explanation:

Deena has a set of 4 coasters, each shaped like a cylinder.

To calculate the surface area of one coaster, we need to find the area of the two circular bases and the area of the side that wraps around the cylinder (the lateral surface area).

The formula for the surface area of a cylinder is SA = 2πr(h+R), where r is the radius, h is the height, and R is the radius of the bases.

Given a height of 0.5 inches and a radius of 2.5 inches:

SA = 2π(2.5)(0.5+2.5)

SA = 2π(2.5)(3)

SA = 2π(7.5)

SA = 15π

The result is the surface area for one coaster. As we have 4 coasters, we multiply this result by 4:

Total surface area for 4 coasters = 4 × 15π ≈ 4 × 47.1239 ≈ 188.4956 square inches.

How do you factor 4x^2+20x-3xy-15y?

Answers

i dont know the answer. but try looking it up on photomath. its an app for your phone, its reallyyyyy helpful

A field is shaped like the figure shown. What is the area of the field? Use 3.14 for pi

Answers

I attached the image for this question.

Answer:
area of field = 146.24 m²

Explanation:
Noticing the figure, we can see that it is composed of three regular shapes:
1- a square
2- a triangle
3- quarter of a circle

We will get the area of each shape separately and then add them to get the total area of the field.

1- getting the area of the square:
area of the square = (side)²
we have the side = 8 m
Therefore:
area of square = (8)² = 64 m² ...............> I

2- getting the area of the triangle:
area of triangle = [tex] \frac{1}{2} [/tex] * base * height
We have the base = 8 m and the height = 8 m
Therefore:
area of triangle = [tex] \frac{1}{2} [/tex] * 8 * 8 = 32 m² ..........> II

3- getting the area of the quarter circle:
area of circle = πr²
area of quarter circle = [tex] \frac{1}{4} [/tex] * πr²
We have π = 3.14 and r = 8 m
Therefore:
area of quarter circle = [tex] \frac{1}{4} [/tex] * 3.14 * (8)² = 50.24 m² .....> III

4- getting the total area of field:
Total area of field = I + II + III
Total area of field = 64 + 32 + 50.24
Total area of field = 146.24 m²

Hope this helps :)

Answer:

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Each small cube is 1 cm3. The length of the large cube is 12 cm. What is the volume of the large cube? A. 24 cm3 B. 144 cm3 C. 864 cm3 D. 1,728 cm3

Answers

The answer to your question is D. 1,728

Answer:

Step-by-step explanation:

Carlos is 63 inches tall. What is his height in feet?

Answers

Carlos is 5feet and 3inches y’all

5 ft. 3in. Is his height (one foot is 12 in.)

How do I find the slope of the line through (–9, –10) and (–2, –5)?

Answers

Finding a slope in a number line is easy just
Steps:
Identify two points on the line
Select one to be (x 1, y 1) and the other to be (x 2, y 2)
Use the slope the slope equation to calculate slope.
Thank you

Answer:
slope of line is 5/7

Explanation:
The slope of the line can be calculated using the following rule:
slope = [tex] \frac{y2-y1}{x2-x1} [/tex]

The given points are:
(-9,-10) which represent (x1,y1)
(-2,-5) which represent (x2,y2)

Substitute with the points in the above equation to get the slope as follows:
slope = [tex] \frac{-5-(-10)}{-2-(-9)} [/tex] = 5/7

Hope this helps :)

In a layout of Mark’s backyard, the ratio is 1 centimeter = 10 meters. The length of the deck on the layout is 4 cm and the width is 4 cm. What is the perimeter of Mark’s deck?

Answers

Think the answer is 10 x 4 x 4 =160

The perimeter of Mark's deck in real life is calculated by converting the layout measurements to actual measurements using the scale factor and then applying the perimeter formula. The perimeter is found to be 160 meters.

To calculate the perimeter of Mark's deck, which is represented on a layout where the scale is 1 centimeter = 10 meters, we use the perimeter formula for a rectangle: Perimeter (P) = 2 * (length + width). Since both the length and width of Mark's deck are provided as 4 centimeters each, and each centimeter corresponds to 10 meters in reality, we first convert the measurements from centimeters to meters by multiplying them by the scale factor.

Length in meters = 4 cm * 10 meters/cm = 40 meters
Width in meters = 4 cm * 10 meters/cm = 40 meters

Now, calculate the perimeter:

P = 2 * (40 meters + 40 meters)
P = 2 * 80 meters
P = 160 meters

The real-life perimeter of Mark's deck is 160 meters.

Iris had $900. She spend 22% of this amount on a mobile phone. How much did she pay for the mobile phone

Answers

you would change 22 to a decimal which would be .22 then multiply that by 900 which would be 198. So she spent 198 dollars on the phone

Answer:

198

Step-by-step explanation:

Of means multiply and 22% as a decimal is 0.22, so multiply 0.22 by 900.

900*0.22=198

find the length of the base indicated for each trapezoid

Answers

Final answer:

To find the base of a trapezoid, you can view the trapezoid as two right triangles joined together. The base of the trapezoid is the sum of the bases of these triangles. Depending on the orientations of the sides of the trapezoid and the height, you might need to use the Pythagorean theorem.

Explanation:

The question asks to find the length of the base for a given trapezoid. In order to do this, we can use concepts from geometry and trigonometry. Consider the trapezoid as two right triangles back to back. The base of the trapezoid is the sum of the bases of the two right triangles.

For example, if you have a trapezoid with sides A and B, and the height is H, you can use these sides and the trigonometric principles to calculate the lengths of the bases. If side A or B is not perpendicular to the base, you would need to use the Pythagorean theorem. That formula is A^2+B^2=C^2, where C is the hypotenuse of the right triangle formed.

In case, the sides A and B are perpendicular to the base, the lengths of the bases would simply be side A + side B. This would give you the total length of the base of the trapezoid.

Learn more about Trapezoid Base Length here:

https://brainly.com/question/31380175

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If a store orders a total of 55 of flour, how many 25-pound bags does the store order

Answers

5. It would be 4, except they need 55, and that 5 extra makes it 5

The store would only have 2 25-pound bags of flour

What is 6.9-(-5)+4.1

Answers

here. the answer is 16

a rectangular field has sides with the length of 120 yards in 62 yards to the nearest tenth of a yard find the length of the diagonal of the field

Answers

The Pythagorean theorem is used for this
.. diagonal = √(120² +62²) = √18244 ≈ 135.1 . . . . yards

Question 19 makes no sense

Answers

Think: To write the equation of a line in slope-intercept form y=mx+b, you must have the slope (m) and the y-intercept (b).

g(0)= -6 tells us that the y-intercept is (0, -6). Thus, b= -6.

(0,-6) and (15,-9) are points on the line. Find the slope of the line. It is:
-9-(-6)
m = ------------- = -3/15 = -1/5
15-0

Thus, the equation of the line in question is y=(-1/5)x - 6.

Important that you understand that g(0)=-6 is also represented by (0,-6), and that -6 is the y-intercept.

What components are important in a linear function?

Answers

In a linear function, slope and the y-intercept are important.

The slope tells us how the line changes as it goes along, and the y-intercept tells where it begins on the positive x-axis.

Which measure is equivalent to 48 quarts? 1 cup = 8 fluid ounces 1 quart = 4 cups

Answers

it would be 12 liquid gallons

If BC -16 and BD-9 then find the length of BA. Show the work you use to find your answer. Draw the two similar triangles separately to help set up the proportion.

Answers

In right triangle the leg is geometrical mean of its projection on the hypotenuse and the hypotenuse. Thus,

[tex] AB^2=BD\cdot BC [/tex].

Substitude BC=16, BD=9, then

[tex] AB^2=9\cdot 16,\\ AB^2=144,\\ AB=12 [/tex].

Answer: AB=12

what are the mean median mode and range of the data set given the altitude of lakes in feet -9, -36, -23, -8, -17, -52, -27, and -36 A. mean=-25; median=-26; mode=-44; range=36 B. mean=-25; median=-36; mode=-36; range=44 C. mean=-26; median=-25; mode=-36; range=44 D. mean=-26; median=-44; mode=-25; range=36

Answers

Hi there!

The first step in solving this should be to determine what mean, median, mode and range are and how to find them.

The "mean" is essentially just the average of a set of numbers, and it's found by finding the sum of the numbers then dividing by the number of add-ins.
The "median" is the middle number in a set. This is found by arranging the numbers from smallest to largest and finding the center number. Sometimes, if there is an even number of items in a set, you'll have to find the median by taking the mean of the two middle number. (For example, if your set is 1, 2, 3, 4 you would find the median by looking or the middle number(s) which in this case would be 2 and 3. To find the median, you'd take 2+3 which equal 5 and divide that by the number of items- 2 which gives you 2.5).
The "mode" is the is simply whichever number occurs the most in a set.
The "range" is the difference between the largest and smallest numbers in a set; in other words, you take the largest number in the set and subtract the smallest number in the set from it (i.e if the largest number is 5 and the smallest number is 3, you'd take 5-3 which gives you the range of 2).

Now the, on to solving your problem!
The set you were given consists of the numbers -9, -36, -23, -8, -17, -52, -27, and -36.

To find the mean, we would take all of the numbers and add them up:
-9+(-36)+(-23)+(-8)+(-17)+(-52)+(-27)+(-36) = 208
Then, you divide the sum by the number of add-ins, in this case, 8 (since there are 8 numbers in the set).
-208/8 = -26.
So the mean = -26.

The median can be found next, by arranging the numbers from smallest to largest:
-52, -36, -36, -27, -23, -17, -9, -8
As we can see, the middle-most numbers are -27 and -23, so what we have to do is add the number which gives us -50 and divide by 2 which gives us the answer of -25.
So the median is -25.

Next is the mode, which can be found by seeing which number occurs the most. As we can see by looking at the set, -36 is the only number that occurs more than once (occurring twice).
Since it occurs the most, the mode is -36.

Finally, we need to find the range. In order to do this, we take the highest number (-8) and subtract the lowest number (-52).
-8-(-52)= 44
Keep in mind that when you're subtracting a negative, it becomes addition.
And there we go, the range is 44.

So now we just have to look at your choices, and you can see that option C matches the answers, since the mean= -26, the median= -25, the mode= -36, and the range = 44.

Truly hope this helps!

Answer:

Step-by-step explanation:

what does it mean to "square a circle"?

Answers

it is an expression like when someone tells you good luck squaring a circle its like good luck doing the impossible

Final answer:

In mathematics, 'squaring the circle' means to create a square with the same area as a given circle, though it is impossible to achieve precisely due to the nature of π. When fitting a circle in a square, the side of the square is twice the radius of the circle, and the circle's area is approximately three-quarters that of the square.

Explanation:

To "square a circle" in mathematical terms traditionally means to create a square with the same area as a given circle, using only a finite number of steps with compass and straightedge. This problem is known to be impossible to solve exactly due to the transcendental nature of π (pi). However, when we discuss fitting a circle within a square, we often refer to having the diameter of the circle, which is twice the radius (2r), fit exactly across the side length of the square (a).

The area of the circle is πr², and when the circle fits within a square the area of the square is a². Since a = 2r, the area of the square can be expressed as 4r². From this, we can approximate that a circle's area would be less than the area of the square but more than half, possibly around three-quarters, thus approximately 3r².

Perimeter-wise, a circle within a square would have a circumference less than the perimeter of the square but significantly longer than a straight line passing through the center of the square. This understanding allows for practical estimations without requiring exact precision, which can be particularly useful in everyday contexts where an approximation suffices.

At a ski resort there are three different chair lifts that will take you to the top of the mountain. There are six ski trails to the bottom of the mountain. How many different paths are there to go up and down the mountain?

Answers

3 * 6 = 18 . . . . . ways up and down.

What do you call a person who makes faces all day long

Answers

A person who makes faces all day is often described as expressive or animated, and may be humorously termed a 'mugger' or 'face-puller'. Facial expressions are crucial for conveying emotion in human interaction and can be particularly striking in literature and dramatic performances.

The question 'What do you call a person who makes faces all day long?' is asking about a term or description for someone who frequently changes their facial expressions. While there is no specific term that describes this behavior in a clinical or professional sense, in a colloquial context, you might call such a person expressive, animated, or if the expressions are humorous or exaggerated, a 'mugger' or 'face-puller'. In literature and drama, characters known for their expressive faces can bring a story to life or provide comic relief.

Facial expressions are an integral part of human communication. They can underscore the emotional state of a person, such as the sense of pride on the face of a monomaniacal king, the devotion seen in a religious fanatic, or the terror described in the imagery of wallpaper patterns appearing to have expressive eyes. Whether in social interactions, literature, or public speaking, the alignment of facial expressions with spoken words is important for conveying meaning and emotion.

What is 1 1/2 subtracted from the sum of 4 2/3 and 5 2/5

Answers

Hi there!

First, we need to add 4 2/3 and 5 2/5.

We should begin with converting the fractions into improper fractions.
4 2/3 + 5 2/5
= 14/3 + 27/5

Now, we find the common denominator.
= 70/15 + 71/15

We can add:
= 141/15

And simplify.
= 9 2/5

Finally, we subtract 1 1/2 from 9 2/5.
9 2/5 - 1 1/2
= 47/5 - 3/2
= 94/10 - 15/10
= 79/10
= 7 9/10

Your answer is 7 9/10.

Hope this helps!

7 9/10 is the correct answer. I use the wonderful world of google to help me with questions like that one

Christopher just found beautiful yarn for 20% off at his favorite yarn store. He can make one scarf from 2/3 of a ball of yarn. If Christopher buys 12 balls of yarn, how many scarves can he make?

Answers

Answer:

Step-by-step explanation:

if 2/3 is used by one scarf, then we can divide 2/3 into 12

12/1 × 2/3

12/1 ×3/2= 36/2

36/2= 16

Christopher can create a total of 12 scarves with the 12 balls of yarn he buys, based on the ratio of 2/3 of a ball needed for one scarf.

Christopher can make scarves from yarn, where it takes 2/3 of a ball of yarn to make one scarf. If he buys 12 balls of yarn, we need to calculate the total number of scarves he can make.

Let's find out how many scarves he can create with 12 balls:

Firstly, determine how many scarf units are in one ball of yarn. Since it takes 2/3 of a ball to make a scarf, 1 ball equals 1.5 scarf units (since 1 / (2/3) = 1.5).Next, multiply the number of scarf units in one ball by the total number of balls. 1.5 scarf units per ball \\times 12 balls = 18 scarf units.Finally, since each scarf uses 2/3 of a ball, we divide the total number of scarf units by 1.5 to get the number of scarves. 18 scarf units / 1.5 = 12 scarves.

Therefore, Christopher can make 12 scarves with 12 balls of yarn.

HELP ME @LOULI pleaseeeeeeeee

Answers

the equation will be y = x - 700

am i right?

Answer:
y = x - 700

Explanation:
The equation of the line in slope-intercept form has the following formula:
y = mx + c
where:
m is the slope
c is the y-intercept

1- getting the slope:
To get the slope, we will need two points from the graph. I will choose (700,0) and (0,-700)
slope = (y2-y1) / (x2-x1) = (-700-0) / (0-700) = 1
The equation now is:
y = 1x + c
y = x + c

2- getting the y-intercept:
The y-intercept means the value of the y at x = 0. We can get this directly from the graph.
From the graph, we can find that:
at x = 0, y = -700
Therefore:
c = -700

Based on the above, the equation of the line is:
y = x - 700

Hope this helps :)

How could you calculate each product? Describe the strategy you would use. Then calculate.

a) 9 x 5

b) 5 x 8

Answers

Hey there!

1. To solve 9 x 5 , I could add the number 9 five times . . .

9 + 9 + 9 + 9 + 9 = 45

9 x 5 = 45 <----

2. I would use the same strategy as the first one, except add the number 8 five times . . .

8 + 8 + 8 + 8 + 8 = 40

8 x 5 = 40 <----

Hope this helps you.
Have a great day!

Ratio of dogs to cats is 9:4. If there are 54 dogs, how many cats?

Answers

There are 24 cats I believe. You multiply 6 to 9 to get 54. So multiply 6 to 4 and KABLAMMO

I need help what is 80% Of 5.00

Answers

To find a percent of a number, convert the percent to a decimal and multiply by the number.

80% of 5.00 =

= 80% * 5.00

= 0.80 * 5.00

= 4.00

480 x 5
100= 4 its four lil bro

When a=-9, and b=-6, which expression has a value of -3?
A. a+b
B. a-b
C. |a+b|
D. |a-b|

Answers

The answer is B. a-b
Hope this helps

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