Find the standard normal area for each of the following(round your answers to 4 decimal places)

To obtain the required equation we divide the equation by sin²Ø.

The first six functions are trigonometric, with the domain value being the angle of a right triangle and the range being a number. The angle, expressed in degrees or radians, serves as the domain and the range of the trigonometric function (sometimes known as the "trig function") of f(x) = sin. Like with all other functions, we have the domain and range. In calculus, geometry, and algebra, trigonometric functions are often utilised.

The given equation is:

sin²Ø+cos²Ø=1

To obtain the required equation we divide the equation with sin²Ø:

sin²Ø/sin²Ø +cos²Ø/ sin²Ø = 1/sin²Ø

1 + cot²Ø = csc²Ø

Hence, to obtain the required equation we divide the equation by sin²Ø.

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Answer:

Required length is 13 feet

Step-by-step explanation:

[tex]{ \rm{length = \sqrt{ {12}^{2} + {5}^{2} } }} \\ \\ { \rm{length = \sqrt{144 + 25} }} \\ \\ { \rm{length = \sqrt{169} }} \\ \\ { \rm{length = 13 \: feet}}[/tex]

Answer:

The value of the derivative at (0, 0) is zero.

Step-by-step explanation:

Given function:

[tex]f(x)=\dfrac{x^2}{x^2+4}[/tex]

To differentiate the given function, use the quotient rule and the power rule of differentiation.

[tex]\boxed{\begin{minipage}{5.4 cm}\underline{Quotient Rule of Differentiation}\\\\If $y=\dfrac{u}{v}$ then:\\\\$\dfrac{\text{d}y}{\text{d}x}=\dfrac{v \dfrac{\text{d}u}{\text{d}x}-u\dfrac{\text{d}v}{\text{d}x}}{v^2}$\\\end{minipage}}[/tex]

[tex]\boxed{\begin{minipage}{5.4 cm}\underline{Power Rule of Differentiation}\\\\If $y=x^n$, then $\dfrac{\text{d}y}{\text{d}x}=nx^{n-1}$\\\end{minipage}}[/tex]

[tex]\boxed{\begin{minipage}{5.4cm}\underline{Differentiating a constant}\\\\If $y=a$, then $\dfrac{\text{d}y}{\text{d}x}=0$\\\end{minipage}}[/tex]

[tex]\begin{aligned}\textsf{Let}\;u &= x^2& \implies \dfrac{\text{d}u}{\text{d}{x}} &=2 \cdot x^{(2-1)}=2x\\\\\textsf{Let}\;v &=x^2+4& \implies \dfrac{\text{d}v}{\text{d}{x}} &=2 \cdot x^{(2-1)}+0=2x\end{aligned}[/tex]

Apply the quotient rule:

[tex]\implies f'(x)=\dfrac{v \dfrac{\text{d}u}{\text{d}x}-u\dfrac{\text{d}v}{\text{d}x}}{v^2}[/tex]

[tex]\implies f'(x)=\dfrac{(x^2+4) \cdot 2x-x^2 \cdot 2x}{(x^2+4)^2}[/tex]

[tex]\implies f'(x)=\dfrac{2x(x^2+4)-2x^3}{(x^2+4)^2}[/tex]

[tex]\implies f'(x)=\dfrac{2x^3+8x-2x^3}{(x^2+4)^2}[/tex]

[tex]\implies f'(x)=\dfrac{8x}{(x^2+4)^2}[/tex]

An extremum is a point where a function has a maximum or minimum value. From inspection of the given graph, the minimum point of the function is (0, 0).

To determine the value of the derivative at the minimum point, substitute x = 0 into the differentiated function.

[tex]\begin{aligned}\implies f'(0)&=\dfrac{8(0)}{((0)^2+4)^2}\\\\&=\dfrac{0}{(0+4)^2}\\\\&=\dfrac{0}{(4)^2}\\\\&=\dfrac{0}{16}\\\\&=0 \end{aligned}[/tex]

Therefore, the value of the derivative at (0, 0) is zero.

Answer:

D

Step-by-step explanation:

the correct answer is D

Answer: False

Step-by-step explanation:

25% of the data is between Q1 and the median.

Answer: 5 standard glasses

A standard glass of wine is 5 ounces. This is typical for any dry white, red, orange, or rosé wine. A standard bottle is 750mL, or about 25 ounces of wine. So, a normal 750mL bottle has 5 standard glasses of wine.

Answer:

3. One solution

Step-by-step explanation:

-x²-9 = 6x

or, x²+6x+9 = 0

or, x²+2.x.3+3² = 0 [using (a+b)² = a²+2ab+b²]

or, (x+3)² = 0

or, x+3 = 0

x = -3

We'll presume that the cash in question were initially split between two accounts since we don't know the answer to question #9: the amount that has been sitting in a savings account for 60 days is $78.00.

Because the FDIC for savings accounts and the NCUA for community bank accounts guarantee all deposit made by consumers, savings are a secure location to put your money.

Savings accounts might assist you avoid overspending by keeping the money away from your spending account. You should save emergency cash in your bank account for easy access. Savings accounts keep money secure because the Deposit Insurance Corporation of the United States insures them for up to $250,000.

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200 people invested in neither stocks nor bonds for retirement.

The inclusion-exclusion principle is a counting method used to determine the size of a set created by joining two or more sets. It is predicated on the notion that if we just sum the set sizes, we can wind up counting certain components more than once (the elements that are in the intersection of the sets). We deduct the sizes of the sets' intersections from the sum of their sizes to prevent double counting.

The total number of people who invested in stocks are:

Total = Stocks + Bonds - Both

Total = 700 + 400 - 300

Total = 800

Using the inclusion- exclusion principle:

neither = Total surveyed - Total

neither = 1000 - 800 = 200

Hence, 200 people invested in neither stocks nor bonds for retirement.

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Therefore , the solution of the given problem of standard deviation comes out to be the group standard deviation in order to use (b) z.

Statistics uses variance as a way to quantify difference. The image of the result is used to compute the average deviation between the collected data and the mean. Contrary to many other valid measures of variability, it includes those pieces of data on their own by comparing each number to the mean. Variations may be caused by willful mistakes, irrational expectations, or shifting economic or business conditions.

Here,

We use the z-distribution if the total standard deviation is known; otherwise, we use the t-distribution.

Additionally, for small sample sizes, the t-distribution is used, whereas for big sample sizes, the z-distribution is used.

The fact that z requires that you know the population standard deviation, and that this is frequently not known in practice, is one reason to use a distribution rather than the traditional .

Normal curve to find critical values when computing a level C confidence interval for a population mean.

You must be aware of the group standard deviation in order to use (b) z.

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Answer:

  (a) 0.0955

  (b) 0.0214

  (c) 0.9545

  (d) 0.3085

Step-by-step explanation:

You want the area under the standard normal PDF curve for intervals (1.22, 2.15), (2.00, 3.00), (-2.00, 2.00), and (0.50, ∞).

The probability functions of a suitable calculator or spreadsheet can find these values for you. The attachment shows one such calculator. Its "normalcdf" function takes as arguments the lower bound and upper bound.

We used 1E99 as a stand-in for "infinity" as recommended by the calculator's user manual. For the purpose here, any value greater than 10 will suffice.

The values of a continuous function f are given which are false is  By the Mean Value Theorem applied to f on the interval (2,5), there is a value c such that f' (c) = 10. So, the correct option is statement (B). Let f be the function defined by f (x) = r - 6x2 + 9x + 4 for 0 < 3 < 3 then f is decreasing on the interval (1,3) because f' (2) < 0 on the interval (1, 3). So, the correct option is D).

For continuous function f the statement (B) is false. Although the Mean Value Theorem guarantees the existence of a point c such that f'(c) = (f(5)-f(2))/(5-2) = 2, there is no guarantee that this value will be exactly 10.

When f (x) = r - 6x2 + 9x + 4 for 0 < 3 < 3 is statement (D) is true. We have f'(x) = -12x + 9, which is negative for x in the interval (1,3). Therefore, f is decreasing on this interval. Statement (A) is false, as f'(2) = 3 is positive, so f is increasing on the interval (0,1).

Statement (B) is also false, as f'(x) is not negative on the interval (0,1). Statement (C) is false, as f" (x) = -12 is negative everywhere, so f is concave down on the entire interval (0,3).

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Answer:

Step-by-step explanation:

long leg = 78  (means that 26√3*√3 = 26√9 = 26*3 = 78

for x: Short leg= 26√3

Hypotenuse= 2*26√3 = 52√3 for y

According to one meaning of the phrase, it merely refers to the selling price of something. For instance, a piece of art would be sold for that amount if bids reached a record high of $10 million. Thus, option A is correct

The sale price of bananas this week is 10% off the original price of $1.40 per pound, which means the sale price is:

$1.40 - 10% of $1.40 = $1.26 per pound

To find the total cost of 6 pounds of bananas this week, we can multiply the sale price by the number of pounds:

$1.26 per pound * 6 pounds = $ [tex]7.56[/tex]

Therefore, the total price of 6 pounds of bananas this week at the grocery store is $ [tex]7.56[/tex] .

However, we need to be careful with the answer choices provided. They all differ from $7.56, so we need to double-check our calculations.

If we add a 10% discount to $1.40 per pound, we get:

$1.40 - (10/100)*$1.40 = $1.26 per pound

And the total cost of 6 pounds at $1.26 per pound is:

$ [tex]1.26 \times 6[/tex] = $ [tex]7.56[/tex]

Therefore,  $ [tex]8.19[/tex] is not a possible answer, and the other options are either miscalculated or rounded incorrectly.

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Answer: A) 1/49

Step-by-step explanation:

Since the spinner has seven equal sections numbered 1 through 7, the theoretical probability of landing on any particular number on a single spin is 1/7.

To find the theoretical probability that the spinner lands on 2 and then an odd number, we can multiply the probability of landing on 2 on the first spin by the probability of landing on an odd number on the second spin.

The probability of landing on 2 on the first spin is 1/7, and the probability of landing on an odd number on the second spin is 3/7 (there are three odd numbers among the remaining six sections).

Therefore, the theoretical probability of the spinner landing on 2 and then an odd number is:

(1/7) x (3/7) = 3/49

So the answer is A) 1/49.

a) F(x,y,z) = y'z'. b) F(x,y,z) = x'y'z' + x'y'z + xy'z'. c) F(x,y,z) = x'y'z'. d) F(x,y,z) = x'y'z + x'yz' + xy'z' + x'y'z'. These are the sum-of-products expansions of the Boolean function F(x, y, z) for the given conditions.

a) When x = 0, F(x,y,z) equals 1 if and only if yz = 0. This can be expressed as the sum of products: F(x,y,z) = y'z' (read as "not y and not z").

b) When xy = 0, F(x,y,z) equals 1 if and only if either x = 0 or y = 0. This can be expressed as the sum of products: F(x,y,z) = x'y'z' + x'y'z + xy'z' (read as "not x and not y and not z" OR "not x and not y and z" OR "x and not y and not z").

c) When x + y = 0, F(x,y,z) equals 1 if and only if x = y = 0. This can be expressed as the sum of products: F(x,y,z) = x'y'z' (read as "not x and not y and not z").

d) When xyz = 0, F(x,y,z) equals 1 if and only if x = 0 or y = 0 or z = 0. This can be expressed as the sum of products: F(x,y,z) = x'y'z + x'yz' + xy'z' + x'y'z' (read as "not x and not y and z" OR "not x and y and not z" OR "x and not y and not z" OR "not x and not y and not z").

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Answer:

the question is incomplete, so I looked for similar questions:

There are 3 sandwiches, 4 drinks, and 2 desserts to choose from.

the answer = 3 x 4 x 2 = 24 possible combinations

Explanation:

for every sandwich that we choose, we have 4 options of drinks and 2 options of desserts = 1 x 4 x 2 = 8 different options per type of sandwich

since there are 3 types of sandwiches, the total options for lunch specials = 8 x 3 = 24

If the numbers are different, all we need to do is multiply them. E.g. if instead of 3 sandwiches there were 5 and 3 desserts instead of 2, the total combinations = 5 x 4 x 3 = 60.

For this question's answer, there are 2 x 5 = 10 lunch specials are possible.

The number of lunch specials possible are 10.

We can use combinations for this case,

Total number of distinguishable things is m.

Out of those m things, k things are to be chosen such that their order doesn't matter.

This can be done in total of

[tex]^mC_k = \dfrac{m!}{k! \times (m-k)!} ways.[/tex]

If the order matters, then each of those choice of k distinct items would be permuted k! times.

So, total number of choices in that case would be:

[tex]^mP_k = k! \times ^mC_k = k! \times \dfrac{m!}{k! \times (m-k)!} = \dfrac{m!}{ (m-k)!}\\\\^mP_k = \dfrac{m!}{ (m-k)!}[/tex]

This is called permutation of k items chosen out of m items (all distinct).

We are given that;

Number of sandwiches=2

Number of drinks=5

Now,

To find the total number of lunch specials, we need to multiply the number of choices for sandwiches by the number of choices for drinks.

Number of sandwich choices = 2

Number of drink choices = 5

Total number of lunch specials = 2 x 5 = 10

Therefore, by combinations and permutations there are 10 possible lunch specials.

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Answer:

Show your solution ( 3. ) C + 18 = 29

Step-by-step explanation:

To solve the equation C + 18 = 29, we want to isolate the variable C on one side of the equation.

We can start by subtracting 18 from both sides of the equation:

C + 18 - 18 = 29 - 18

Simplifying the left side of the equation:

C = 29 - 18

C = 11

Therefore, the solution to the equation C + 18 = 29 is C = 11.

Cardinality of given set is 10.

The cardinality of a mathematical set refers to the number of entries in the set. It may be limited or limitless. For instance, if set A has six items, its cardinality is equivalent to 6: 1, 2, 3, 4, 5, and 6. A set's size is often referred to as the set's cardinality. The modulus sign is used to indicate it on either side of the set name, |A|.

A set that can be counted and has a finite number of items is said to be finite. On the other hand, an infinite set is one that has an unlimited number of components and can either be countable or uncountable.

Possible set of A=14+4+1+9=28

Possible set of C=1 +6+9+9=25

n(A∩ C)=10

Hence, Cardinality of given set is 10.

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The correct standard form of the equation of the parabola is:

[tex]y = -x^2 - 1[/tex].

To find the standard form of the equation of the parabola that passes through the given points (-3, 3) and (1, -1), we can use the general form of the equation of a parabola:

[tex]y = ax^2 + bx + c[/tex] ___________(1)

Substituting the coordinates of the two given points into this equation, we get a system of two equations in three unknowns (a, b, and c):

[tex]3 = 9a - 3b + c[/tex]

[tex]-1 = a + b + c[/tex]

To solve for a, b, and c, we can eliminate one of the variables using subtraction or addition. Subtracting the second equation from the first, we get:

[tex]4 = 8a - 4b[/tex]

Simplifying this equation, we get:

[tex]2 = 4a - 2b[/tex]

Dividing both sides by 2, we get:

[tex]1 = 2a - b[/tex]___________(2)

Now we can substitute this expression for b into one of the earlier equations to eliminate b. Using the first equation, we get:

[tex]3 = 9a - 3(2a - 1) + c[/tex]

Simplifying this equation, we get:

[tex]3 = 6a + c + 3[/tex]

Subtracting 3 from both sides, we get:

[tex]0 = 6a + c[/tex]

Solving for c, we get:

c = -6a __________(3)

Substituting this expression for c into the second equation, we get:

[tex]-1 = a + (2a - 1) - 6a[/tex]

Simplifying this equation, we get:

[tex]-1 = -3a - 1[/tex]

Adding 1 to both sides, we get:

[tex]-3a =0[/tex]

Solving for a, we get:

[tex]a = 0[/tex]

Substituting this value of a into the equation(3) for c, we get:

c = 0

Substituting a = 0 into the equation(2) for b that we found earlier, we get:

[tex]1 = 0 - b[/tex]

Solving for b, we get:

[tex]b = -1[/tex]

Putting the values of a, b and c in (1), we get

[tex]y = -x^2 - 1[/tex]

Therefore, the equation of the parabola that passes through the given points (-3, 3) and (1, -1) is:

[tex]y = -x^2 - 1[/tex]

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Complete question:

A parabola goes through (-3, 3) & (1, -1). A point is below the parabola at (-3, 2). A line above the parabola goes through (-3, 4) & (0, 4). A point on the parabola is labeled (x, y).

What is the correct standard form of the equation of the parabola?

The figure is in the image attached below

The end balance is negative, so Mr. Manuel lost money.

We know that Mr. Manuel spended $800,000 in a product, and it can be sold with an extra 60% over the cost. Then the revenue here is:

$800,000*(1.6) = $1,280,000

We also know that there are costs of $75,680, $247.000 and $185.460.

Now we know that profit is defined as the difference between the revenue and the costs, so to get the profit we need to solve the equation below:

P = $1,280,000 - $800,000 - $75,680 - $247.000 - $185.460

P = -$28,140

So we can see that Mr. Manuel had a loss at the end.

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The expressiοn 5x(x²) is equal tο 5 * x¹ * x² (5 * x³ = 5x³). Thus, οptiοn (c) 5x3 is the cοrrect respοnse.

The cοefficients (the numbers in frοnt οf the variables) οf the expressiοn 5x(x²) can be multiplied, and the expοnents οf the variables can be added, tο determine the prοduct.

The first cοefficient we have is 5 times 1, giving us 5. Sο, using the secοnd x², we have x tο the pοwer οf 2 multiplied by x tο the pοwer οf 1 (frοm the first x). Expοnents are added when variables with the same base are multiplied. Sο, x¹ multiplied by x² results in x³.

Cοmbining all οf the parts, the phrase becοmes:

5x(x²) is equal tο 5 * x¹ * x² (5 * x³ = 5x³).

Thus, οptiοn (c) 5x³ is the cοrrect respοnse.

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In linear equation, 11.85 pounds is the weight of the wire.

What is  linear equation?

A linear equation is a first-order (linear) term plus a constant in the algebraic form y=mx+b, where m is the slope and b is the y-intercept. The variables in the previous sentence, y and x, are referred to as a "linear equation with two variables" at times.

Total weight of pool having 16 wires     =13.6 pounds

Weight of the pool                                 =1.75

Therefore the weight of the wire alone = 13.6 - 1.75

                                                             =  11.85 pounds

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Answer:

Traffic signs are regulated by the Manual on Uniform Traffic Control Devices (MUTCD). The perimeter of a rectangular traffic sign is 126 inches. Also, its length is 9 inches longer than its widthFind the dimensions of this sign.

Step-by-step explanation:

Let's say the width of the sign is x inches. Then, according to the problem, the length of the sign is 9 inches longer than the width, which means the length is x + 9 inches.

The perimeter of a rectangle can be found by adding up the length of all its sides. For this sign, the perimeter is given as 126 inches. So we can set up an equation:

2(length + width) = 126

Substituting the expressions for length and width in terms of x, we get:

2(x + x + 9) = 126

Simplifying and solving for x:

2(2x + 9) = 126

4x + 18 = 126

4x = 108

x = 27

So the width of the sign is 27 inches, and the length is 9 inches longer, or 36 inches. Therefore, the dimensions of the sign are 27 inches by 36 inches.

Answer: 66

Step-by-step explanation:

all 3 of them should equal to 180

so 79+35 is 114

180-114 will give us the answer which is 66

i. The total height of the tent including the spire is 150 m.

ii. The length of the side of the tent  x is 132.7 m.

Trigonometric functions are required functions in determining either the unknown angle of length of the sides of a triangle.

Considering the given question, we have;

a. To determine the total height of the tent, let its height from the ground to the top of the tent be represented by x. Then:

Tan θ = opposite/ adjacent

Tan 42.7 = h/ 97.5

h = 0.9228*97.5

  = 89.97

h = 90 m

The total height of the tent including the spire = 90 + 60

                                           = 150 m

b. To determine the length of the side of the tent x, we have:

Cos θ = adjacent/ hypotenuse

Cos 42.7 = 97.5/ x

x = 97.5/ 0.7349

  = 132.67

The length of the side of the tent x is 132.7 m.

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Distance from a point in space to the moon is a continuous random variable and Number of coins in a stack is a discrete random variable.

A discrete random variable is one that has a finite number of possible values or one that can be countably infinitely numerous. A discrete random variable is, for instance, the result of rolling a die because there are only six possible outcomes.

A continuous random variable, on the other hand, is one that is not discrete and "may take on uncountably infinitely many values," like a spectrum of real numbers.

1. The Name of the people in the car that crosses the bridge - Not a Variable

2. Continuous random variable measuring the interval between each car crossing the bridge.

3. The Categorical Random Variable for the type of vehicles crossing the bridge

4. The number of cars that use the bridge in one hour - Continuous Random Variable

For Question 2:

1. Type of coin - Categorical Random Variable

2. The distance from a given location in space to the moon - Continuous Random Variable

3. Number of coins in a stack - Discrete Random Variable

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Complete question is:

Label each of the following as Discrete Random Variable, Continuous Random Variable, Categorical Random Variable, or Not a Variable.

1. The Name of the people in the car that crosses the bridge Not a Variable

2. The time between each car crossing the bridge Continuous Random Variable

3. The type of cars that cross the bridge Categorical Random Variable

4. The number of cars that use the bridge in one hour Continuous Random Variable

Question 2: Which of these are Continuous and which are Discrete Random Variables?

1. Type of coin

2. Distance from a point in space to the moon

3. Number of coins in a stack

There are 792 strings across a,b, and 27,720 in a,b,c.

(a) We must select five slots for a's in an alphabet of "a,b" before filling the remaining spaces with "b's." Hence, the binomial coefficient is what determines how many strings of length 12 that include precisely five as:

C(12,5) = 792

As a result, there are 792 distinct strings of length 12 that include exactly five a's across the letters a, b.

(b) We may use the same method as before for an alphabet consisting of the letters "a,b,c." The first five slots must be filled with a's, followed by three b's, and the final four positions must be filled with c's. The number of strings of length 12 that contain exactly five a's across the letters "a," "b," and "c" is thus given by:

C(12,5) * C(7,3) = 792 * 35 = 27720

Thus, there are 27,720 distinct strings.

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Answer:

Part A: To determine the exact value of cos 2θ, we can use the double-angle identity for cosine:

cos 2θ = 2 cos^2 θ - 1

We already know sin θ, so we can use the Pythagorean identity to find cos θ:

cos^2 θ = 1 - sin^2 θ

cos^2 θ = 1 - (2√2/5)^2

cos^2 θ = 1 - 8/25

cos^2 θ = 17/25

cos θ = ± √(17/25)

cos θ = ± (1/5) √17

Since θ is in the third quadrant (π/2 < θ < π), cos θ is negative, so we take the negative root:

cos θ = -(1/5) √17

Substituting into the double-angle identity:

cos 2θ = 2 cos^2 θ - 1

cos 2θ = 2 [-(1/5) √17]^2 - 1

cos 2θ = 2 (1/25) (17) - 1

cos 2θ = 34/25 - 1

cos 2θ = 9/25

Therefore, the exact value of cos 2θ is 9/25.

Part B: To determine the exact value of sin (θ/2), we can use the half-angle identity for sine:

sin (θ/2) = ± √[(1 - cos θ)/2]

We already know cos θ, so we can substitute it in:

cos θ = -(1/5) √17

sin (θ/2) = ± √[(1 - cos θ)/2]

sin (θ/2) = ± √[(1 - (-1/5) √17)/2]

sin (θ/2) = ± √[(5 + √17)/10]

sin (θ/2) = ± (1/2) √(5 + √17)

Since θ is in the third quadrant (π/2 < θ < π), sin θ is negative, so we take the negative root:

sin (θ/2) = -(1/2) √(5 + √17)

Therefore, the exact value of sin (θ/2) is -(1/2) √(5 + √17).

The exact values of the sine and cosine given are -(1/2) √(5 + √17) and 9/25.

The sine of an angle in a right triangle is the ratio of the hypotenuse to the side opposite the angle.

The cosine of an angle in a right triangle is the ratio of the hypotenuse to the side adjacent the angle.

Part A: To determine the exact value of cos 2θ, we can use the double-angle identity for cosine:

cos 2θ = 2 cos² θ - 1

Using the Pythagorean identity to find cos θ:

cos² θ = 1 - sin² θ

cos² θ = 1 - (2√2/5)²

cos² θ = 1 - 8/25

cos² θ = 17/25

cos θ = ± √(17/25)

cos θ = ± (1/5) √17

Since θ is in the third quadrant (π/2 < θ < π), cos θ is negative, so we take the negative root:

cos θ = -(1/5) √17

Substituting into the double-angle identity:

cos 2θ = 2 cos² θ - 1

cos 2θ = 2 [-(1/5) √17]² - 1

cos 2θ = 2 (1/25) (17) - 1

cos 2θ = 34/25 - 1

cos 2θ = 9/25

Therefore, the exact value of cos 2θ is 9/25.

Part B: To determine the exact value of sin (θ/2), we can use the half-angle identity for sine:

sin (θ/2) = ± √[(1 - cos θ)/2]

We already know cos θ, so we can substitute it in:

cos θ = -(1/5) √17

sin (θ/2) = ± √[(1 - cos θ)/2]

sin (θ/2) = ± √[(1 - (-1/5) √17)/2]

sin (θ/2) = ± √[(5 + √17)/10]

sin (θ/2) = ± (1/2) √(5 + √17)

Since θ is in the third quadrant (π/2 < θ < π), sin θ is negative, so we take the negative root:

sin (θ/2) = -(1/2) √(5 + √17)

Therefore, the exact value of sin (θ/2) is -(1/2) √(5 + √17).

Hence, the exact values of the sine and cosine given are -(1/2) √(5 + √17) and 9/25.

Learn more about sine and the cosine of an angle, click;

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