Solve The Equation. Check For Extraneous Solutions.3|4 W-1|-5=10 (2024)

Exact values:

Cosine of -30°: √3/2, Sine of -30°: -1/2

Decimal values (rounded to the nearest hundredth):

Cosine of -30°: 0.87, Sine of -30°: -0.50

The unit circle can be used to precisely calculate the cosine and sine at a -30° angle. A circle with a radius of one and a center at the origin of a Cartesian coordinate system is the unit circle.

Let's start by figuring out the reference angle. The reference plot for - 30° will be 30° in light of the fact that it is the positive intense point framed between the terminal side of - 30° and the x-hub.

The values of the cosine and sine can then be determined using the 30° reference angle. On the unit circle, the x-coordinate addresses the cosine esteem, and the y-coordinate addresses the sine esteem.

For the 30° reference angle:

Since the reference angle is in the fourth quadrant, the cosine value for -30° is the same as for 30°, which is 3/2. The cosine value is positive.

Because the reference angle is in the fourth quadrant, the sine value is negative. Consequently, the sine an incentive for - 30° is the negative of the sine an incentive for 30°, which is - 1/2.

Let's now round the decimal numbers to the nearest hundredth when calculating:

The cosine of -30° has a decimal value of approximately 0.87.

The sine of -30 degrees has a decimal value of roughly -0.50.

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To make the left side of the equation x² + kx + 64 a perfect square trinomial, we can rewrite it in the form (x + a)², where a is a constant. The values of k that would make the left side of the equation x² + kx + 64 a perfect square trinomial are k = 16 and k = -16.

Expanding (x + a)² gives us x² + 2ax + a². Comparing this with the given equation, we can equate the corresponding terms:

x² + kx + 64 = x² + 2ax + a²

By comparing the coefficients, we can determine the value of k:

k = 2a

64 = a²

To find the value of a, we can solve the second equation:

a² = 64

a = ±√64

a = ±8

Now we can find the corresponding value of k:

k = 2a

k = 2(8) = 16

k = 2(-8) = -16

Therefore, the values of k that would make the left side of the equation x² + kx + 64 a perfect square trinomial are k = 16 and k = -16.

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In statistics, hypothesis testing is one of the most important techniques used to test claims or statements regarding population parameters. The hypothesis testing process consists of six steps, including state the null and alternative hypotheses.

Null Hypothesis, H0: p ≤ 0.65 Alternative Hypothesis, Ha: p > 0.65 Where p is the proportion of computer chips that do not fail in the first 1000 hours of their use. Now we have to check whether there is sufficient evidence at the 0.01 level to support the company's claim or not. For this, we use the following formula to compute the z-score for the sample proportion.

[tex]$$z=\frac{p-_0}{\sqrt{_0(1−_0)/}}$$[/tex] where

p0 = 0.65 (assumed under null hypothesis)

p = 0.69 (sample proportion)

n = 900 (sample size) Now, we have,

[tex]$$z=\frac{0.69-0.65}{\sqrt{0.65(1−0.65)/900}}$$ $$\implies z=\frac{0.04}{0.0156}=2.564$$[/tex]

The null and alternative hypotheses for the given scenario are H0: p ≤ 0.65 and Ha: p > 0.65. At the 0.01 level, there is sufficient evidence to support the company's claim that more than 65% of computer chips do not fail in the first 1000 hours of their use.

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The probability of Josh and Sokka winning the football game tickets is 2/500. This means that there is a very low chance of them winning compared to the total number of participants.

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You can find the angle θ in degrees between the calculated vector and the x-axis, measured counterclockwise from the x-axis.

To determine the angle θ in degrees between a calculated vector and the x-axis, measured counterclockwise from the x-axis, you can use trigonometry. The x-component and y-component of the vector are needed for this calculation.

Calculate the ratio of the y-component to the x-component of the vector:

y/x = tan(θ)

Use the inverse tangent function (arctan or atan) to find the angle θ:

θ = atan(y/x)

Convert the angle from radians to degrees by multiplying by 180/π:

θ_degrees = θ * (180/π)

By following these steps, you can find the angle θ in degrees between the calculated vector and the x-axis, measured counterclockwise from the x-axis.

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If the supply and demand curves in the provided graph represent the market supply and demand for a purely competitive industry, then the demand curve that an individual firm in the industry faces is perfectly elastic.

This is because firms in a perfectly competitive market are price takers. They have to accept the market price because they are too small to influence it. Therefore, the market demand curve is also the demand curve for the firm because it can sell any amount of output at the market price.

The supply curve for the firm is also the marginal cost curve because the firm produces where marginal cost equals the price. Hence, the individual firms in a perfectly competitive industry face a perfectly elastic demand curve, while the market demand curve is downward sloping.

If the supply and demand curves in the provided graph represent the market supply and demand for a purely competitive industry, then the demand curve that an individual firm in the industry faces is perfectly elastic. This is because firms in a perfectly competitive market are price takers.

They have to accept the market price because they are too small to influence it. Therefore, the market demand curve is also the demand curve for the firm because it can sell any amount of output at the market price. The supply curve for the firm is also the marginal cost curve because the firm produces where marginal cost equals the price.

Hence, the individual firms in a perfectly competitive industry face a perfectly elastic demand curve, while the market demand curve is downward sloping.In a perfectly competitive market, an individual firm can sell all of its output at the market price.

Since the market price is fixed, the firm faces a perfectly elastic demand curve. The reason for this is that the firm is too small to influence the market price. Therefore, the market demand curve is also the demand curve for the firm. As for the supply curve for the firm, it is equal to the marginal cost curve because the firm produces where marginal cost equals the price.

If the market price rises, the firm will produce more because it can cover its costs. If the market price falls, the firm will produce less because it will not be able to cover its costs.

Hence, individual firms in a perfectly competitive industry face a perfectly elastic demand curve, while the market demand curve is downward sloping.

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The probability of a football player weighing more than 240 pounds, given a normally distributed weight with a mean of 200 pounds and a standard deviation of 20 pounds, is approximately 0.0228 or 2.28%. This can be found by standardizing the weight using z-scores and using the standard normal distribution table to find the probability.

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True. Symbolic logic, also known as formal logic or mathematical logic, encompasses three fundamental requirements.

Firstly, it involves expressing propositions, which are statements that can be either true or false. These propositions serve as the building blocks of logical reasoning.

Secondly, symbolic logic enables the expression of relationships between propositions through logical connectives such as conjunction (AND), disjunction (OR), negation (NOT), implication (IF-THEN), and biconditional (IF AND ONLY IF). These connectives allow for the construction of compound propositions and the evaluation of their truth values.

Lastly, symbolic logic provides mechanisms to describe how new propositions can be inferred from other propositions assumed to be true. This is achieved through logical rules and deduction techniques such as modus ponens, modus tollens, and proof by contradiction. These inference rules facilitate logical reasoning and allow for the derivation of new propositions based on existing ones.

In summary, symbolic logic encompasses expressing propositions, expressing relationships between propositions, and describing how new propositions can be inferred from assumed true propositions.

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The nth term of the given pattern can be determined using the formula: Tn = 9n + 9.

In this pattern, each term is obtained by multiplying n by 9 and adding 9. Let's break it down step by step:

First term (n = 1):

T1 = (9 × 1) + 9 = 18

Second term (n = 2):

T2 = (9 × 2) + 9 = 27

Third term (n = 3):

T3 = (9 × 3) + 9 = 36

Fourth term (n = 4):

T4 = (9 × 4) + 9 = 45

Fifth term (n = 5):

T5 = (9 × 5) + 9 = 54

As you can see, each term is obtained by multiplying n by 9 and adding 9. This pattern continues for any value of n.

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Answer:

f(x) = 5x - 5

Step-by-step explanation:

the equation of a line in slope- intercept form is

y = mx + c ( m is the slope and c the y- intercept )

calculate m using the slope formula

m = [tex]\frac{y_{2}-y_{1} }{x_{2}-x_{1} }[/tex]

with (x₁, y₁ ) = (2, 5 ) and (x₂, y₂ ) = (3, 10 )

m = [tex]\frac{10-5}{3-2}[/tex] = [tex]\frac{5}{1}[/tex] = 5 , then

y = 5x + c ← is the partial equation

to find c substitute either of the 2 points into the partial equation

using (2, 5 )

5 = 5(2) + c = 10 + c ( subtract 10 from both sides )

- 5 = c

y = 5x - 5 , that is

f(x) = 5x - 5 ← in function notation

A quartic function with the real zeros x = -3 and x = -4 can be expressed as f(x) = (x + 3)(x + 4)(ax² + bx + c), where a, b, and c are coefficients to be determined.

To find the quartic function with the given real zeros x = -3 and x = -4, we start by setting up the equation f(x) = (x + 3)(x + 4)(ax² + bx + c). The factors (x + 3) and (x + 4) account for the real zeros.

Expanding the equation, we have f(x) = (x² + 7x + 12)(ax² + bx + c).

To determine the coefficients a, b, and c, we can use additional information or conditions. Without any additional information, we cannot uniquely determine the quartic function. However, we can still find a quartic function with the given real zeros by selecting arbitrary values for a, b, and c.

For example, let's assume a = 1, b = 2, and c = 1. Plugging in these values, we have f(x) = (x² + 7x + 12)(x² + 2x + 1).

Therefore, a quartic function with the real zeros x = -3 and x = -4 can be expressed as f(x) = (x + 3)(x + 4)(x² + 2x + 1).

Given the real zeros x = -3 and x = -4, we can find a quartic function by multiplying the factors (x + 3) and (x + 4) with another quadratic factor ax² + bx + c. Since we do not have any additional information or conditions, we can choose arbitrary values for a, b, and c. In the example above, we assumed a = 1, b = 2, and c = 1 to construct the quartic function f(x) = (x + 3)(x + 4)(x² + 2x + 1). Keep in mind that there are infinitely many quartic functions with the given real zeros, and the specific values of a, b, and c will vary based on the desired characteristics or constraints of the function.

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The volume of the hemisphere, rounded to the nearest tenth, is 2144.7 cubic centimeters.

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To determine the relationship between n and f(n), we can analyze the given values. As n increases, f(n) decreases. Specifically, it looks like f(n) is halving each time n increases by 1.

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The function that best describes the relationship between "n" and "f(n)" is [tex]f(n) = 56 / (2^{(n-1)})[/tex].

= 56 / 2

= 28
f(3) = f(2) / 2

= 28 / 2

= 14
f(4) = f(3) / 2

= 14 / 2

f(1) =[tex] 56 / (2^{(1-1)})[/tex]

= 56 / 1

= 56
For n = 2:

f(2) = [tex] 56 / (2^{(2-1)})[/tex]

= 56 / 2 = 28
For n = 3:

f(3) =[tex] 56 / (2^{(3-1)}) [/tex]

= 56 / 4

= 14
For n = 4:

f(4) = [tex] 56 / (2^{(4-1)})[/tex]

= 56 / 8

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By the AA (Angle-Angle) similarity postulate, we can conclude that △lmn ∼ △opn.

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The given statement "The Millennium Scholarship requirements indeed include 4 credits of math, specifically Algebra II or a higher-level math course, and 3 credits of natural science" is true because the Millennium Scholarship is a merit-based scholarship program offered in certain states or regions to eligible high school graduates.

It aims to support students pursuing higher education by providing financial assistance. The requirement of 4 credits of math, specifically Algebra II or a higher-level math course, reflects the importance of strong mathematical skills in college and career readiness. Algebra II is often considered a crucial subject for developing advanced problem-solving and critical thinking abilities.

Additionally, the requirement of 3 credits of natural science highlights the significance of scientific knowledge and understanding. It ensures that students have a foundational understanding of scientific principles and concepts, which can be applied to various fields of study.

By setting these specific requirements, the Millennium Scholarship program aims to encourage students to pursue rigorous coursework in math and science, preparing them for success in higher education and future careers. These subjects are recognized as fundamental pillars of education that provide essential skills and knowledge applicable in a wide range of academic and professional pursuits.

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The null hypothesis in a one-way ANOVA test is that there is no treatment effect. This means that the mean scores of the groups being compared are not significantly different. The one-way ANOVA test is a statistical method used to compare the means of three or more groups.

The one-way ANOVA test requires that certain assumptions be met, including that the data is normally distributed and that the variances of the groups being compared are equal. If these assumptions are met, then the test can be used to determine whether there is a significant difference between the means of the groups. The test works by comparing the variability between the groups to the variability within the groups.

If the variability between the groups is larger than the variability within the groups, then it is likely that there is a significant difference between the means of the groups. The alternative hypothesis in a one-way ANOVA test is that there is some treatment effect, which means that at least one of the group means is different from the others.

If the null hypothesis is rejected, then the alternative hypothesis is accepted, and it can be concluded that there is a significant difference between the means of the groups. The one-way ANOVA test is useful for determining whether there is a significant difference between the means of several groups, but it does not provide information about which specific group means are different. To determine which group means are different, post hoc tests can be performed.

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In commutative ring theory, if every ideal of a commutative ring with identity (denoted as r) can be generated by a finite or denumerable subset, then the same is true for the ring r[x].

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A geodesic is the shortest path between two points on a curved surface, such as a sphere. To find a geodesic, an integral is set up to determine the length of a path on the surface.

This integral is similar to the one used for flat surfaces but may be more complex and involve different coordinates depending on the nature of the curved surface. In the case of a sphere, spherical polar coordinates are commonly used.

The length of a path on a curved surface can be expressed using an integral similar to Equation (6.2) for flat surfaces. However, when dealing with a sphere, the integral may involve different coordinates, specifically spherical polar coordinates. Spherical polar coordinates consist of three components: radius (r), polar angle (θ), and azimuthal angle (φ). By expressing the path in terms of these coordinates, the integral can be set up to calculate the length of the geodesic.

To illustrate this, let's consider the example of finding the geodesic on the surface of a sphere. The integral would involve integrating the square root of the sum of squared differentials of the three spherical polar coordinates, weighted by the appropriate metric coefficients. The specific form of the integral would depend on the metric tensor associated with the spherical polar coordinate system. By minimizing this integral, the geodesic between two points on the sphere can be determined. In summary, finding the geodesic, or shortest path, between two points on a curved surface involves setting up an integral to calculate the path length. The integral is similar to the one used for flat surfaces but may be more complex and use different coordinates, such as spherical polar coordinates for a sphere. By minimizing the integral, the geodesic on the curved surface can be found.

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Hello!

4x + 2y = 6

2y = 6 - 4x

y = 3 - 2x

Hello !

Answer:

[tex]\Large \boxed{\sf y=-2x+3}[/tex]

Step-by-step explanation:

We want to isolate y in the following expression :

[tex]\sf 4x+2y=6[/tex]

First, substract 4x from both sides :

[tex]\sf 4x+2y-4x=6-4x\\2y=-4x+6[/tex]

Now let's divide both sides by 2 :

[tex]\sf \frac{2y}{2} =\frac{-4x+6}{2}[/tex]

Finally, simplify the expression :

[tex]\sf \frac{2y}{2} =\frac{-4x+6}{2}\\y=\frac{-4x}{2}+\frac{6}{2} \\\boxed{\sf y=-2x+3}[/tex]

Have a nice day ;)

To calculate the mean, variance, and standard deviation of a discrete probability distribution, you need to follow these steps:

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To determine a rejection region for a test at level α using the fact that the sum of independent Poisson random variables follows a Poisson distribution, we calculate the critical values based on the desired significance level α and compare them with the observed sum of Poisson variables.

To determine a rejection region for a test at level α using the fact that the sum of independent Poisson random variables follows a Poisson distribution, we can follow these steps:

Specify the null and alternative hypotheses: Determine the null hypothesis (H0) and the alternative hypothesis (Ha) for the statistical test. These hypotheses should be stated in terms of the parameters being tested.

Choose the significance level (α): The significance level α represents the maximum probability of rejecting the null hypothesis when it is true. It determines the probability of making a Type I error (rejecting H0 when it is actually true). Common choices for α are 0.05 or 0.01.

Determine the test statistic: Select an appropriate test statistic that follows a Poisson distribution based on the data and hypotheses being tested. The test statistic should be able to capture the effect or difference being examined.

Calculate the critical region: The critical region is the set of values of the test statistic for which the null hypothesis will be rejected. To determine the critical region, we need to find the values of the test statistic that correspond to the rejection region based on the significance level α.

Use the Poisson distribution: Since the sum of independent Poisson random variables follows a Poisson distribution, we can utilize the Poisson distribution to determine the probabilities associated with different values of the test statistic. We can calculate the probabilities for the test statistic under the null hypothesis.

Compare the probabilities: Compare the probabilities calculated under the null hypothesis with the significance level α. If the calculated probability is less than or equal to α, it falls in the rejection region, and we reject the null hypothesis. Otherwise, if the probability is greater than α, it falls in the acceptance region, and we fail to reject the null hypothesis.

It is important to note that the specific details of determining the rejection region and performing hypothesis testing depend on the specific test being conducted, the data at hand, and the nature of the hypotheses being tested. Different tests and scenarios may require different approaches and considerations.

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Matrices can be used to represent and manipulate data, such as payoffs or incidence relationships in a network. Matrix operations allow us to perform calculations on the data and analyze patterns and relationships.

To represent and manipulate data using matrices, we can use matrix operations. Matrices are rectangular arrays of numbers, and they can be used to represent various types of information. One application of matrices is to represent payoffs or incidence relationships in a network. By using matrices, we can perform calculations and transformations on the data, which can help us analyze and understand the underlying patterns and relationships.

For example, let's consider a matrix representing payoffs in a game. Suppose we have a 2x3 matrix A, where each element represents the payoff of a player in a particular scenario. We can perform operations on this matrix, such as addition, subtraction, and multiplication, to analyze the payoffs.

Matrices can also be used to represent incidence relationships in a network. In this context, a matrix is used to describe the connections between different nodes or vertices in the network. The elements of the matrix indicate whether there is a connection (or an edge) between two nodes. By manipulating this matrix, we can determine important properties of the network, such as the number of connections, the shortest path between nodes, or the centrality of a particular node.

In summary, matrices are powerful tools for representing and manipulating data. They can be used to represent payoffs in games, describe incidence relationships in networks, and perform various operations to analyze the data. By understanding and utilizing matrix operations, we can gain insights into the underlying structure and relationships of the data.

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To find two positive numbers whose product is 81 and whose sum is a minimum, we can use the concept of the arithmetic mean-geometric mean inequality. The two positive numbers whose product is 81 and whose sum is a minimum are 3 and 27.

Step 1: Let's call the two positive numbers x and y.

Step 2: We know that the product of x and y is 81, so we can write the equation: x * y = 81.

Step 3: To find the sum of x and y, we can use the formula for the arithmetic mean, which is (x + y)/2.

Step 4: To minimize the sum, we want to minimize the arithmetic mean.

Step 5: According to the arithmetic mean-geometric mean inequality, the arithmetic mean is always greater than or equal to the geometric mean.

Step 6: The geometric mean of x and y is the square root of their product, so we can write the equation: √(x * y) = √81.

Step 7: Simplifying, we get √(x * y) = 9.

Step 8: Taking the square root of both sides, we find that x * y = 9.

Step 9: Since the product of x and y is the same as before, we know that x * y = 81.

Step 10: So, we have two equations: x * y = 9 and x * y = 81.

Step 11: Solving these equations, we find that the values of x and y are (3, 27) or (27, 3).

Step 12: Therefore, the two positive numbers whose product is 81 and whose sum is a minimum are 3 and 27.

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The next number in the pattern 101, 92, 83, 74 is 65.

In the given pattern, each number is obtained by subtracting 9 from the previous number. Starting with 101, we subtract 9 to get 92, then subtract 9 again to get 83, and so on. This pattern of subtracting 9 from the previous number continues. Therefore, to find the next number, we subtract 9 from 74, resulting in 65.

This pattern follows a common arithmetic sequence where each term is obtained by subtracting a constant value (in this case, 9) from the previous term. By identifying the pattern and observing the regularity of the differences between consecutive terms, we can predict the next term in the sequence.

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The correct term to fill in the blank is "a) sequestered."

Fossilized carbon, which is found in ancient plant and animal remains, is said to be sequestered.

This means that the carbon is trapped or stored within these remains over long periods of time. Fossilization occurs when organic material undergoes a process called carbonization, where the carbon in the remains is preserved. This carbon then becomes fossilized and is no longer part of the carbon cycle.

It is important to note that fossilized carbon is different from carbon that is transferred, eroded, or absorbed.

These terms refer to processes that involve the movement or interaction of carbon in various forms, whereas sequestering specifically refers to the trapping and preservation of carbon within fossils.

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The pattern of monthly payments for an investment follows an increasing trend of $2 per month. Using this pattern, the payment for the tenth month is determined to be $68. The pattern can be described by the explicit formula: Payment = $50 + (Month - 1) * $2.

Based on the given pattern, it seems that the payment amount is increasing by $2 each month.

To find the payment for the tenth month, we can continue this pattern:

Month 1: $50

Month 2: $52 ($50 + $2)

Month 3: $54 ($52 + $2)

Month 4: $56 ($54 + $2)

Month 5: $58 ($56 + $2)

Month 6: $60 ($58 + $2)

Month 7: $62 ($60 + $2)

Month 8: $64 ($62 + $2)

Month 9: $66 ($64 + $2)

Month 10: $68 ($66 + $2)

Therefore, in the tenth month, you would receive $68.

b. We can write an explicit formula to describe the pattern of payments:

Payment = Initial Payment + (Month - 1) * Increment

In this case, the initial payment is $50, and the increment is $2. So, the explicit formula would be:

Payment = $50 + (Month - 1) * $2

This formula can be used to calculate the payment for any given month in the series.

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The error described in this situation is called selection bias. Selection bias occurs when the sample used for a study or survey is not representative of the entire population.

In this case, using the phonebook as a sampling frame may result in a biased sample because people who are more safety conscious and have chosen not to be listed in the phonebook are not included.

This can lead to an underrepresentation of individuals who are more safety conscious in the sample. Consequently, any conclusions or estimates drawn from this sample may not accurately reflect the entire population of the city.

To minimize selection bias, alternative sampling methods that include the entire population should be considered.

In conclusion, using the phonebook as a sampling frame may introduce selection bias and may not provide an accurate estimate of the population's safety consciousness.

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Next, we can use the inverse tangent function to find the angle whose tangent is equal to √3/3.

In this case, we want to find the principal values of t.
Using the inverse tangent function on both sides of the equation, we get: 2t = arctan(√3/3)

arctan(√3/3) and (1/2) *

(arctan(√3/3) + π).

To get the values rounded to the nearest hundredth, you can substitute the value of √3/3 into the arctan function and calculate the approximate values.

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