Students were asked to prove the identity (sec x)(csc x) = cot x + tan x. Two students' work is given.
Part A: Did either student verify the identity properly? Explain why or why not. (10 points)

Part B: Name two identities that were used in Student A's verification and the steps they appear in. (5 points)

The measure of angle TYK is given as follows:

46º.

Two angles are defined as complementary if the sum of their measures is of 90º.

In this problem, we have that angle Y is an angle of 90º, which is then divided into two angles, given as follows:

Then 44 and TYK are complementary angles, and thus the measure of angle TYK is given as follows:

m < TYK + 44 = 90

m < TYK = 90 - 44

m < TYK = 46º.

(the angle addition postulate is also applied for the complementary angles in this problem, as the sum of the two smaller smaller angles combined is of 90º).

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Answer:

a) 0.32 = 32/100 = 8/25

b) 3.003 = 3 + 3/1000 = 3000/1000 + 3/1000 = 3003/1000

c) 13.134 = 13 + 134/1000 = 13000/1000 + 134/1000 = 13134/1000

d) 5.2303 = 5 + 2303/10000

He will have $1322.5 in 2 years.

Compound Interest is the interest calculated on the principal and the interest accumulated over the previous period. It is also the interest-based on the initial principal amount and the interest collected over the period of time.

The formula is A = P(1 + r/n)^nt

Where A = Amount compounded annually

P = Principal = $1000

r = Rate of interest = 15%

n = Number of times interest is compounded per year

t = Time in years

So, A = 1000(1 + 15%/1)^1*2

A = 1000(1 + 0.15)^2

A = 1000(1.15)^2

A = 1000(1.3225)

A = $1322.5

Therefore, the amount he will have in 2 years is $1322.5

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Using the zero product property

To use the zero product property to determine the values of x for which the expression equals 0, we need to set each factor equal to 0 and solve for x.

Setting each factor equal to 0 gives us:

x+7 = 0 or x-2 = 0

Solving each equation for x, we get:

x = -7 or x = 2

Hence, the values of x are x=-7 and x=2.

Setting each factor equal to 0 gives us:

2x+7 = 0 or x-6 = 0

Solving each equation for x, we get:

x = -7/2 or x = 6

Hence, the values of x  are x=-7/2 and x=6.

Setting each factor equal to 0 gives us:

-3x+11 = 0 or 4x+18 = 0

Solving each equation for x, we get:

x = 11/3 or x = -9/2

Hence, the values of x  are x=11/3 and x=-9/2.

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The z-scores for 17, 33, and 53 are -0.82, 0.22, and 1.52, respectively.

To find the z-scores for 17, 33, and 53, we first need to calculate the mean and standard deviation of the data set.

Mean = (10 + 13 + 17 + 19 + 22 + 23 + 29 + 33 + 34 + 35 + 35 + 38 + 53 + 68)/14 = 29.57

Standard deviation = √[(10-29.57)² + (13-29.57)² + (17-29.57)² + (19-29.57)² + (22-29.57)² + (23-29.57)² + (29-29.57)² + (33-29.57)² + (34-29.57)² + (35-29.57)² + (35-29.57)² + (38-29.57)² + (53-29.57)² + (68-29.57)²]/13 = 15.37

Now we can calculate the z-scores using the formula:

z-score = (data point - mean)/standard deviation

Z-score for 17 = (17-29.57)/15.37 = -0.82

Z-score for 33 = (33-29.57)/15.37 = 0.22

Z-score for 53 = (53-29.57)/15.37 = 1.52

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The discriminant is positive (D = 8), the equation x² - 6x + 7 = 0 has two distinct real solutions, which supports Jason's claim.

The quadratic equation's roots are represented geometrically by the discriminant. The equation has two separate real roots if the discriminant is positive, and as a result, the graph of the quadratic function meets the x-axis twice. The quadratic function's graph crosses the x-axis precisely one time if the discriminant is zero, which indicates that the equation has one real root.

To find the discriminant of the given equation, we can substitute the values of a, b, and c into the formula:

D = (-6)² - 4(1)(7) = 36 - 28 = 8

Since the discriminant is positive (D = 8), the equation x² - 6x + 7 = 0 has two distinct real solutions, which supports Jason's claim.

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The complete question is:

13 inches of wire will cost 52 cents at the same rate as 18 inches of wire costs 72 cents.

To solve this word problem on proportions using a unit rate, we need to first find the unit rate for the cost of the wire. The unit rate is the cost per one inch of wire.

We can find this by dividing the cost by the number of inches:

Unit rate = 72 cents / 18 inches = 4 cents per inch

Now that we have the unit rate, we can use it to find the cost of 13 inches of wire.

We simply multiply the unit rate by the number of inches:

Cost = 4 cents per inch × 13 inches = 52 cents

Therefore, 13 inches of wire will cost 52 cents at the same rate as 18 inches of wire costs 72 cents.

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The zeros of the function [tex]f(t) = t^(2) + 7t + 12[/tex] are -3 and -4.

To find the zeros of a quadratic function, we can either factor the equation or use the quadratic formula. In this case, we can easily factor the equation to find the zeros.

First, we need to find two numbers that multiply to give us 12 and add to give us 7. These numbers are 3 and 4.

Next, we can rewrite the equation using these numbers:

[tex]f(t) = t^(2) + 7t + 12 = (t + 3)(t + 4)[/tex]

Now, we can set each factor equal to zero and solve for t:

[tex]t + 3 = 0  ->  t = -3[/tex]

[tex]t + 4 = 0  ->  t = -4[/tex]

So, the zeros of the function are -3 and -4.

In conclusion, the zeros of the function [tex]f(t) = t^(2) + 7t + 12[/tex] are -3 and -4.

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Answer:

He realizes the casino is a trap when he found out people from 1977 are in the casino. They wasted five days in the casino.

Answer:

First Option, (A) $29,922.

Step-by-step explanation:

To calculate the future value of Heather's savings account after 5 years, we can use the formula for compound interest:

FV = P((1+i)^n - 1)/i

where:

FV = future value

P = principal (the initial amount Heather deposits)

i = interest rate per period (monthly in this case)

n = number of periods (months in this case)

P = $75 (the amount of Heather's monthly payments)

i = 3.6% / 12 = 0.003 (the monthly interest rate, calculated by dividing the annual interest rate by 12)

n = 5 x 12 = 60 (the total number of months in 5 years)

Substituting these values into the formula, we get:

FV = $75((1+0.003)^60 - 1)/0.003

FV = $75(1.21879)/0.003

FV = $29,922.02 (rounded to the nearest cent)

Therefore, Heather will have approximately $29,922.02 in her savings account after the 5 year period. The answer is option A: $29,922.

The slope-intercept form of the equation of the line that passes through the point P (5, 4) and makes an angle, θ = 30°, with the x-axis is the option A.

A. y = ((sqrt(3))/3)·x - ((5·sqrt(3))/3 - 4)

The slope-intercept form of linear equation is an equation of the form; y = m·x + c, where;

m = The slope of the graph of the equation

c = The y-intercept of the graph of the equation.

The point through which the line passes, P = (5, 4)

The angle θ the line makes with the positive x-axis = 30°

The slope of the line = The tangent of the angle the line makes with the positive x-axis, therefore;

(y - 4)/(x - 5) = tan(30°) = 1/√3

y - 4 = (x - 5)/√3 = (x - 5)/√3 × (√3/√3) = (x - 5)·√3/3

y = (x - 5)·√3/3 + 4

The above equation can be converted into the slope-intercept form of a linear equation; y = m·x + c by simplifying the equation and rearranging the equation, into the required form;

y = (x - 5)·√3/3 + 4

y = (√3/3)·x - 5·√3/3 + 4 = (√3/3)·x - (5·√3/3 - 4)

y = (√3/3)·x - (5·√3/3 - 4)

The equation in slope-intercept form, is therefore;

A. y = (sqrt(3))/3)·x - (5·sqrt(3))/3 - 4)

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The most accurate comparison is that a gamma ray has more energy than a radio wave because it has a shorter wavelength and higher frequency.

The most energetic and high frequency particles are gamma rays. On the other side, radio waves are the EM radiation types with the lowest energies, longest wavelengths, and lowest frequencies.

All electromagnetic radiation travels in a vacuum at the speed of light (c), which is the same for all electromagnetic radiation types, including microwaves, visible light, and gamma rays.

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The basic subtraction of rational expression simplified is (7)/(12d).

To subtract these rational expressions, we need to find a common denominator. The least common denominator (LCD) of 4d and 6d is 12d. We can then rewrite the expressions with the LCD as the denominator:

(3)/(4d) = (3 * 3)/(4d * 3) = (9)/(12d)
(1)/(6d) = (1 * 2)/(6d * 2) = (2)/(12d)

Now we can subtract the numerators and keep the same denominator:

(9)/(12d) - (2)/(12d) = (9 - 2)/(12d) = (7)/(12d)

We can simplify this fraction by dividing both the numerator and denominator by their greatest common factor (GCF), which is 1:

(7)/(12d) = (7/1)/(12d/1) = (7)/(12d)

So the final answer is (7)/(12d).

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1 and 3

Step-by-step explanation:

pick one from the junior side and 3 from seniors because the senior is more that the junoir

Answer:

990 different combinations

Step-by-step explanation:

There are 6 juniors and 12 seniors for a total of 16 students

Out of 6 juniors we have to pick 2 juniors

Out of 12 seniors we have to pick 2 seniors

The number of items r that we can pick from a larger set of n items is given by C(n, r) pronounced n choose r.. This is sometimes written as nCr

The formula forC(n, r) is

[tex]\boxed{C(n,r) = \dfrac{n!}{r! (n - r)! }}[/tex]

where n! = n factorial = n x (n-1) x (n-2) x .... x 3 x 2 x 1

We can choose 2 juniors out of 6 juniors in C(6, 2) ways
and
2 seniors out of 12 seniors in C(12, 2) ways

[tex]C(6, 2) = = \dfrac{6!}{ 2! (6 - 2)! }\\\\= \dfrac{6!}{2! \times 4! }\\\\= 15[/tex]

[tex]C(12, 2) = \dfrac{12!}{ 2! (12 - 2)! }\\\\ = \dfrac{12!}{2! \times 10! }\\\\= 66[/tex]

Therefore the total number of ways you can select 2 juniors and 2 seniors from a pool of 6 juniors and 12 juniors

= 15 x 66 = 990

The soap from Seventh Generation brand cost less than Method brand.

The unitary technique involves first determining the value of a single unit, followed by the value of the necessary number of units.

For Example, Let's say Ram spends 36 Rs. for a dozen (12) bananas.

12 bananas will set you back 36 Rs. 1 banana costs 36 x 12 = 3 Rupees.

As a result, one banana costs three rupees. Let's say we need to calculate the price of 15 bananas.

This may be done as follows: 15 bananas cost 3 rupees each; 15 units cost 45 rupees.

The Method brand that is $5. 49 for 36 oz

So, the unit rate of soap

= 5.49 / 36

= $ 0.1525

and, The Seventh Generation brand that is $3. 39 for 25 oz

So, the unit rate of soap

= 3.39/ 25

= $0.1356

So, the soap from Seventh Generation brand is cheaper.

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The correct answer is B. 56 4/7 in². This is determined by multiplying 22/7 by the square of the radius, which is 4 in this case. The answer is equal to 56 4/7 in².

The area of a circle is calculated by the formula A=πr2, where r is the radius of the circle. Therefore, to determine the area of a circle, one must know the radius of the circle. Once the radius is known, the area of the circle can be determined by multiplying pi (π) by the radius squared (r2).

The area of a circle is equal to pi multiplied by the square of the radius of the circle. Pi is equal to 22/7, so the area of the circle can be calculated by multiplying 22/7 by the square of the radius. The correct answer is B. 56 4/7 in².

This can be calculated by multiplying 22/7 by the square of the radius, which is 4 in this case. 22/7 multiplied by 4 squared is equal to 56 4/7 in².

To summarize, the correct answer is B. 56 4/7 in2. This is determined by multiplying 22/7 by the square of the radius, which is 4 in this case. The answer is equal to 56 4/7 in².

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One way is to divide both numbers (the numerator and denominator) by 5. You will get 2/5.

Another way is to multiply both numbers (the numerator and denominator) 10. You will get 100/250.

In trigonometry, temperatures can be converted between different scales, such as Fahrenheit and Celsius, using equations. For example, to convert from Fahrenheit to Celsius, you can use the equation C = (F - 32) * (5/9), where C is the temperature in Celsius and F is the temperature in Fahrenheit.

It seems like there are a lot of typos and irrelevant information in this question, making it difficult to understand what is being asked. However, I will do my best to provide an answer based on the key terms provided.

In trigonometry, temperatures can be converted between different scales, such as Fahrenheit and Celsius, using equations. For example, to convert from Fahrenheit to Celsius, you can use the equation C = (F - 32) * (5/9), where C is the temperature in Celsius and F is the temperature in Fahrenheit.

For the final aubmision, it is important to make sure that your work is accurate and complete before converting it to a PDF. This will ensure that your modified portfolio is professional and easy to understand.

Once you have completed your work, you can download it as a PDF and submit it in your portfolio. This will allow you to keep a record of your work and show your understanding of trigonometry and teffeeratures.

I hope this helps! If you have any further questions, please feel free to ask.

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The pair of functions that are not inverses of each other is (A).

To determine if two functions, g(x) and h(x), are inverses of each other, we need to check if the composition of the two functions, g(h(x)) and h(g(x)), both result in x.

A. g(x) = 2 + 9 = 11, h(x) = 1/2x - 9

g(h(x)) = g(1/2x - 9) = 2 + 9 = 11

h(g(x)) = h(11) = 1/2(11) - 9 = -3/2

Since g(h(x)) ≠ x and h(g(x)) ≠ x, the functions g(x) and h(x) are not inverses of each other.

B. g(x) = x - 1, h(x) = x + 1

g(h(x)) = g(x + 1) = (x + 1) - 1 = x

h(g(x)) = h(x - 1) = (x - 1) + 1 = x

Since g(h(x)) = x and h(g(x)) = x, the functions g(x) and h(x) are inverses of each other.

C. g(x) = 3x - 6, h(x) = 1/3x + 2

g(h(x)) = g(1/3x + 2) = 3(1/3x + 2) - 6 = x

h(g(x)) = h(3x - 6) = 1/3(3x - 6) + 2 = x

Since g(h(x)) = x and h(g(x)) = x, the functions g(x) and h(x) are inverses of each other.

D. g(x) = 3x + 4, h(x) = x - 4/3

g(h(x)) = g(x - 4/3) = 3(x - 4/3) + 4 = 3x - 4

h(g(x)) = h(3x + 4) = (3x + 4) - 4/3 = 3x + 8/3

Since g(h(x)) ≠ x and h(g(x)) ≠ x, the functions g(x) and h(x) are not inverses of each other.

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a. The inverse of a linear transformation T₁,  which is obtained is:

T₁⁻¹(x,y) = (37x - 9y)/2 , (-8x + 2y)/2

b. The inverse of T₂ does not exist.

c. Entering the values of T₁ into the equations gives:  

d. Entering the values of T₂ into the equations gives:  

a. To find the inverse of a linear transformation T₁ , we need to solve the system of equations:

2x + 9y = a
8x + 37y = b

We can use the determinant of the matrix associated with this system to find the inverse:

[tex]\left[\begin{array}{cc}2&9\\8&37\end{array}\right][/tex]

The determinant Δ is:

Δ = (2)(37) - (9)(8)

Δ = 74 - 72

Δ =  2

The inverse of T₁ is:

T₁⁻¹(a,b) = (1/2)(|37 -9| |a|) = (37a - 9b)/2 , (-8a + 2b)/2

T₁⁻¹(x,y) = (37x - 9y)/2 , (-8x + 2y)/2

b. To find the inverse of a linear transformation T₂, we need to solve the system of equations:

x + z = a
x + y = b
y + z = c

We can use the determinant of the matrix associated with this system to find the inverse:

[tex]\left[\begin{array}{ccc}1&0&1\\1&1&0\\0&1&1\end{array}\right][/tex]

The determinant Δ is:

Δ =  (1)(1)(1) + (0)(0)(1) + (1)(1)(0) - (1)(0)(0) - (1)(1)(1) - (0)(1)(1)

Δ = 1 - 1

Δ = 0

Since the determinant is 0, the inverse of T₂ does not exist.

c. To find x and y given T₁(x,y) = (5,-1), we can plug in the values into the equations for T₁:

2x + 9y = 5
8x + 37y = -1

We can use substitution to solve for x and y. From the first equation, we can solve for x:

x = (5 - 9y)/2

Plugging this into the second equation:

8(5 - 9y)/2 + 37y = -1

Simplifying:

20 - 36y + 37y = -2

y = -22

Plugging this back into the first equation to solve for x:

x = (5 - 9(-22))/2

x = 101/2

d. To find x, y, and z given T₂(x,y,z) = (5,-3,-1), we can plug in the values into the equations for T₂:

x + z = 5
x + y = -3
y + z = -1

We can use substitution to solve for x, y, and z. From the first equation, we can solve for x:

x = 5 - z

Plugging this into the second equation:

5 - z + y = -3

Simplifying:

y = -8 + z

Plugging this back into the third equation:

-8 + z + z = -1

2z = 7

z = 7/2

Plugging this back into the equations to solve for x and y:

x = 5 - 7/2

x = 3/2

y = -8 + 7/2

y = -9/2

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Answer:

6

Step-by-step explanation:

First, we need to calculate the average number of toys produced per hour during the busy season and the regular season.

During the busy season, the factory operates for 2 x 30 = 60 days or 60 x 12 = <<2*30=60>>720 hours.

The average number of toys produced during this time is 528 toys per 12-hour shift, or 528/12 = <<528/12=44>>44 toys per hour.

During the regular season, the factory operates for the rest of the year, which is 12 - 2 = 10 months, or 10 x 4 = 40 weeks, or 40 x 5 = <<1045=200>>200 days, or 200 x 8 = <<200*8=1600>>1600 hours.

The average number of toys produced during this time is 304 toys per 8-hour shift, or 304/8 = <<304/8=38>>38 toys per hour.

The difference between the average number of toys produced per hour during the busy season and the regular season is:

44 - 38 = <<44-38=6>>6 toys per hour.

Therefore, the factory produces 6 more toys per hour during the busy season than during the regular season. Answer: 6.

Answer:

To calculate the number of more toys the factory produces in an hour during the busy season than during the regular season, we can divide the difference between the two shifts by the total hours of each shift:

More toys per hour = (528-304)/(12-8) = 224/4 = 56 toys per hour

The simplification form of the expression "(z)/(z^(2)+8z+12)+(2)/(z^(2)+8z+12)" is = "1/(z+6)".

To add the two fractions, we need to have a common denominator. Since both fractions already have the same denominator of z^(2)+8z+12, we can simply add the numerators together and keep the same denominator.

So, (z)/(z^(2)+8z+12)+(2)/(z^(2)+8z+12) = (z+2)/(z^(2)+8z+12)

Now, we need to simplify the fraction as much as possible. We can do this by factoring the denominator and seeing if there are any common factors that can be canceled out.

The denominator can be factored as (z+6)(z+2).

So, (z+2)/(z^(2)+8z+12) = (z+2)/(z+6)(z+2)

Now, we can see that there is a common factor of (z+2) in both the numerator and denominator, so we can cancel them out.

So, the final simplified answer is 1/(z+6).

Therefore, (z)/(z^(2)+8z+12)+(2)/(z^(2)+8z+12) = 1/(z+6).

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The answer is V equals 355 over 113 times 6 squared times 8.

What is the volume of the cylinder?

The volume of a cylinder is given by the formula V = πr²h, where r is the radius of the base of the cylinder, and h is the height of the cylinder.

Alternatively, the volume of a cylinder can be found by multiplying the area of the base (πr²) by the height (h).

The correct method to calculate the number of cubic units of cake batter needed to fill the mold is:

[tex]V = \pi r^2h[/tex], where r is the radius and h is the height of the cylinder.

The diameter of the cake mold is 12 inches, so the radius is half of that, which is 6 inches.

Therefore, the volume of the cake batter needed is:

V = (355/113) x 6² x 8

V = (355/113) x 36 x 8

V = 904.96 cubic inches

So the answer is: V equals 355 over 113 times 6 squared times 8.

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So, the final answers are:

a.) f(g(-2)) = 1/3

b.) g(f(1)) = 8

Question 1: Use the following functions to evaluate each expression f(x) = 1/(x + 2) and g(x) = 3x + 7.

a.) f(g(-2)) = f(3(-2) + 7) = f(1) = 1/(1 + 2) = 1/3

b.) g(f(1)) = g(1/(1 + 2)) = g(1/3) = 3(1/3) + 7 = 1 + 7 = 8

So, the final answers are:

a.) f(g(-2)) = 1/3

b.) g(f(1)) = 8

In summary, to evaluate the expression f(g(x)) or g(f(x)), we need to first find the value of the inner function and then substitute it into the outer function. This process is called function composition. It is important to follow the order of operations and simplify the expression as much as possible.

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The possible rational zeros for the polynomial function P(x)=21x^(3)-38x^(2)+44x-10 are ±1, ±2, ±5, ±10, ±1/3, ±2/3, ±5/3, ±10/3, ±1/7, ±2/7, ±5/7, ±10/7, ±1/21, ±2/21, ±5/21, ±10/21.

The possible rational zeros of a polynomial function can be determined using the Rational Zero Theorem. This theorem states that if a polynomial function has rational zeros, they will be in the form of p/q, where p is a factor of the constant term and q is a factor of the leading coefficient.

In the given polynomial function, P(x)=21x^(3)-38x^(2)+44x-10, the constant term is -10 and the leading coefficient is 21. The factors of -10 are ±1, ±2, ±5, ±10 and the factors of 21 are ±1, ±3, ±7, ±21.

Using the Rational Zero Theorem, the possible rational zeros are:

p/q = ±1/1, ±2/1, ±5/1, ±10/1, ±1/3, ±2/3, ±5/3, ±10/3, ±1/7, ±2/7, ±5/7, ±10/7, ±1/21, ±2/21, ±5/21, ±10/21

Simplifying these fractions gives us the possible rational zeros:

±1, ±2, ±5, ±10, ±1/3, ±2/3, ±5/3, ±10/3, ±1/7, ±2/7, ±5/7, ±10/7, ±1/21, ±2/21, ±5/21, ±10/21

Therefore, the possible rational zeros for the polynomial function P(x)=21x^(3)-38x^(2)+44x-10 are ±1, ±2, ±5, ±10, ±1/3, ±2/3, ±5/3, ±10/3, ±1/7, ±2/7, ±5/7, ±10/7, ±1/21, ±2/21, ±5/21, ±10/21.

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The equivalent expression to the given one is:

y⁸*y³*x⁰*x⁻² = y¹¹*x⁻²

We start with the expression:

y⁸*y³*x⁰*x⁻²

Remember the exponent rule for products of powers with the same base, it says that we can write:

xᵃ*xᵇ = xᵃ⁺ᵇ

So we just add the two exponents.

Using exponent rules, we can rewrite the product as

y⁸*y³*x⁰*x⁻² = y³⁺⁸*x⁰⁻² = y¹¹*x⁻²

That is the equivalent expression.

Then we can write:

y⁸*y³*x⁰*x⁻² = y¹¹*x⁻²

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The area of shaded region is 0.15 ft².

The area of the shaded region is calculated by subtracting the area of the triangle from the area of the entire sector.

The angle subtended by the sector is calculated as;

θ = ¹/₂ (55⁰)

θ = 27.5⁰

The area of the triangle is calculated as;

A₁ = ¹/₂r² sinθ

where;

A₁ = ¹/₂ x 4² sin(27.5)

A₁ = 3.69 ft²

Area of the sector is calculated as;

A_t = θ/360 x πr²

A_t = ( 27.5 / 360) x π x 4²

A_t = 3.84 ft²

Area of shaded region = 3.84 ft² - 3.69 ft² = 0.15 ft².

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The amount of cement needed is 609 cubic ft.

The amount of cement needed for the job can be calculated by finding the difference between the volume of the outer cistern and the volume of the inner cistern.

The volume of the outer cistern can be found using the formula for the volume of a cylinder, V = πr²h, where r is the radius and h is the height. The outer radius is half the outer diameter, or 10ft/2 = 5ft. The outer height is 20ft. So the volume of the outer cistern is:

V = π(5ft)²(20ft) = 1570.8 cubic ft

The volume of the inner cistern can be found using the same formula, but with the inner radius and inner height. The inner radius is the outer radius minus the thickness of the walls, or 5ft - 1ft = 4ft. The inner height is the outer height minus the thickness of the bottom, or 20ft - 1ft = 19ft. So the volume of the inner cistern is:

V = π(4ft)²(19ft) = 961.6 cubic ft

The difference between the two volumes is the amount of cement needed:

1570.8 cubic ft - 961.6 cubic ft = 609.2 cubic ft

To the nearest cubic foot, the amount of cement needed is 609 cubic ft.

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The correct answer is c) \( 12 / 5 \).

We can use the relationship between the sine, cosine, and tangent of an angle to find the value of the tangent. The formula is:

\( \tan \alpha = \frac{\sin \alpha}{\cos \alpha} \)

Plugging in the given values for the sine and cosine of alpha, we get:

\( \tan \alpha = \frac{12 / 13}{5 / 13} \)

Simplifying the fraction, we get:

\( \tan \alpha = \frac{12}{5} \)

Therefore, the correct answer is c) \( 12 / 5 \).

In conclusion, if \( \sin \alpha=12 / 13 \), and \( \cos \alpha=5 / 13 \), then \( \tan \alpha=12 / 5 \).

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