The equation of the plane passing through the origin, containing the vectors \( \left[\begin{array}{c}1 \\ -1 \\ 1\end{array}\right] \) and \( \left[\begin{array}{l}1 \\ 0 \\ 3\end{array}\right] \) ha

The exact value of cos(5π/12), using the cosine summation identity, is (√6 - √2)/4.

The exact value of cos(5π/12) can be found using the cosine sum identity, which is:

cos(a + b) = cosa · cosb - sina · sin b

In this case, we can rewrite 5π/12 as (π/4) + (π/6) and use the identity:

cos(5π/12) = cos[(π/4) + (π/6)] = cos(π/4) · cos (π/6) - sin(π/4) · sin (π/6)

Using the values of cos(π/4) = √2/2, cos(π/6) = √3/2, sin(π/4) = √2/2, and sin(π/6) = 1/2, we can plug them into the equation:

cos(5π/12) = (√2/2) · (√3/2) - (√2/2) · (1/2) = √6/4 - √2/4 = (√6 - √2)/4

Therefore, the exact value of cos(5π/12) is (√6 - √2)/4.

See more about cosine at https://brainly.com/question/24305408.

#SPJ11

Given:-

[tex] \: [/tex]

[tex] \: [/tex]

Solution:-

[tex] \: [/tex]

[tex] \: [/tex]

[tex] \: [/tex]

[tex] \: [/tex]

[tex] \: [/tex]

hope it helps! :)

The correct answer is c. 1/x^2 - 2/x + 1.

To find f(g(x)), we need to plug in the function g(x) into the function f(x).

f(x) = x^2

g(x) = 1/x-1

So, f(g(x)) = (1/x-1)^2

Using the distributive property, we can expand this expression:

f(g(x)) = (1/x-1)(1/x-1)

f(g(x)) = 1/x^2 - 1/x - 1/x + 1

f(g(x)) = 1/x^2 - 2/x + 1

Therefore, the correct answer is c. 1/x^2 - 2/x + 1.

Learn about Distributive property

brainly.com/question/5637942

#SPJ11

The correct answer is option C. Only Table 1 and Table 3 represent a function.

Each input value should be paired with only one output value, as stated in the definition of a function. Therefore, we must verify that each input value is paired with only one output value in order to determine which tables represent a function.

For each input value, Table 1 contains unique output values and unique input values. As a result, a function is represented by Table 1.

For the same input value (input 1), there are two distinct output values in Table 2. As a result, there is no function represented in Table 2.

For each input value, Table 3 contains unique output values and unique input values. As a result, a function is represented by Table 3.

For the same input value (input -2) in Table 4, there are two distinct output values. Subsequently, Table 4 doesn't address a capability.

Consequently, c is the correct response because Tables 1 and 3 only depict functions.

Learn more about Functions :

https://brainly.com/question/17043948
#SPJ4

Complete Question:

Which table(s) represent(s) a function? A. Table 1 only B. Table 2 only C. Tables 1 and 3 only D. Tables 1, 3, and 4 only

The required,
(a) The missing terms of the first sequence are -7, 6, and 19.
(b) The missing terms of the second sequence are 71, 88, and 105.
(c) The missing terms of the third sequence are 2, -6, and -14.

Arithmetic progression is the series of numbers that have common differences between adjacent values

To find the missing terms of each arithmetic sequence, we need to use the formula for the nth term of an arithmetic sequence:

an = a₁ + (n - 1)d

where an is the nth term, a₁ is the first term, n is the number of terms, and d is a common difference.

Sequence 1,

a₁ = -20, an = 19

n = ? (unknown)

d = (an - a1) / (n - 1)

19 = -20 + (n - 1)d
39 = (n - 1)d

d = 39 / (n - 1)

We also know that a₂, a₃, and a₄ are missing, and there are four terms in total. Therefore, n = 4.

d = 39 / (4 - 1) = 13

Using the formula, we can now find the missing terms:

a₂ = a₁ + d = -20 + 13 = -7

a₃ = a₂ + d = -7 + 13 = 6

a₄ = a₃ + d = 6 + 13 = 19

So the missing terms of the first sequence are -7, 6, and 19.

Similarly,

The missing terms of the second sequence are 71, 88, and 105.
The missing terms of the third sequence are 2, -6, and -14.

Learn more about arithmetic progression here: https://brainly.com/question/20334860
#SPJ9

This equation has two solutions: λ = 0 and λ = -1

The determinant of a matrix is given by the formula:

det(A) = a11(a22a33 - a23a32) - a12(a21a33 - a23a31) + a13(a21a32 - a22a31)

In the given matrix, the values of a11, a22, and a33 are λ, (λ+1), and λ respectively. The values of a12, a13, a21, a23, a31, and a32 are all 0. Substituting these values into the formula for the determinant gives:

det(A) = λ((λ+1)λ - 0) - 0(0 - 0) + 0(0 - 0) = λ^3 + λ^2

To find the values of λ for which the determinant is zero, we can set the determinant equal to zero and solve for λ:

λ^3 + λ^2 = 0
λ^2(λ + 1) = 0

This equation has two solutions: λ = 0 and λ = -1. Therefore, the values of λ for which the determinant is zero are 0 and -1. The answer can be written as a comma-separated list as follows:

λ = 0, -1

Learn more about determinant of a matrix

brainly.com/question/4470545

#SPJ11

The interest earned on a deposit of $18,000 for five years at 5% APR compounded semiannually.

To calculate the interest earned on a deposit of $18,000 for five years at 5% APR compounded semiannually, we will use the compound interest formula:

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

Where:
A = final amount
P = principal amount (initial investment)
r = annual interest rate (as a decimal)
n = number of times interest is compounded per year
t = number of years

Plugging in the given values:

A = 18,000(1 + 0.05/2)^(2*5)

A = 18,000(1.025)^10

A = 23,386.28

To find the interest earned, we subtract the initial investment from the final amount:

Interest earned = A - P

Interest earned = 23,386.28 - 18,000

Interest earned = $5,386.28

Therefore, the interest earned on a deposit of $18,000 for five years at 5% APR compounded semiannually is $5,386.28.

Learn more about interest

brainly.com/question/30393144

#SPJ11

The decimal (base-10) equivalent of the binary (base-2) value (111)_(2) is (7)_(10).

To convert a binary (base-2) value into its decimal (base-10) equivalent, we need to multiply each digit in the binary value by the corresponding power of 2, and then add the results together.

Here is how to do it step-by-step for the given binary value (111)_(2):

1. Start with the rightmost digit (the least significant bit) and work your way to the left:
(1 * 2^0) + (1 * 2^1) + (1 * 2^2)

2. Simplify the powers of 2:
(1 * 1) + (1 * 2) + (1 * 4)

3. Multiply the digits by the powers of 2:
1 + 2 + 4

To know more about binary value click on below link:

https://brainly.com/question/19802955#

#SPJ11

The equation of the line in slope-intercept form is y = -x + 1

To write an equation for the line in point-slope form, we can use the formula y - y1 = m(x - x1), where m is the slope of the line and (x1, y1) is a point on the line.

Since the x-intercept and y-intercept are both 1, we know that the line passes through the points (1,0) and (0,1).

To find the slope of the line, we can use the formula m = (y2 - y1) / (x2 - x1). Plugging in the coordinates of the two points, we get:

m = (1 - 0) / (0 - 1) = -1

Now we can plug in the slope and one of the points into the point-slope form equation:

y - 0 = -1(x - 1)

Simplifying, we get:

y = -x + 1

This is the equation of the line in point-slope form.

To write the equation in slope-intercept form, we can use the formula y = mx + b, where m is the slope of the line and b is the y-intercept.

We already found the slope to be -1, and the y-intercept is given as 1. So we can plug these values into the slope-intercept form equation:

y = -1x + 1

Simplifying, we get:

y = -x + 1

This is the equation of the line in slope-intercept form.

To know more about slope-intercept form click on below link:

https://brainly.com/question/29146348#

#SPJ11

After 5 years, the balance of your account with an initial investment of $2,500, 6% annual interest compounded quarterly, will be $3,325.81. This can be calculated using the following formula:

Balance = P(1 + r/n)nt

Where:
P = Principal Amount
r = Annual Interest Rate
n = Number of Compounding Periods
t = Number of Years

Therefore, for your given example, the equation would be:
Balance = 2500(1 + 0.06/4)4 x 5
Balance = 2500(1.015)20
Balance = 2500(1.32581)
Balance = $3,325.81

You can read more about compound interest at https://brainly.com/question/28020457

#SPJ11

The null and alternative hypothesis for the given scenario is (H0: μ ≤ 1.5)  and (Ha: μ > 1.5).

The null hypothesis states that a population parameter is equal to a value. The null hypothesis is often an initial claim that researchers specify using previous research or knowledge.

Null Hypothesis (H0): The average time that teens use their phones before upgrading is 1.5 years or less (H0: μ ≤ 1.5).

Alternative Hypothesis (Ha): The average time that teens use their phones before upgrading is more than 1.5 years (Ha: μ > 1.5).

In words, the null hypothesis states that the average time that teens use their phones before upgrading is 1.5 years or less, which is consistent with the statement made by the phone company.

The alternative hypothesis states that the average time is more than 1.5 years, which is consistent with your mother's belief.

We use the symbol μ to represent the population means of the time that teens use their phones before upgrading.

Hence, the null and alternative hypothesis for the given scenario is (H0: μ ≤ 1.5)  and (Ha: μ > 1.5).

To learn more about the null hypothesis and alternative hypothesis visit:

https://brainly.com/question/25263462

#SPJ1

The Mean, Mode, and Median for the number of wins are 37.8; 29, 42; 38 respectively. The standard deviation and variation are 11.15 and 124.26 respectively. 9 teams had over 41 wins. 16.67% of teams scored below 30 wins. Our data set for age does not represent a normal curve, it is positively skewed. The lowest and highest Z-score for wins are -1.78 and 1.99 respectively. The Z-score for the number of wins for 37 is -0.07.

To create a frequency table, we need to categorize the number of wins into intervals and count the number of teams in each interval. We can use the following intervals: 15-24, 25-34, 35-44, 45-54, 55-64.

| Interval | Frequency |
|---------|----------|
| 15-24   | 3        |
| 25-34   | 5        |
| 35-44   | 8        |
| 45-54   | 5        |
| 55-64   | 2        |

To calculate the mean, we add up all the values and divide by the number of values:

Mean = (60 + 44 + 39 + 29 + 23 + 57 + 50 + 43 + 37 + 27 + 49 + 42 + 37 + 29 + 19 + 56 + 51 + 40 + 33 + 26 + 48 + 42 + 31 + 25 + 18 + 53 + 44 + 40 + 29 + 23)/30 = 37.8

To find the mode, we look for the value that appears most often. In this case, it is 29 and 42, both appearing twice.

Mode = 29, 42

To find the median, we need to order the values from smallest to largest and find the middle value. If there is an even number of values, we take the average of the two middle values.

Ordered values: 18, 19, 23, 23, 25, 26, 27, 29, 29, 31, 33, 37, 37, 39, 40, 40, 42, 42, 43, 44, 44, 48, 49, 50, 51, 53, 56, 57, 60

Median = (37 + 39)/2 = 38

To calculate the standard deviation, we first find the variance by subtracting the mean from each value, squaring the result, and then taking the average of those squared differences. The standard deviation is the square root of the variance.

Variance = [(60-37.8)^2 + (44-37.8)^2 + ... + (23-37.8)^2]/30 = 124.26

Standard deviation = sqrt(124.26) = 11.15

To find the number of teams with over 41 wins, we can count the values that are greater than 41:

Number of teams with over 41 wins = 9

To find the percent of teams with below 30 wins, we can count the values that are less than 30 and divide by the total number of teams:

Percent of teams with below 30 wins = (5/30)*100 = 16.67%

To create a histogram, we can use the intervals and frequencies from the frequency table and plot them on a graph with the intervals on the x-axis and the frequencies on the y-axis. We can also add a normal curve by calculating the mean and standard deviation and using a normal distribution function.

The data set for the number of wins does not represent a normal curve, as it is slightly positively skewed (there are more values on the lower end of the distribution).

The lowest Z-score is for the value 18, which is (18-37.8)/11.15 = -1.78. The highest Z-score is for the value 60, which is (60-37.8)/11.15 = 1.99.

The Z-score for the value 37 is (37-37.8)/11.15 = -0.07. This means that the value 37 is 0.07 standard deviations below the mean.

For more such questions on Data Set and Statistics.

https://brainly.com/question/30218856#

#SPJ11

The expression by using property is 1000(1+0.03)

A number system is defined as a system of writing to express numbers.

Using the distributive property, we can factor out 1000 from the expression for year 1:

Y = 1000 + 0.03(1000) - 1030.00

= 1000(1 + 0.03) - 1030.00

Using the identity property of 1, we can rewrite (1 + 0.03) as:

(1 + 0.03) = 1.03

Substituting back into the expression for year 1, we get:

Y = 1000(1.03) - 1030.00

Hence, the expression by using property is 1000(1+0.03)

To learn more on Number system click:

https://brainly.com/question/22046046

#SPJ1

Answer:

Step-by-step explanation:

A cylinder has a base diameter of 12 foot and a height of 12 foot what is its volume in cubic feet to the nearest 10s place

The volume of a cylinder can be calculated using the formula:

V = πr^2h

where:

V is the volume of the cylinder

π is a mathematical constant, approximately equal to 3.14159

r is the radius of the base of the cylinder

h is the height of the cylinder

We are given the diameter of the base, which is 12 feet. The radius (r) is half of the diameter, so we have:

r = 12/2 = 6 feet

The height (h) is also given as 12 feet.

Substituting the values we have into the formula, we get:

V = π(6)^2(12) ≈ 1357.17

Rounding to the nearest 10's place, the volume is approximately 1360 cubic feet. Therefore, the volume of the cylinder is 1360 cubic feet to the nearest 10's place.

Answer: Using 45π cm^3

a) h = 5 cm

b) approximately $2.83

c) h = 9/5 cm

d) approximately $9.90

Step-by-step explanation:

a) The volume of a cylinder is given by the formula [tex]V = \pi r^2h[/tex]. Given that the volume is equal to [tex]45\pi cm^{3}[/tex] and radius is [tex]3cm[/tex], we can plug in.

[tex]45\pi = 3^2h\pi \\45\pi = 9h\pi\\\frac{45\pi }{9\pi } = h\\ h = 5[/tex]

b) Since coffee costs $0.02 per cubic centimeter, and there are [tex]45\pi[/tex] cubic centimeters, 0.02 x [tex]45\pi[/tex] = 2.8274338, and rounding gives $2.83

c) Using the same formula from part a, we plug in [tex]5cm[/tex] for r instead of [tex]3cm[/tex].

[tex]45\pi = 5^2h\pi \\45\pi =25h\pi \\\frac{45\pi }{25\pi } = h\\h = \frac{45}{25} = \frac{9}{5}[/tex]

d) Because hot chocolate powder costs $0.07 per cubic centimeter, and there are [tex]45\pi[/tex] cubic centimeters, 0.07 x [tex]45\pi[/tex] = 9.89601685, and rounding makes it $9.90

The surface area of the pyramid. A drawing of a square pyramid. The length of the base is 4.5 meters is 74.25 square meters.

To find the area of ​​a pyramid, we need to find the area of ​​the square base and the areas of the four triangular faces, then add them together.

Calculate the area of ​​the base of the square:

The area of ​​the square is the length of one side multiplied by itself, so the area of ​​the base of the square is 4.5 x 4.5 = 20 .25 square meters.

Find the area of ​​each triangle face:

The area of ​​a triangle is the base times the height divided by 2, so the area of ​​each triangle face is 0.5 x 4.5 x 6 = 13.5 square meters.

Add the area of ​​:

The area of ​​the pyramid is the sum of the areas of the square base and the four triangular faces, so the area is 20.25 + 4 x 13.5 = 20.25 + 54 = 74.25 square meters.

The area of ​​the pyramid is therefore 74.25 square meters.

Learn more about the surface area of the pyramid at

https://brainly.com/question/1503383

#SPJ4

a)H0:p=0.4
H1:p>0.4

b)right-tailed test

c)0.075

d)0.0074
e)reject the null hypothesis

In this problem, we are testing the claim that the proportion of people who own cats is larger than 40% at the 0.025 significance level. The null and alternative hypothesis can be written as:
H0:p=0.4
H1:p>0.4
This is a right-tailed test. Using the sample data, the test statistic is 0.075 and the p-value is 0.0074. Therefore, we reject the null hypothesis and conclude that the proportion of people who own cats is greater than 40% at the 0.025 significance level.

Learn more about null hypothesis

brainly.com/question/28920252

#SPJ11

d

2/3 is the same as 4/6 and 25/100 is the same as 1/4. -1.3 is transferred to the beginning of the equation

The expressions ordered from least to greatest are: -2 < b < (a + c)

In maths, an expressiοn is a cοmbinatiοn οf numbers, variables, functiοns (such as additiοn, subtractiοn, multiplicatiοn οr divisiοn etc.)

Expressiοns can be thοught οf as similar tο phrases. In language, a phrase οn its οwn may include an actiοn, but it dοesn't make a cοmplete sentence.

According to the question:

-1 < a < 0

2 < b < 3

4 < c < 5

Now, we have the expression a + c

= (-1 < a < 0) + (4 < c < 5)

= (-1 + 4) < (a + c) < (0 + 5)

= 3 < (a + c) <  5

Thus, we have the expressions ordered from least to greatest are:

-2 < 2 < b < 3 < (a + c) <  5

-2 < b < (a + c)

To know more about problems related to mathematical expressions, click here:

https://brainly.com/question/4344214

#SPJ1

Complete question:

Answer: y = 5/6x

Step-by-step explanation:

It is a direct variation.

The simplified expression is \(2 \cos x\).

The question is asking us to simplify the expression \( \frac{\sin x+\tan x \cos x}{\tan x} \) and show the steps to reach the final result.

First, we can use the identity \(\tan x = \frac{\sin x}{\cos x}\) to rewrite the expression:

\( \frac{\sin x+\tan x \cos x}{\tan x} = \frac{\sin x+\frac{\sin x}{\cos x} \cos x}{\frac{\sin x}{\cos x}} \)

Next, we can simplify the numerator by canceling out the \(\cos x\) terms:

\( \frac{\sin x+\frac{\sin x}{\cos x} \cos x}{\frac{\sin x}{\cos x}} = \frac{\sin x+\sin x}{\frac{\sin x}{\cos x}} \)

Now, we can combine the \(\sin x\) terms in the numerator:

\( \frac{\sin x+\sin x}{\frac{\sin x}{\cos x}} = \frac{2 \sin x}{\frac{\sin x}{\cos x}} \)

Finally, we can simplify the expression by canceling out the \(\sin x\) terms:

\( \frac{2 \sin x}{\frac{\sin x}{\cos x}} = \frac{2 \sin x}{\sin x} \cdot \frac{\cos x}{1} = 2 \cos x \)

So the final result is:

\( \frac{\sin x+\tan x \cos x}{\tan x} = 2 \cos x \)

Therefore, the simplified expression is \(2 \cos x\).

Learn more about fraction

brainly.com/question/10354322

#SPJ11

Answer:

Step-by-step explanation:

Make a model

Answer: The answer is 9/25. (36%)

Step-by-step explanation:

Since we know that 7/25 workers walk and 2/25 walk, we can do simple addition to find the total Percentage. This equals 9/25. To find the percentage, do the following:

Write 9/25 into a percentage: Divide

9 / 25 x (100%) = 36%

0.36 (100%) = 36%

Therefore 36% is your answer.

The value of the discriminant is 0 and the number of real solutions is 1

For the given equation, 36x^(2)-24x+4=0, we can find the value of the discriminant and the number of real solutions using the quadratic formula.

The quadratic formula is x = (-b ± √(b^(2)-4ac))/(2a), where a, b, and c are the coefficients of the equation.

The discriminant is the part of the formula under the square root, b^(2)-4ac.

In this equation, a = 36, b = -24, and c = 4. Plugging these values into the discriminant, we get:

Discriminant = (-24)^(2)-4(36)(4) = 576-576 = 0

Since the discriminant is equal to 0, there is only one real solution to this equation.

Therefore, the value of the discriminant is 0 and the number of real solutions is 1.

To know more about discriminant and real value refer here:

https://brainly.com/question/17858049#

#SPJ11

The scores of students were normally distributed.

a)Approximately 68% of the students scored between 40 and 62 .

b) The approximately 95% of the students scored between 29 and 73.

We have a nationwide test taken by high school students, Mean score, μ = 51

Standard deviations, s = 12

The scores were normally distributed, that is

X~ N(M= 51, s = 12)

Lower bound of confidence interval= 40

Upper bound of CI = 62

As we know according to empirical rule the percentage of data falls within one,two and three

standard deviations are 68%,95% and 99.7%

respectively.

a) Mean + standard deviations = 51 + 12 = 63 close to 62

= Upper bound

For 2 standard deviations, 51 + 2×12 = 75

Mean - standard deviations= 51 - 12 = 39 close to 40 = lower bound

For 2 standard deviations, 51 - 2×12 = 27

Thus, data falls within one standard deviation

that is under 68% and so, approximately 68% of students scored between 40 and 62.

b) Similarly, the empirical rule demonstrates that 95% of scores falls within two standard deviation.

mean - 2× standard deviations= 51 - 2× 11

= 51 - 22 = 29

mean + 2× standard deviations= 51 + 2×11

= 51 + 22 = 73

Therefore, approximately 95% of students scored

between 29 and 73.

For more information about empirical rule, visit :

https://brainly.com/question/21417449

#SPJ4

Complete question:

On a nationwide test taken by high school students, the mean score was 51 and the standard deviation was 11

The scores were normally distributed. Complete the following statements.

(a) Approximately ?% of the students scored between 40 and 62 .

(b) Approximately 95% of the students scored between ? and ?

Answer:

vertex = (2, - 10 )

Step-by-step explanation:

given a quadratic function in standard form

ax² + bx + c ( a ≠ 0 )

then the x- coordinate of the vertex is

[tex]x_{vertex}[/tex] = - [tex]\frac{b}{2a}[/tex]

x² - 4x - 6 ← is in standard form

with a = 1 and b = - 4 , then

[tex]x_{vertex}[/tex] = - [tex]\frac{-4}{2}[/tex] = 2

substitute x = 2 into the function for corresponding y- coordinate

2² - 4(2) - 6

= 4 - 8 - 6

= - 10

vertex = (2, - 10 )

The vertex of the function is (-3, -20), the axis of symmetry is x = -3, the x-intercepts are (-8.44, 0) and (1.44, 0), the y-intercept is (-11), the range is (-∞, -20], and the domain is (-∞, ∞).

To find the vertex, axis of symmetry, x-intercept, y-intercept, range, and domain of the function f(x) = x² + 6x - 11, we can use the following formulas:

- Vertex: (-b/2a, f(-b/2a))

- Axis of symmetry: x = -b/2a

- X-intercept: Solve f(x) = 0

- Y-intercept: f(0)

- Range: All real numbers for a parabola that opens up or down

- Domain: All real numbers


Using these formulas, we can find the following:

- Vertex: (-3, -20)

- Axis of symmetry: x = -3

- X-intercept: (-8.44, 0) and (1.44, 0)

- Y-intercept: (-11)

- Range: (-∞, -20]

- Domain: (-∞, ∞)


Therefore, the vertex of the function is (-3, -20), the axis of symmetry is x = -3, the x-intercepts are (-8.44, 0) and (1.44, 0), the y-intercept is (-11), the range is (-∞, -20], and the domain is (-∞, ∞).

See more about domain and range at: https://brainly.com/question/28801359

#SPJ11

The value of g is  9√3 in

In mathematics, the radical form refers to expressing a mathematical expression or equation using radicals, which are symbolized by the radical sign (√). The radical sign indicates the root of a number, which is a value that when multiplied by itself, produces the original number. The most common type of radical is the square root (√), which represents the positive root of a number.

We have that;

Sin 30 = g/18√3

1/2 = g/18√3

g = 1/2 * 18√3

g = 9√3 in

Learn more about radical form:https://brainly.com/question/30260621

#SPJ1

Answer:

t=5.2, w=6

Step-by-step explanation:

tan(30)=[tex]\frac{3}{t}[/tex]

[tex]t=\frac{3}{tan(30)} \\t=5.2[/tex]

[tex]sin(30)=\frac{3}{w}\\w=\frac{3}{sin(30)} \\w=6[/tex]

The side w is the hypotenuse and is equal to 6, while f is the adjacent and is equal to 3√3 in radical form and 5.2 expressed as a decimal.

The trigonometric ratios is concerned with the relationship of an angle of a right-angled triangle to ratios of two side lengths.

The basic trigonometric ratios includes;

sine, cosine and tangent.

recall that sin30° = 1/2 and cos30° = √3/2

sin 30° = 3/w {opposite/hypotenuse}

w = 3/sin 30° {cross multiplication}

w = 3 ÷ 1/2

w = 3 × 2

w = 6

cos 30° = f/6 {adjacent/hypotenuse}

f = 6 × cos 30° {cross multiplication}

f = 6 × √3/2

f = 3√3

Therefore, the side lengths of w and f of the right triangle are 6 and 3√3 respectively using the trigonometric ratio of sine and cosine of angle 30°.

Know more about trigonometric ratios here: https://brainly.com/question/3457795

#SPJ1