Answer:
f(-1) = 7
Step-by-step explanation:
Hello!
You can evaluate for f(-1) by substituting -1 for x in the equation.
Evaluate f(-1)f(x) = -x² + 3x + 11f(-1) = -(-1)² + 3(-1) + 11f(-1) = -1 -3 + 11f(-1) = -4 + 11f(-1) = 7f(-1) is 7.
Answer:
f(-1) = 7
Step-by-step explanation:
Hello!
You can evaluate for f(-1) by substituting -1 for x in the equation.
Evaluate f(-1)f(x) = -x² + 3x + 11f(-1) = -(-1)² + 3(-1) + 11f(-1) = -1 -3 + 11f(-1) = -4 + 11f(-1) = 7f(-1) is 7.
A circular pool measures 12 feet across. One cubic yard of concrete is to be used to create a circular border of uniform width around the pool. If the border is to have a depth of 6 inches, how wide will the border be?
SOLUTION:
Step 1:
In this question, we are given the following:
A circular pool measures 12 feet across.
One cubic yard of concrete is to be used to create a circular border of uniform width around the pool.
If the border is to have a depth of 6 inches, how wide will the border be?
Step 2:
From the question, we can see that:
[tex]6\text{ inches = 0. 5 feet}[/tex][tex]1\text{ cubic yard = 3 ft x 3ft x 3ft = }27ft^3[/tex][tex]\begin{gathered} \text{Let the radius of the pool = ( 6+x ) feet} \\ \text{Let the width of the concrete that is used to } \\ \text{create the circular border = 6 feet} \end{gathered}[/tex][tex]\text{Let the depth of the border = 6 inches = }\frac{6}{12}=\text{ 0. 5 inches}[/tex]Step 3:
[tex]\begin{gathered} U\sin g\text{ } \\ \pi R^2h\text{ - }\pi r^2\text{ h = 27} \\ \pi(6+x)^2\text{ 0. 5 - }\pi(6)^2\text{ 0. 5 = 27} \\ \text{0. 5}\pi(x^2\text{ + 12x + 36 - 36 ) = 27} \\ 0.\text{ 5 }\pi(x^2\text{ + 12 x) = 27} \\ \text{Divide both sides by 0. 5 }\pi\text{ , we have that:} \end{gathered}[/tex][tex]x^2\text{ + 12 x - (}\frac{27}{0.\text{ 5}\pi})=\text{ 0}[/tex]Solving this, we have that:
CONCLUSION:
From the calculations above, we can see that the value of the x:
( which is the width of the border ) = 1. 293 feet
(correct to 3 decimal places)
[tex]x \geqslant - 2[/tex]
PLEASE HELP!!
A)
B)
C)
D)
Answer:
B
Step-by-step explanation:
[tex]x\geq -2[/tex] means that [tex]x[/tex] can be all values that are greater than -2, and the line under the inequality sign adds that [tex]x[/tex] can be equal to it as well.
Since B represents all values of [tex]x[/tex] that are greater than -2 along with -2 itself due to the closed circle, it is the correct answer.
Answer:
it is c i took the test i hope this helps
If the expression 1/ square root of x was placed in form x^a, then which of the following would be the value of a?
4) -1/2
1) Rewriting the expression:
[tex]\frac{1}{\sqrt[]{x}}[/tex]2) As a power we can write this way, considering that we can rewrite any radical as a power and that when we have a radical on the denominator we can rewrite it as a negative rational exponent. So we can write it out:
[tex]\frac{1}{\sqrt[]{x}}=\frac{1}{x^{\frac{1}{2}}}=x^{-\frac{1}{2}}[/tex]3) Hence, the answer is 4) -1/2
Find the greatest common factor of the following monomials. 28g^5h^2 12g^6h^5
The GCF of these monomials i.e, 28g^5h^2 and 12g^6h^5 is 4h^2g^5
What is monomials?
Monomial expressions include only one non-zero term. Numbers, variables, or multiples of numbers and variables are all examples of monomials.
First take the coefficient ie, 28 and 12 to find the GCF
The GCF of 28 and 12 is 4
Now, find out the GCF of the variables for that you take the lowest exponent from both the variables g and h
for g variable it will be g^5 and,
for h variable it will be h^2
Therefore, the GCF of these monomials is 4h^2g^5
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A pool is built in the shape of an ellipse, centered at the origin. The maximum vertical length is 40 feet, and the maximum horizontal width is 18 feet. Which of the following equations represents the pool?
If the maximum vertical length of pool is 40 feet, and the maximum horizontal width of pool is 18 feet , then the equation that represent the pool is x/81 + y/400 = 1 , the correct is option is (B) .
In the question ,
it is given that
the shape of the pool is ellipse .
the maximum vertical length is 40 feet
the maximum horizontal length is 18 feet .
the general equation of the ellipse , is given by x/a + y/b = 1 ,
where a is the length of horizontal axis from origin
and b is the length of the vertical axis from the origin ,
So , the equation that represents the pool is x/81 + y/400 = 1
Therefore , if the pool is in the shape of ellipse , then the equation of the pool is x/81 + y/400 = 1 .
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Write standard form for the equation of the line: y = 1/2x - 5*
Explanation
the standard form of a line is in the form Ax + By = C where A is a positive integer, and B, and C are integers
so, we need to write in this form
[tex]Ax+By=C[/tex]Step 1
subtrac 1/2x in both sides
[tex]\begin{gathered} y=\frac{1}{2}x-5 \\ \\ y-\frac{1}{2}x=\frac{1}{2}x-5-\frac{1}{2}x \\ \\ y-\frac{1}{2}x=-5 \\ \text{reorder} \\ -\frac{1}{2}x+y=-5 \end{gathered}[/tex]I hope this helps you
StatusRecovery8Help ResourcessAABC ~ AXYZFind the missing side length, s.B.3 65А&Х-ZCross multiplySE ][?] = [ ]153s
Since triangles ABC and XYZ are similar, the ratio between their corresponding sides is constant; thus,
[tex]\begin{gathered} \frac{AB}{XY}=\frac{BC}{YZ} \\ \Rightarrow\frac{3}{5}=\frac{6}{s} \end{gathered}[/tex]Solving for s,
[tex]\begin{gathered} \frac{3}{5}=\frac{6}{s} \\ \Rightarrow\frac{3}{5}\cdot s=\frac{6}{s}\cdot s \\ \Rightarrow\frac{3s}{5}=6 \\ \Rightarrow\frac{3s}{5}\cdot5=6\cdot5 \\ \Rightarrow3s=30 \\ \Rightarrow s=\frac{30}{3} \\ \Rightarrow s=10 \end{gathered}[/tex]Thus, the result of the cross multiplication is 3s=30 and the answer is s=10
12. Suppose you roll a pair of six-sided dice.(a) What is the probability that the sum of the numbers on your dice is exactly 4? (b) What is the probability that the sum of the numbers on your dice is at most 2? (c) What is the probability that the sum of the numbers on your dice is at least 12?
Probability is computed as follows:
[tex]\text{probability}=\frac{\text{ number of favorable outcomes}}{\text{ total number of outcomes}}[/tex]When rolling a pair of six-sided dice, the total number of outcomes is 36 (= 6x6)
(a) number of favorable outcomes: 3 (dice: 1 and 3, 2 and 2, 3 and 1)
Then, the probability that the sum of the numbers on your dice is exactly 4 is:
[tex]\text{probability }=\frac{3}{36}[/tex](b) number of favorable outcomes: 1 (dice: 1 and 1)
Then, the probability that the sum of the numbers on your dice is at most 2 is:
[tex]\text{probability }=\frac{1}{36}[/tex](c) number of favorable outcomes: 1 (dice: 6 and 6)
Then, the probability that the sum of the numbers on your dice is at least 12 is:
[tex]\text{probability }=\frac{1}{36}[/tex]an art teacher makes a batch of green paint by mixing 5/8 cup of yellow paint with 5/8 cup of blue paint if she mixes 29 batches how many cups will she have with green paint
1 lote = 5/8 cup yellow + 5/8 cup blue
29 lotes = 29(5/8) +29(5/8) cups
29 lotes = 58(5/8)= (58*5)/8=290/8=145/4
145/4 =35.25 cups of paint
Calculate Sample Variance for the following data collection: 10, 11, 12, 13, 14,18.
The Variance of a set of data is defined as the average of the square of the deviation from the mean.
The first step is to calculate the mean of the data.
[tex]\frac{10+11+12+13+14+18}{6}=13[/tex]Now we take the difference from the mean, square it, and then average the result.
[tex]\frac{(10-13)^2+(11-13^2)+(12-13)^2+(13-13)^2+(14-13)^2+(18-13)^2}{6}[/tex][tex]\Rightarrow\frac{9+4+1+0+1+25}{6}[/tex][tex]\Rightarrow6.67[/tex]Hence, the variance of the data is 6.7 (rounded to the nearest tenth)
Question 2-22
A cake is cut into 12 equal slices. After 3 days Jake has eaten 5 slices. What is his wealty rate of eating the cale
5
36
.
B
TH
cakesliveek
er
9
Using the concept of Fraction, the weekly rate of Jake eating the cake is 11.2.
What is Fraction?Fraction represents parts of a whole or group of objects. A fraction consists of two parts. The numerator is the number at the beginning of the line. It specifies the number of equal parts taken from the whole or collection. The number below the line is the denominator. It shows the total number of equal parts into which the whole is divided or the total number of identical objects in a collection.
We know that,
The cake is cut into 12 equal slices.
After 3 days Jake eats 5 slices then,
For 1 day = [tex]\frac{5}{3}[/tex]
= 1.6
Then for 7 days,
1.6 × 7 = 11.2
Hence, Jake's weekly rate of eating the cake is 11.2.
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The complete question would be
'A cake is cut into 12 equal slices. After 3 days Jake has eaten 5 slices. What is his weekly rate of eating the cake?'
find 2 numbers if their ratio is 9:11 and their difference is 6 the numbers can be _, _ or _, _ HELP ASAP
Answer:
27: 33
you could also do -27 and -33 ig
Step-by-step explanation:
That's the only one possible.
Answer:
The only two numbers that your ratio is 9:11 and their differences is 6 are:
33 and 27
Step-by-step explanation:
9a = 11b Eq. 1
a - b = 6 Eq. 2
From Eq. 2:
a = 6 + b Eq. 3
Replacing Eq. 3 in Eq. 1:
9(6+b) = 11b
9*6 + 9*b = 11b
54 + 9b = 11b
54 = 11b - 9b
54 = 2b
54/2 = b
27 = b
From Eq. 3:
a = 6 + 27
a = 33
Check:
From Eq. 1:
9*33 = 11*27 = 297
Can anyone help? I’ve asked this same question 6 times!
Answer: 54080
Since the first number cannot be 0 or 1, there would be only 8 possible numbers for the first number. For the second number, we can now have all 10 numbers.
The number of different combinations of numbers would then be:
[tex]8\times10=80[/tex]Then, for the first letter, we have 26 possible letters, as well as the second letter. The number of different combinations of letters would then be:
[tex]26\times26=676[/tex]So, for a license plate that has 2 numbers and 2 letters, where the first number cannot be 0 or 1, there would be:
[tex]8\times10\times26\times26=54080[/tex]Hello! I need some guidance please. Having trouble with which graph is correct
Given:
[tex]y\ge3x+3[/tex]Required:
to show which graph is correct for the inequality.
Explanation:
Given graph is correct for the equation.
Required answer:
The given graph is correct.
Which problems can be solved using the equation 3 X 6 = |? Circle all the correct answers. A Cleo has 3 times as many bananas as Arianna has. Cleo has 6 bananas. A How many bananas does Arianna have? B Dylan has 6 oranges. Jane has 3 times as many oranges as Dylan. В How many oranges does Jane have? C Liam has 3 cherries. Brian has 6 times as many cherries as Liam. How many cherries does Brian have? D Tina has 3 more peaches than Charlie. Charlie has 6 peaches. How many peaches does Tina have? E Jaclyn has 6 times as many grapes as Kaitlyn. Kaitlyn has 3 grapes. How many grapes does Jaclyn have?
A. Cleo (C) has 3 times as many bananas as Arianna (A) has. Cleo has 6 bananas. How many bananas does Arianna have? Equation:
[tex]\begin{gathered} 3A=C \\ \\ C=6 \\ \\ 3A=6 \\ \\ ?=\frac{6}{3} \end{gathered}[/tex]It doesn't correspond to the given equation
_______________
B. Dylan (D) has 6 oranges. Jane (J) has 3 times as many oranges as Dylan. How many oranges does Jane have? Equation:
[tex]\begin{gathered} D=6 \\ J=3D \\ \\ J=3\times D \\ \\ \text{?}=3\times6 \end{gathered}[/tex]It correspond to the given equation
___________
C. Liam (L) has 3 cherries. Brian (B) has 6 times as many cherries as Liam. How many cherries does Brian have? Equation:
[tex]\begin{gathered} L=3 \\ B=6L \\ \\ \text{?}=6\times3 \end{gathered}[/tex]It correspond to the given equation
_______________
Tina (T) has 3 more peaches than Charlie (C). Charlie has 6 peaches. How many peaches does Tina have? Equation:
[tex]\begin{gathered} T=3+C \\ C=6 \\ \\ T=3+6 \\ \\ \text{?}=3+6 \end{gathered}[/tex]It doesn't correspond to the given equation
___________________
E. Jaclyn (J) has 6 times as many grapes as Kaitlyn (K). Kaitlyn has 3 grapes. How many grapes does Jaclyn have? Equation:
[tex]\begin{gathered} J=6K \\ K=3 \\ \\ J=6\times3 \\ \\ \text{?}=6\times3 \end{gathered}[/tex]It correspond to the given equation
____________
Answer: B, C and EWrite a recursive formula for the following sequence. You are welcome to submit an image of handwritten work. If you choose to type then use the following notation to indicate terms; a_n and a_(n-1). To earn full credit be sure to share all work/calculations and thinking.a_n = { \frac{3}{5}, \frac{1}{10}, \frac{1}{60}, \frac{1}{360} }
Answer:
[tex]a_n=a_{n-1}\left(\frac{1}{6}\right)[/tex][tex]a_n=\frac{3}{5}\left(\frac{1}{6}\right){}^{n-1}[/tex]
Explanation:
we can see for the fractions with 1 as the numerator that the denominator is multiplied by 6 and the numerator remains the same, that corresponds to multiply the previous fraction by 1/6 and when verifying with the first fraction we observe that applies for all the terms.
x = 3y for y how should we solve it
If x=3y is the equation then y = x/3.
What is Equation?Two or more expressions with an Equal sign is called as Equation.
The given expression x equal to three y.
Here x and y are two variables.
The value of x is three times of y.
The value of y is x over three. If we know the value of x we can substitute in place of x and we can calculate it.
Divide both sides by 3.
y=x/3.
Hence the value of y is x/3.
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Find the lateral area of the cylinder .The lateral area of the given cylinder is _ M2(Round to the nearest whole number as needed .)
The lateral area of a cylinder is:
[tex]LA=2\pi rh[/tex]r is the radius
h is the height
For the given cylinder:
As the diameter is 4m, the radius is half of the diameter:
[tex]r=\frac{4m}{2}=2m[/tex]h=12m
[tex]\begin{gathered} SA=2\pi(2m)(12m) \\ SA=48\pi m^2 \\ SA\approx151m^2 \end{gathered}[/tex]Then, the lateral area of the given cylinder is 151 square metersIn solving for the inverse function for y = sqrt(3x + 2) - 1 , which of the following represents the first step?
we know that
The first step to find out the inverse of the function is to exchange the variables (x for y and y for x)
therefore
the answer is the second optionWhat is the area of the shaded region if the radius of the circle is 6 in.
Then, the area of 1/4 of the circle is:
[tex]\begin{gathered} A=\text{ }\frac{\theta}{360}\text{ x }\pi r^2 \\ A=\text{ }\frac{90}{360}\text{ x }\pi r^2 \\ A\text{ = }\frac{1}{4}\pi\text{ 6}^2 \\ A=\text{ 9}\pi \\ \\ \end{gathered}[/tex]The area of the triangle is:
[tex]\begin{gathered} A=\text{ }\frac{b\text{ x h }}{2} \\ A\text{ = }\frac{6\text{ x 6}}{2} \\ A=\text{ 18in}^2 \end{gathered}[/tex]The area of the shaded region is the area of 1/4 of the circle minus the area of the triangle:
[tex]\begin{gathered} A\text{ = 9}\pi\text{ - 18 in}^2 \\ A=\text{ 28.27in}^2\text{ - 18in}^2 \\ A=\text{ 10.27in}^2 \end{gathered}[/tex]0> -2x^2+4x+4Solve each inequality by graphing. Sketch it.
To solve the inequality we need to find the x-values that are the roots of the quadratic equation, let's use the quadratic formula:
[tex]\begin{gathered} \text{For an equation in the form:} \\ ax^2+bx+c=0 \\ The\text{ quadratic formula is:} \\ x=\frac{-b\pm\sqrt[]{b^2-4ac}}{2a} \\ \text{Then a=-2, b=4 and c=4} \\ x=\frac{-4\pm\sqrt[]{4^2-4(-2)(4)}}{2(-2)} \\ x=\frac{-4\pm\sqrt[]{16+32}}{-4} \\ x=\frac{-4\pm\sqrt[]{48}}{-4} \\ x=\frac{-4\pm6.93}{-4} \\ \text{Then} \\ x1=\frac{-4+6.93}{-4}=\frac{2.93}{-4}=-0.732 \\ x2=\frac{-4-6.93}{-4}=\frac{-10.93}{-4}=2.732 \end{gathered}[/tex]Now, let's try values less or greater than these roots:
If x=-1:
[tex]\begin{gathered} 0>-2(-1)^2+4(-1)+4 \\ 0>-2\cdot1-4+4 \\ 0>-2\text{ This is right, then number less than -0.732 are solutions of the inequality} \end{gathered}[/tex]Now let's try x=3:
[tex]\begin{gathered} 0>-2(3)^2+4(3)+4 \\ 0>-2\cdot9+12+4 \\ 0>-18+16 \\ 0>-2\text{ This is correct two, then the values greater that 2.732 are solutions to the inequality too} \end{gathered}[/tex]Then, the graph of the inequality is:
The red-shaded area are the solution to the inequality, then in interval notation we have:
[tex](-\infty,-0.732)\cup(2.732,\infty)[/tex]In builder notation it would be:
[tex]x|x<-0.732orx>2.732[/tex]Select from these metric conversions1 kg = 1000 g1 g = 1000mgand use dimensional analysis to convert 4.59 kg to g.4.59 kg X 1
Since
[tex]1kg=1000g,[/tex]then:
[tex]1=\frac{1000g}{1kg}.[/tex]Then:
[tex]4.59kg=\frac{4.59kg}{1}\times\frac{1000g}{1kg}=4590g.[/tex]Answer:
[tex]\frac{4.59kg}{1}\times\frac{1000g}{1kg}=4590g.[/tex]Write the first six terms of each arithmetic sequence,Please see the photo
Answer: - 9, - 3, 3, 9, 15, 21
Explanation:
The given formula is
an = a(n - 1) + 6
a1 = - 9
where
n, n - 1 and 1 are subscripts
This is a recursive formula. Each term is defined with respect to the term before it.
From the information given,
first term = a1 = - 9
Second term = a2 = a(2 - 1) + 6 = a1 + 6 = - 9 + 6
a2 = - 3
Third term = a3 = a(3 - 1) + 6 = a2 + 6 = - 3 + 6
a3 = 3
Fourth term = a4 = a(4 - 1) + 6 = a3 + 6 = 3 + 6
a4 = 9
Fifth term = a5 = a(5 - 1) + 6 = a4 + 6 = 9 + 6
a5 = 15
Sixth term = a6 = a(6 - 1) + 6 = a5 + 6 = 15 + 6
a6 = 21
Thus, the first six terms are
- 9, - 3, 3, 9, 15, 21
A binomial probability experiment is conducted with the given parameters. Compute the probability of x successes in the n independent trials of the experiment n=10, p=0.2, x=2
The binomial probability of x successes is 0.302.
How to calculate the probability of x successes?Since we are dealing with a binomial probability experiment. We are going to use the binomial distribution formula for determining the probability of x successes:
P(x = r) = nCr . p^r . q^n-r
Given: n=10, p=0.2, x=2
The failures can be calculated using q = 1 - p = 1 - 0.2 = 0.8
P(x = 2) = 10C2 x 0.2² x 0.8¹⁰⁻²
= 10!/(10-2)! 2! x 0.2² x 0.8⁸
= 10!/(8!2!) x 0.2² x 0.8^8
= 10x9x8!/(8!2!) x 0.2² x 0.8⁸
= 45 x 0.2² x 0.8⁸
= 0.302
Therefore, the probability of x successes in 10 trials is 0.302
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Find the area of the sector interms of pi.2460°Area = [?]
Answer:
Area= 24π.
Explanation:
The area of a sector is calculated using the formula below:
[tex]A=\frac{\theta}{360\degree}\times\pi r^2[/tex]From the diagram:
• The central angle, θ = 60°
Diameter of the circle = 24
• Therefore, Radius, r = 24/2 = 12
Substitute these values into the formula:
[tex]\begin{gathered} A=\frac{60\degree}{360\degree}\times\pi\times12^2 \\ =24\pi\text{ square units} \end{gathered}[/tex]The area of the sector in terms of pi is 24π square units.
Preston drove to his new college and then back home.Round trip he traveled 642 miles. Preston drives aHonda Civic and gets 38 miles for every gallon of gas. IfPreston needs to make 15 round trips a year how muchwill it cost him in gas assuming the price of gas stays at$2.48 a gallon for all his trips?$Round all answers to the nearest hundredthsDo not put a label, just the numeric value
1) Gathering the data
Preston
642 miles
38 miles/gallon
15 round trips
1 gallon = $2.48
2) Considering that each round trip consists of 642 miles
So Preston in 15 roundtrips is going to make
15 x 642 miles =9,630 miles
His car gets 38 miles per gallon. So we can write a proportion for that:
38 miles ---------1 gallon
9,630 miles ----- x
Cross multiplying it:
38x = 9,630 Divide by 38
x =9630/38
x=253.42 gallons
Finally, let's set another proportion to find out the cost of it
1 gallon -------------- $2.48
253.42 -------------- y
y= 253.42 x 2.48
y=628.4816
3) Rounding off to the nearest hundredth
$628. 48 That's how much Preston will spend.
4. Which of the following represent the distance
formula? Select all that apply.
A d = √(x₁-x₂)² + (y₁ − y₂)²
B d = √(x₂− ×₂)² + (⁄₂ − y,}²
C d = √(x₂+x₂)² + (y₂ + y,)²
D d=√√₂-X₁1² + VY₂ − Y₁1²
A appears to be the only correct answer
a^2+b^2=c^2
you are solving for c when finding distance, so (a^2 + b^2) must be square rooted, as a whole, not separately
and a = (x1-x2)
and b = (y1-y2)
you can flip the 1 and 2 but you have to flip for both x and y
like x1-x2 means you have to do y1-y2
like x2-x1 means you have to do y2-y1
so both above are correct as long as the order of 1 and 2 stays the same for both x and y
Write the Distance Formula
Replace c with d to write the distance formula. Use the Distance Formula to Find the Distance Between Two Points
Find the distance, d, between G and H using the distance formula.
The distance between any two points (x1,y₁) and (x2,y2) on a
coordinate plane can be found by using the distance formula. Let (x,y)= (-2,1) and (x2,y2) =(4,-3). Substitute these values into the
distance formula and evaluate.
The distance between the two points is [tex]2\sqrt{13} units[/tex]
What is distance formula?
Distance formula is the measurement of distance between 2 points. It calculates the straight line distance between the given points. The formula can be given as [tex]distance=\sqrt{(c-a)^{2} +(d-b)^{2} }[/tex] Where A(a, b) B(c, d) Are the coordinates.
We are given the coordinates as (-2, 1) and (4, -3)
We substitute the values in the distance formula we get
[tex]distance=\sqrt{(c-a)^{2} +(d-b)^{2} } \\distance=\sqrt{(4+2)^{2} +(-3-1)^{2} }\\ distance=\sqrt{36+16 } \\distance=\sqrt{52 } \\distance =2\sqrt{13}[/tex]
Hence the distance between two points is [tex]2\sqrt{13} units[/tex]
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help meeeeeeeeee pleaseee !!!!!
The composition will be:
(g o h)(x) = 5*√x
By evaluating in x = 0, we get:
(g o h)(0) = 0
How to evaluate the composition?Here we have the two functions:
g(x) = 5x
h(x) = √x
And we want to get the composition:
(g o h)(x) = g( h(x))
So we need to evaluate g(x) in h(x), we will get:
g( h(x)) = 5*h(x) = 5*√x
And now we want to evaluate this in x = 0, we will et:
(g o h)(0) = 5*√0 = 0
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Find the exact value of the expression. No decimal answers. Show all work.Hint: Use an identity to expand the expression.
Given the expression:
[tex]\cos (\frac{\pi}{4}+\frac{\pi}{6})[/tex]You can expand it by using the following Identity:
[tex]\cos \mleft(A+B\mright)\equiv cos(A)cos(B)-sin(A)sin(B)[/tex]You can identify that, in this case:
[tex]\begin{gathered} A=\frac{\pi}{4} \\ \\ B=\frac{\pi}{6} \end{gathered}[/tex]Then, you can expand it as follows:
[tex]\cos (\frac{\pi}{4}+\frac{\pi}{6})=cos(\frac{\pi}{4})cos(\frac{\pi}{6})-sin(\frac{\pi}{4})sin(\frac{\pi}{6})[/tex]By definition:
[tex]\cos (\frac{\pi}{4})=\frac{\sqrt[]{2}}{2}[/tex][tex]\cos (\frac{\pi}{6})=\frac{\sqrt[]{3}}{2}[/tex][tex]\sin (\frac{\pi}{4})=\frac{\sqrt[]{2}}{2}[/tex][tex]\sin (\frac{\pi}{6})=\frac{1}{2}[/tex]Then, you can substitute values:
[tex]=(\frac{\sqrt[]{2}}{2})(\frac{\sqrt[]{3}}{2})-(\frac{\sqrt[]{2}}{2})(\frac{1}{2})[/tex]Simplifying, you get:
[tex]\begin{gathered} =(\frac{\sqrt[]{2}}{2})(\frac{\sqrt[]{3}}{2})-(\frac{\sqrt[]{2}}{2})(\frac{1}{2}) \\ \\ =\frac{\sqrt[]{6}}{4}-\frac{\sqrt[]{2}}{4} \end{gathered}[/tex][tex]=\frac{\sqrt[]{6}-\sqrt[]{2}}{4}[/tex]Hence, the answer is:
[tex]\frac{\sqrt[]{6}-\sqrt[]{2}}{4}[/tex]