Find the exact value of tan π/12.
Answers
[tex]\textit{Half-Angle Identities}\qquad tan\left(\cfrac{\theta}{2}\right)= \begin{cases} \pm \sqrt{\cfrac{1-cos(\theta)}{1+cos(\theta)}} \\\\ \cfrac{sin(\theta)}{1+cos(\theta)}~~\leftarrow \textit{let's use this one} \\\\ \cfrac{1-cos(\theta)}{sin(\theta)} \end{cases} \\\\\\ 2\cdot \cfrac{\pi }{12}\implies \cfrac{\pi }{6}~\hspace{10em} \cfrac{~~ \frac{\pi }{6}~~}{2}\implies \cfrac{\pi }{6}\cdot \cfrac{1}{2}\implies \cfrac{\pi }{12}[/tex]
so let's use the double of the angle in the half-angle identities, check the picture below for the cosine, sine pairs
[tex]tan\left( \cfrac{~~ \frac{\pi }{6}~~}{2} \right)=\cfrac{sin\left( \frac{\pi }{6} \right)}{1+cos\left( \frac{\pi }{6} \right)}\implies tan\left( \cfrac{~~ \frac{\pi }{6}~~}{2} \right)=\cfrac{~~ \frac{1}{2}~~}{1+\frac{\sqrt{3}}{2}}[/tex]
[tex]tan\left( \cfrac{~~ \frac{\pi }{6}~~}{2} \right)=\cfrac{~~ \frac{1}{2}~~}{~~\frac{2+\sqrt{3}}{2}~~}\implies tan\left( \cfrac{~~ \frac{\pi }{6}~~}{2} \right)=\cfrac{1}{2}\cdot \cfrac{2}{2+\sqrt{3}} \\\\[-0.35em] ~\dotfill\\\\ ~\hfill tan\left( \cfrac{~~ \frac{\pi }{6}~~}{2} \right)=\cfrac{1}{2+\sqrt{3}}~\hfill[/tex]
Related Questions
The average of eight different numbers is 5. If 1 is added to the largest number, what is the resulting average of the eight numbers?
Answers
Answer:
Step-by-step explanation:
X/8 = 5
X = 40
Adding 1 to the largest number (or any other of the numbers, for that matter) makes the sum 41 and the average 41/8 or 5 1/8.
The average of a given number is the sum of the given numbers divided by the number of numbers in the given set.
The resulting average of the eight numbers after adding 1 to the largest number is 5.125.
What is a mean?It is the average value of the set given.
It is calculated as:
Mean = Sum of all the values of the set given / Number of values in the set
Example:
2, 3, 4, 5
The average number of 1, 2, 4, and 5 is 3.
i.e 12 ÷ 4 = 3
We have,
The average of eight different numbers is 5.
Let the sum of 8 different numbers be M.
Average = M / 8
5 = M / 8
M = 8 x 5
M = 40
Now,
If 1 is added to the largest number.
This means,
40 + 1 = 41
The resulting average of the eight numbers:
= 41/8
= 5.125
Thus,
The resulting average of the eight numbers after adding 1 to the largest number is 5.125.
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Select the correct answer.
Simplify the following expression.
Answers
Answer:
B. 1/3
Step-by-step explanation:
This question is designed to show if you know the exponent rules. Exponent Rules are short cuts. If you are multiplying two terms and they both have the same base (the big bottom number), then you could ADD the exponents (the small font raised numbers) Here the base is 3 in both terms. We keep that base. Then add 2/5 and -7/5 to get -5/5. This is just -1.
Another Exponent Rule is how to use negative exponents. So far we have 3^-1 (that means three to the negative one power)
In order to make that negative exponent positive, you just move it across the fraction bar. So we would lower it below the fraction bar to make it a positive one exponent. So you end up with 1/3
I NEED HELP ON QUESTION B AND C ASAP.
Answers
Answer:
See belowStep-by-step explanation:
Part BAssumed the dimensions of the top and bottom parts are identical.
Since the cylindrical part has total height of 1.8 cm and the hemisphere volume is transferred to bottom part and the cone part is still full, the value of h is the difference of the total height of cylinder and full part of the top section of cylinder:
h = 1.8 cm - 0.3 cm = 1.5 cmPart CFind the volume of sand in the bottom part. It consists of a hemisphere and a cylinder of 1.5 cm height.
V(cylinder) = πr²h = 3.14*(2.6/2)²*1.5 ≈ 7.96 cm³V(hemisphere) = 2/3πr³ = 2/3*3.14*(2.6/2)³ ≈ 4.6 cm³Total sand in the bottom part:
7.96 + 4.6 = 12.56 cm³Time taken:
12.56 / 0.05 = 251.2 seconds = 4 min 11.2 secondsSami has a bag full of 4900 marbles of three different colors which include blue, green and black in the ratio 16:13:20. Find the number of each colored marbles in her bag.
Answers
Answer:
See the calculation below
Step-by-step explanation:
[tex]16 + 13 + 20 = 49[/tex]
[tex]blue = \frac{16}{49} \times 4900 \\ = 1600[/tex]
[tex]green = \frac{13}{49} \times 4900 \\ = 1300[/tex]
[tex]black = \frac{20}{49} \times 4900 \\ = 2000[/tex]
A restaurant is serving a special dinner combo meal that includes a drink, a main dish, and a dessert. Cus
4 drinks, 5 main dishes, and 3 desserts.
How many different combo meals are possible?
Select from the drop-down menu to correctly complete the statement.
Customers can create Choose... w different dinner combo meals.
1
2
3
4
Choose...
12
45
60
75
5
6
7
Answers
Answer:
Your Answer Is 60
Step-by-step explanation:
I Took The Test Just Now
The number of different combo meals are possible in a restaurant which is serving a special dinner combo meal that includes a drink, a main dish, and a dessert is 60.
What is arrangement?Arrangement of the things or object is mean to make the group of them in a systematic order, in all the possible ways.
The number of possible ways to arrange is the n!. Here, n is the number of objects.
A restaurant is serving a special dinner combo meal that includes a drink, a main dish, and a dessert.
Dinner includes 4 drinks,5 main types of dishes.3 types of desserts.The number of different combo meals are possible can be find out by simply multiplying all the meals. Thus,
[tex]n=4\times5\times3\\n=60[/tex]
Thus, the number of different combo meals are possible in a restaurant which is serving a special dinner combo meal that includes a drink, a main dish, and a dessert is 60.
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Approximately what portion of the circle is shaded blue?
Answers
Answer: C. (1/6)
Step-by-step explanation: If you split the circle in half horizontally, the top part would be 1/3 and the bottom part would be 3/3. 1/3 + 3/3 = 4/6 and 4/6 simplified is 1/6.
The jazz band has 14 members. To raise money for a tour, each member sold r raffle tickets. Write an expression that shows the number of raffle tickets sold.
Answers
Answer:
14r
Step-by-step explanation:
since there are 14 members and each sold r tickets r x 14 which is 14r
Which value of a makes the equation true?
Answers
Answer:
a = -2
Step-by-step explanation:
3(a + 1.5) = -1.5
3a + 4.5 (-4.5) = -1.5 (-4.5)
3a = -6
a = -2
Evaluate the expression when c=4. c^4 + 6
Answers
Answer:
262
Step-by-step explanation:
4x4x4x4+6
256+6
262
HELPPPP ASAPPPPPPPPPPPP
Answers
Answer:
C
Step-by-step explanation:
?/5 =8/10 type the missing number that makes this fraction equal
Answers
Answer:
[tex]?=4[/tex]
Step-by-step explanation:
[tex]\frac{?}{5}=\frac{8}{10}[/tex]
[tex]\frac{2?}{10}=\frac{8}{10}[/tex]
[tex]2?=8[/tex]
[tex]?=4[/tex]
Examine the graphic organizer above. Which of the choices below BEST fills in the blank on the graphic organizer?
Answers
A statistics teacher was interested in the relationship between the number of days students waited to start a project
and the score that project received (out of 100 points). Information was collected on several students and was used
to obtain the regression equation ý= -3.64x + 96.5, where x represents the number of procrastination days and ý is
the predicted grade. What is the predicted grade of a student who procrastinated for 1 week?
O24.59
71.02
O 92.86
O. 96.5
Answers
Answer:
71.02
Step-by-step explanation:
x is the number of days
1 week = 7 days
-3.64 (7) + 96.5 = 71.02
Faith earned $565 last week. She earned the same amount each
day. If she worked 5 days, how much did she earn each day? Draw
models to help you find the quotient.
Answers
Answer:
113 each day
Step-by-step explanation: i cant draw it out but just put it into a bubble or something sorry
Divide $565 by 5 and you get $113
So you would have 5 groups of 113
someone please answer this
Answers
The slope is the number in front of the x. And the y-intercept is the number after the x.
A negative y-intercept would be subtracted from the equation.
Answer is B. Y = 7.5x - 2
Answer:
B
Step-by-step explanation:
y = 7.5x - 2
The slope is 7.5
y- intercept is -2
A line with a slope of 2 passes through the point (3,9).
Write an equation for this line in point-slope form.
Answers
Answer:
y-9=2(x-3)
Step-by-step explanation:
y-y1=m(x-x1)
y-9=2(x-3)
If f(x) and g(x) are inverse functions, find f(g(x)).
A)
-1
B)
0
C)
1
D)
х
Answers
Answer:
x.
Step-by-step explanation:
f(g(x)) = x.
Devaughn is 15 years younger than sydney. the sun of their ages is 63. what is sydney’s age?
Answers
Answer:
39
Step-by-step explanation:
d = s - 15
d + s = 63
Replace d with s - 15
s - 15 + s = 63
2s -15 = 63
2s = 78
s = 39
20% percent tip on 8 dollar
Answers
Answer:
9.60 dollars tip is $1.60
Step-by-step explanation:
Helppp it’s due today
Answers
Answer:
128
Step-by-step explanation:
Can somebody, anybody please helppp
Simplifiy the expression.
x^2 + 3x - 28/ x^2 - 7x + 12
show your work! (this part is REALLY important) thanks(:
Answers
Answer:
Step-by-step explanation:
3x3 then 4 x v ?
-16-|-2-4|+7 please solve this for me and explain. i would really appreciate it.
Answers
Answer:
-15
Step-by-step explanation:
-16 -|-6| +7
-16 - 6 + 7
= - 15
Absolute value is just taking the positive value of a number.
Step-by-step explanation:
-16-(-2-4)+7=
-16-(-6)+7=
-16+6+7=
-16+13=
-3
I need help ASAPPP this is very important
Answers
Answer:
D
Step-by-step explanation:
Answer:
D
Step-by-step explanation:
Helpppp for a math exammm
Answers
Answer:
2. angle A is congruent to angle X
Step-by-step explanation:
Which list of angle measures could be the angle measures of a triangle?
Answers
1 minute is equal to 60 seconds. How many seconds are in 3 minutes, 37 seconds?
Answers
Answer:
217 Seconds.
Step-by-step explanation:
The answer is 217 seconds because 60x3+37=217.
To make four cakes, Dena used 6 cups of sugar. Using the same recipe, how much sugar does she need to make one cake?
Answers
6 cups/ 4 cakes
= 1.5 cups per cake
Answer:
Step-by-step explanation:
So 4 cakes = 6 sugars, right so try taking it slow and think to yourself, 3=? then take it even slower until you go to 1=0.67 repeated
Select the interval where g(2) < 0.
(graphed on khan academy)
Answers
Answer:
the number 0 Originally called finding the vertical asymptote.
Which quadrilateral is a trapezoid?
Answers
Answer:
Hey mate!Step-by-step explanation:
A trapezoid is a quadrilateral with exactly one pair of parallel sides. (There may be some confusion about this word depending on which country you're in. In India and Britain, they say trapezium ; in America, trapezium usually means a quadrilateral with no parallel sides.)
pls mark me as brainliest!!4. There are 20
students in a rath
class. The ratio of
boys to girls is 3:2.
a) How many boys
are in the class?
b) How many girls
are in the class?