How did early mathematicians define trigonometric functions?
Answers
Answer:
There are six trigonometric function, the first three are mainly used and other three are not use often, they are:
Sine (sin)Cosine (cos)Tangent (tan)Cosecant (cosec or csc)Secant (sec)Cotangent (cot)They are mainly the ratio of two side of triangle.i.e.
[tex]Sin x = \dfrac{Perpendicular}{Hypotenuse}[/tex]
[tex]Cos x = \dfrac{Base}{Hypotenuse}[/tex]
[tex]Tan x = \dfrac{Perpendicular}{Base}[/tex]
[tex]Cosec x = \dfrac{Hypotenuse}{Perpendicular}[/tex]
[tex]Sec x = \dfrac{Hypotenuse}{Base}[/tex]
[tex]Cot x = \dfrac{Base}{Perpendicular}[/tex]
On the basis of these ratios there are different relations defined in Trigonometry Chapter.
Answer:
length of chords
Explanation:
In the study of the earth and astronomy, mathematicians developed trigonometric functions as lengths of chords in a circle, not as ratios. The mathematician Ptolemy calculated chord lengths by inscribing regular polygons in a circle. Remarkably, he was able to calculate these lengths to an accuracy of about five decimal places. Ptolemy's table of chords is equivalent to our table of sine values. In this lesson, you will begin to see the relationship between the trigonometric functions and the circle.
Related Questions
How do you do question number 5? If you do it can you please explain how to do it? Thank you!!
Answers
H(2) = -5(2) + 7 = -10 + 7 = -3.
F(-3) + H(2) = -15 + -3 = -18.
The solution is -18.
Answer:
F(-3) = 4(-3) - 3 = -12 - 3 = -15.
H(2) = -5(2) + 7 = -10 + 7 = -3.
F(-3) + H(2) = -15 + -3 = -18.
The solution is -18.F(-3) = 4(-3) - 3 = -12 - 3 = -15.
H(2) = -5(2) + 7 = -10 + 7 = -3.
F(-3) + H(2) = -15 + -3 = -18.
Step-by-step explanation:
Mr. Valenzuela’s new bottle of shampoo contains 60 ounces of shampoo. He uses ounce of shampoo each time he washes his hair. Which equation can be used to find x, the number of washes after which 24 ounces will remain in the bottle?
Answers
Answer:the number of washes after which 24 ounces will remain in the bottle is:
36 washes.
Step-by-step explanation:
The correct answer is 18.
To find the equation that can be used to determine the number of washes (x) after which 24 ounces of shampoo will remain in the bottle, we need to set up an equation based on the given information.
Given information:
- Initial amount of shampoo in the bottle (I) = 60 ounces
- Amount used per wash (U) = 2 ounces
- Remaining amount after x washes (R) = 24 ounces
[tex]I - (U \times x) = R[/tex]
[tex]60 - (2 \times x) = 24[/tex]
[tex]60 - 24 = 2x[/tex]
[tex]36 = 2x[/tex]
[tex]x = \frac{36}{2}[/tex]
[tex]x = 18[/tex]
Therefore, the equation that can be used to find x is:
[tex]60 - (2 \times x) = 24[/tex]
And the number of washes after which 24 ounces of shampoo will remain in the bottle is 18.
what is the answer to this word problem??
Answers
So,
1st service
10m + 20
1. 10 + 20 =30
2. 10x2=20+20=40
3.10x3=30+20=50
4.10x4=40+20=60
5.10x5=50+20=70
2nd service
11m + 15
1. 11+15=26
2. 11x2=22+15=37
3 11x3=33+15=48
4.11x4=44+15=59
5.11x5=55+15=70
So, the 5th month the services cost the same
Hope this helps! ;)
List 3 values that would make this inequality true 4n <_ 12
Answers
[tex]4n \le 12[/tex]
[tex]\frac{4n}{4} \le \frac{12}{4}[/tex]
[tex]n \le 3[/tex]
So we can pick n to be 3 or it can be less than 3
Three values we can pick are: n = 0, n = 1, n = 2
Or we could pick: n = 1, n = 2, n = 3
How much money does the average professional football fan spend on food at a single football game? that question was posed to 10 randomly selected football fans. the sample results provided a sample mean and standard deviation of $14.00 and $2.50, respectively. use this information to construct a 90% confidence interval for the mean?
Answers
When the mean ([tex]\bar{x}[/tex]), the number of samples (n) and standard deviation (σ) are known, you can apply the following equation to determine confidence intervals. The value of z at 90% confidence can be found in tables (It is 1,645).
[tex](\bar{x}-z* \frac{\sigma}{ \sqrt{n} },\bar{x}+z* \frac{\sigma}{ \sqrt{n} }) \\ \\ (14-1,645* \frac{2,50}{ \sqrt{10} },14+1,645* \frac{2,50}{ \sqrt{10} } \\ \\ (12,70\$;15,30\$) [/tex]
Have a nice day!
Suppose that after he makes his first shot, he shoots exactly 5 more shots. what is the probability that, out of these 5 attempts, he makes an even number of shots?
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.. 0.31250 +0.15625 = 0.46875
_____
p(2 of 5) = (5C2)/2^5 = 10/32
p(4 of 5) = (5C4)/2^5 = 5/32
For the above, we have assumed the probability of success is 1/2.
The tangent line has the form y=mx+b . use tl above to find b +juliabox
Answers
You can rake a 20 square meter lawn in 10 minutes. If your mother asks you to rake your own lawn, which is 30 square meters, how long will it take you to do this?
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10 square meter lawn...............................5 min
30 square meter lawn in.........................5*3=15 min
Elevator 1 moved up 10 feet from the ground level. Its position is labeled as +10. Elevator 2 moved down 5 feet from the ground level. Its position is labeled as _____. (Use the hyphen for negative such as -1)
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Paula is going to choose the size, color, phrase, and picture for a birthday card for her friend. There are 2 sizes, 4 colors, 7 phrases, and 4 pictures for her to choose from. (The printing company charges a fee to add extra design elements, so she will choose only one of each.) How many different card designs are possible?
Answers
Answer:
The different card designs possible are:
224
Step-by-step explanation:
It is given that:
There are 2 sizes, 4 colors, 7 phrases, and 4 pictures for her to choose from. (The printing company charges a fee to add extra design elements, so she will choose only one of each.)
As she has to choose one from each given item this means she will use a combination to choose the item.
We know that the combination for r choices out of total n choices is given as:
[tex]n_C_r=\dfrac{n!}{r!\times (n-r)!}[/tex]
Hence, the total combinations made are given as:
[tex]2_C_1\times 4_C_1\times 7_C_1\times 4_C_1\\\\\\=\dfrac{2!}{1!\times (2-1)!}\times \dfrac{4!}{1!\times (4-1)!}\times \dfrac{7!}{1!\times (7-1)!}\times \dfrac{4!}{1!\times (4-1)!}\\\\=2\times 4\times 7\times 4\\\\=224[/tex]
Hence, the different card designs that are possible are:
224
The different card designs that are possible will be 224. The formula for the combination is used to solve the problem.
What are permutation and combination?A permutation is an act of arranging the objects or elements in order.
Combinations are the way of selecting objects or elements from a group of objects or collections, in such a way the order of the objects does not matter.
The given data in the problem is;
Paula is going to choose 2 sizes, 4 colors, 7 phrases, and 4 pictures . If the one item has to be chosen;
The total combination is;
[tex]= \rm 2C_1 \times 4C_1 \times 7C_1 \times 4 C_1 \\\\ = \frac{2!}{1!\times (2-1)!} \times \frac{4!}{1! \times (4-1)!} \times \frac{7!}{1!\times (7-1)!} \times \frac{4!}{1!\times (4-1)!} \\\\ =2 \times 4 \times 7 \times 4 \\\\ =224[/tex]
Hence different card designs are possible will be 224.
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What is the expression for "the sum of 8 times a number and two"? 8 × 2N 8N + 2 8(N + 2)
Answers
Answer:
Option B. 8N + 2
Step-by-step explanation:
Expression given in the question is " the sum of 8 times a number and two" will be = (8 times N + 2)
= 8N + 2
Therefore, option B. is the answer.
Edgar has a bird bath that holds 2 gallons of water. He only want to fill it 3/4 full. How much water in gallons should Edgar put in the bird bath?
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2 × 3/4 = 6/4 = 1 1/2
Edgar should put 1 1/2 gallons of water in the bath.
Hope this helps!
How to find turning point of (c-2)^3+8
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The function has a point of inflection at (2, 8), where the curve is horizontal, but does not reverse direction.
to find the turning point, you must have a function, so let's set this expression equal to y.
y=(c-2)^3+8
y'=3(c-2)^2
Now set y' equal to 0...
3(c-2)^2=0
3(c-2)(c-2)=0
c=2
Now that we know c=2, plug this in to the original equation to get the y-value for the turning point.
The turning point is (2,y) we will find y now...
y=(2-2)^3+8
y=0+8
y=8
The turning point is at (2,8)
The explanation for this is that the x values of the zeros of the derivative function are the same as the x values as the turning points on the original function.
I really hope this helps!
Best wishes :)
If a family has three pets, what is the theoretical probability that they have three dogs or three cats?
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If f(x) = 0.5x and g(x) = 0.25x. Which statement is true?
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PLEASE SEE ATTACHED IMAGE.
First, we rewrite both functions correctly:
f (x) = 0.5 ^ x
g (x) = 0.25 ^ x
We observe that they are exponential functions, where, rewriting:
f (x) = (1/2) ^ x
g (x) = (1/4) ^ x
We observe that both graphs present an exponential decay where, f (x)> g(x), since, the rate of change of f (x) (1/2) is greater than the rate of change of g (x) ( 1/4).
Answer:
as -> infinity, f (x)> g (x) *
The lifetimes of batteries produced by the juicy battery company are normally distributed with a mean of 100 hours and a standard deviation of 10 hours. find the number of hours where at least 90% of the batteries will continue to operate at.
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P(0.9)=1.28
sup[pose the number of hours whereby at least 90% of the batteries will continue to operate is x.
the z-score is given by:
z=(x-μ)/σ
where:
μ is the mean
σ is the standard deviation
substituting the values we shall have:
1.28=(x-100)/10
solving for x we get:
x=112.8 hours
the function f(x)=a^x-4 will never cross the x-axis. true or false
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Answer: False
Solution:
The exponential functions with the form y=a^x never cross the x-axis (x-axis is an asymptote), but in this case we have the exponential function translated 4 units downward, then this function crosses the x-axis.
As it is not given a is positive real number or negative real number.
By supposing a is any real number, we will draw the graph of the curve
[tex]a^x-4[/tex] by taking different values of x.
As i have taken different values of a ,
I found that for a< 0, the graph of the curve [tex]y=a^x-4[/tex] does not exist.
For any other value of a, the curve cuts the x axis at one point.
For values of 1<a<0, the curve will cut negative side of X axis.
For value of a, a≥1, the curve will cut positive side of X axis.
For , a=0, the curve will not cut the x axis.
So, The function , f(x)=[tex]a^x -4[/tex] will cuts the x axis at one point for a >0 and for negative values of a, and for a< 0 ,it will not cut the x axis.
P= s-c/s solve for c
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Therefore, the solution for c in terms of P and s is c = s(1 - P).
To solve for c in the equation P = (s - c) / s, you want to isolate c on one side. Here's the step-by-step:
Multiply both sides by s to eliminate the denominator: P × s = s - c.
Distribute P within s: Ps = s - c.
Now, get all terms containing c on one side: c = s - Ps.
So, the solution for c in terms of P and s is c = s - Ps, or c = s(1 - P).
The relationship between the number of hours a plumber works and the total work fee she charges is proportional. Her fee for 5 hours of work is $350. Which of the following could be combinations of values for the plumber's work hours and total work fee she charges?
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Answer: I saw someone else answer this and it doesn't match up with the choices so I decided to do this my self and for anyone else who might need this in the future. For those on Khan academy the answers are B, C, AND D.
Step-by-step explanation: I found the unit rate of 350 and 5, because we can use a calculator (as stated on top) I just punched in 350 divided by 5 and got 70. Now i know this may not sound that "professional", but nothing needs to always be professional as long as its efficient and most importantly gets you the right answer. I then took the time to multiply each number, (the ones on the left : 3, 3.5, 6, 7.25, and 8) by 70. If the answer you got on the calculator match up to the number on the right, (240, 420, 507.50, and 490) its correct, if it doesn't match up, you can tell that its most likely not the answer. You should end up with three answers that all end up having the unit rate of 70. Hope this helps anyone who needed it.
ANSWERS ARE B, C, AND D.
Following could be combinations of values for the plumber's work hours and total work fee she charges : 8 hours and $490, 7.25 hours and $500, 270 for 3 hours work. So, the correct choice of options for the image are B, C, and D.
Since the relationship between the number of hours a plumber works and the total work fee he charges is proportional, we can use the following equation to calculate the total work fee:
Total work fee = Number of hours worked * Hourly rate
Therefore, the following combinations of values for the plumber's work hours and total work fee are proportional:
8 hours and $490 (the hourly rate is $490/8 hours = $61.25)
7.25 hours and $500 (the hourly rate is $500/7.25 hours = $68.75)
6 hours and $420 (the hourly rate is $420/6 hours = $70)
The other two options, A and E, are not proportional.
For example, if the plumber works for 3 hours and charges $245, the hourly rate would be $245/3 hours = $81.67.
However, if the plumber works for 3.5 hours and charges $270, the hourly rate would be $270/3.5 hours = $77.14.
This shows that the hourly rate is not constant, so the relationship between the number of hours worked and the total work fee is not proportional.
The correct choice of options for the image are B, C, and D.
For similar question on hours
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If the dimensions of a square pyramid are measured in yards, then the volume of the pyramid is measured in
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The volume is measured in cubic yards.
Answer: The volume of the pyramid is measured in cubic yards.
Step-by-step explanation: Given that the dimensions of a square pyramid are measured in yards.
We are to find the unit in which the volume of the pyramid is measured.
We know that
the volume of a square pyramid with length of base edge 'a' units and height 'h' units is given by
[tex]V=\dfrac{1}{3}a^2h~\textup{cubic units}.[/tex]
Here, length of base edge 'a' and the height 'h' of the given square pyramid is measured in yards.
Therefore, the volume of the pyramid will be measured in
[tex]\textup{yards}\times \textup{yards}\times \textup{yards}\\\\=\textup{cubic yards}.[/tex]
Thus, the volume of the square pyramid is measured in cubic yards.
There were three trains made of the cars with exactly the same number of passenger's seats. The first train had 418 seats, the second 456, and the third 494. How many cars were in each train, if it is known that each train had not more than 50 cars?
1 train=?
2 train=?
3 train=?
Answers
.. 456 -418 = 38
.. 494 -456 = 38
So, if 38 divides these numbers evenly, that is the number of seats in each car. (It does).
train 1 had 418/38 = 11 cars
train 2 had 456/38 = 12 cars
train 3 had 494/38 = 13 cars
_____
The number of seats in each car could be a sub-multiple of 38, say 19. In that case, the number of cars in each train would be twice the number shown above. It is slightly unusual for a train car to have an odd number of seats, but is not unheard of.
Hannah keeps her marbles in bags with for marbles in each bag she writes 3/4 to show the number of red marbles in each bag find another fraction to name the number of red marbles in 2 bags
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She keeps 4 marbles in every bag and mark it with 3/4 to represent red marbles in the bag. It means there are 3 Red marbles out of total 4 marbles in the bag.
It says to write another fraction to represent number of red marbles in two bags.
Since one-bag contains 3 Red marbles out of total 4 marbles in the bag.
Then two-bags would contain 2x3 Red marbles out of total 2x4 marbles in the bag i.e. 6 Red marbles out of total 8 marbles in the bag.
So, [tex] \frac{6}{8} [/tex] is the another fraction to show number of red marbles.
find the area of the right triangle with the base length 30 inches and height 3 feet
Answers
.. = (1/2)(2.5 ft)*(3 ft) . . . . . . 30 inches = 2.5 feet
.. = 3.75 ft^2
Help with this algebra math question please
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We have then:
cot (x) = 1 / (tan (x))
Rewriting we have:
cot (x) = 1 / ((sin (x)) / (cos (x)))
We do double c:
cot (x) = cos (x) / sin (x)
Answer:
cot (x) = cos (x) / sin (x)
option 3
Help! I’ve been stuck for like minuteeesss!!!
Answers
b. For each gallon, you can drive 348/14.5=24 miles. If you have 132 gallons, you can drive 132*24=3168 miles.
One side of a triangle is increasing at a rate of 8 cm/s and a second side is decreasing at a rate of 3 cm/s. if the area of the triangle remains constant, at what rate does the angle between the sides change when the first side is 24 cm long, the second side is 35 cm, and the angle is π/6?
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A' = (a'*b*sin(θ) +a*b'*sin(θ) +a*b*cos(θ)*θ')/2
0 = (8 cm/s)*(35 cm)*sin(π/6) +(24 cm)*(-3 cm/s)*sin(π/6) +(35 cm)*(24 cm)*cos(π/6)*θ'
0 = 140 cm^2/s -36 cm^2/s +420√3 cm^2*θ'
θ' = -104/(420√3) . . . . rad/s
.. = -26(√3)/315 . . . . . . rad/s
.. ≈ -8.19°/s
_____
The figure offers a sort of sanity check on the numbers.
what id is 29/4 improper
Answers
By doing this, we would be able to find the improper fraction.
In the decimal form, this would be 7.25
And in the mixed number, it would look like:
[tex]\boxed{7 \frac{1}{4} }[/tex]
The bottom denominator always stays the same!
[tex]7 \frac{1}{4} [/tex] is the fraction as a mixed number
7.25 is the fraction as a decimal
725% is the fraction as a percent
hope this helps
Please help asap!!!
Which expression is equivalent to -9x^-1y^-9/-15^5y^-3
A.3/5x^5y^3
B.3/5x^6y^6
C.5/3x^5y^3
D.5/3x^6y^6
Answers
Our original expression was
[tex]\frac{-9x^{-1}y^{-9}}{-15x^5y^{-3}}[/tex]
Negative exponents tell us to switch sides of the fraction; therefore they get moved as such:
[tex]\frac{-9y^3}{-15x^5x^1y^9}[/tex]
We can reduce the fraction we have:
[tex]\frac{3y^3}{5x^5x^1y^9}[/tex]
Using the product property of exponents we can combine the exponents on the denominator:
[tex]\frac{3y^3}{5x^6y^9}[/tex]
Using the division property of exponents, we subtract the exponents of y and get:
[tex]\frac{3}{5x^6y^6}[/tex]
solve the system of equations algebraically.
Answers
a 0.149-kilogram baseball, initially moving at 15 meters per second, is brought to rest in 0.040 seconds by a baseball glove on a catchers hand. the magnitude of the average force exerted on the ball by glove is
Answers
Using the relationship between force and acceleration, the average force exerted on the ball by the glove is :55.875 N
Given the Parameters :
Mass, m = 0.149 kg Velocity, V = 15 m/s Time, t = 0.040 secondsRecall :
Force = Mass × acceleration Acceleration = Δv / tForce = (mΔv ÷ t)
Inputting the Values into the equation :
Force = [0.149 × (15 - 0)] ÷ 0.040
Force = (0.149 × 15) ÷. 0.040
Force = 2.235 ÷ 0.040
Force = 55.875 N
Therefore, the force exerted is 55.875 N
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