The Big Ideas or Focal Points in the mathematics curriculum for grade 6 are: 1. Ratios and Proportional Relationships
2. The Number System. 3. Expressions and Equations. 4. Geometry. 5. Statistics and Probability.
1. Ratios and Proportional Relationships: Students should be able to understand and use ratios to describe relationships between quantities, and use proportional reasoning to solve problems.
2. The Number System: Students should be able to understand and use negative numbers, and perform operations with multi-digit numbers and decimals.
3. Expressions and Equations: Students should be able to write and evaluate expressions, and use equations to solve problems.
4. Geometry: Students should be able to understand and use the properties of shapes and the relationships between them.
5. Statistics and Probability: Students should be able to use data to make predictions and understand the concept of probability.
These Big Ideas or Focal Points are the key concepts that students should master in grade 6 mathematics in order to be prepared for higher-level math courses.
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How many times as great as the value of 4 in 4,670 is the value of 4 in 5,040
The value of the digit 4 in 5,040 is 100 times more than that of the digit 4 in 4,670.
At the unit place of both numerals, there is a digit 4. The place value of a digit determines how valuable it is in a number. The position of the digit from the number's rightmost side determines its place value.
The value of the digit 4 in the tens place of the number 4,670 is 4 x 10 = 40.
The value of the digit 4 in the thousands position of the number 5,040 is 4 x 1000, or 4000.
Therefore, As 4000/40 = 100, the value of the digit 4 in 5,040 is 100 times more than that of the digit 4 in 4,670.
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During the summer, Olympic swimmer Adam Johnson swims every day. On sunny summer days, he goes to an outdoor pool, where he may swim for no charge. On rainy days, he must go to a domed pool. At the beginning of the summer, he has the option of purchasing a $15 season pass to the domed pool, which allows him use for the entire summer. If he doesn't buy the season pass, he must pay $1 each time he goes there. Past meteorological records indicate that there is a 60% chance that the summer will be sunny (in which case there is an average of 6 rainy days during the summer) and a 40% chance the summer will be rainy (an average of 30 rainy days during the summer).
Before the summer begins, Adam has the option of purchasing a long-range weather forecast for $1. The forecast predicts a sunny summer 80% of the time and a rainy summer 20% of the time. If the forecast predicts a sunny summer, there is a 70% chance that the summer will actually be sunny. If the forecast predicts a rainy summer, there is an 80% chance that the summer will actually be rainy. Assuming that Adam's goal is to minimize his total expected cost for the summer, what should he do? Also find EVSI and EVPI
Adam should purchase the long-range weather forecast for $1. By doing so, he will have the best chance of minimizing his total expected cost for the summer.
The expected value of the sample information (EVSI) is the expected cost if Adam does not purchase the forecast and goes with the meteorological records, which is $16.6 ($15 for the season pass plus an average of $1.6 per rainy day).
The expected value of perfect information (EVPI) is the expected cost if Adam purchases the forecast and uses it to make the decision, which is $15.6 ($15 for the season pass plus an average of $0.6 per rainy day).
Thus, purchasing the long-range weather forecast is the best decision for Adam, as it will save him $1 in expected costs and allow him to minimize his total expected cost for the summer.
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Of the 650 juniors at Jefferson High School, 468 are enrolled in Algebra II, 292 are enrolled in
Physics, and 180 are taking both courses at the same time. If one of the 650 juniors was picked
at random, what is the probability they are taking Physics, if we know they are in Algebra II
(Physics given AII)? Round to the nearest hundredth.
Answer:
0.38
Step-by-step explanation:
HELP! PLEASE ND THANK YOU
The equation that gives the cost of each juice bottle is 24j + 2.19 = 40.83
What is an equation?An equation is an expression that shows the relationship between numbers and variables using mathematical operators such as addition, subtraction, exponent, multiplication and division.
Let j represent the cost of each juice.
Sophie bought a popcorn for $2.19 and 24 juice bottles for a total of $40.83.
The equation that represents this problem is:
24j + 2.19 = 40.83
The equation is 24j + 2.19 = 40.83
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34.34. A P-value is
A. the correlation between two variables
B. the ratio between the test statistic and the standard error
C. the probability of incorrectly rejecting the null hypothesis
D. None of the above
E. the same as the significance level
The correct answer is C. the probability of incorrectly rejecting the null hypothesis.
A P-value is used in hypothesis testing to determine the likelihood of observing a test statistic as extreme as the one observed under the null hypothesis. It is a measure of the strength of evidence against the null hypothesis.
A small P-value (typically ≤ 0.05) indicates strong evidence against the null hypothesis, so you reject the null hypothesis. A large P-value (> 0.05) indicates weak evidence against the null hypothesis, so you fail to reject the null hypothesis.
It is important to note that a P-value is not the same as the significance level (option E), which is the threshold used to determine whether to reject or fail to reject the null hypothesis.
The P-value is also not the same as the correlation between two variables (option A), which measures the strength of a linear relationship between two variables.
The P-value is also not the ratio between the test statistic and the standard error (option B), which is used to calculate the P-value but is not the same thing.
Therefore, the correct answer is option C, the probability of incorrectly rejecting the null hypothesis.
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The original price of a polo shirt is $20. How much will Bert pay if he buys it during the sale?
Answer:
We would need to know the percentage of discount offered during the sale to determine the sale price of the polo shirt. Here's an example calculation for a 25% discount:
If the discount offered is 25%, Bert will only need to pay 75% of the original price. We can calculate this as:
Sale price = (100% - 25%) x $20
Sale price = 75% x $20
Sale price = 0.75 x $20
Sale price = $15
Therefore, if the discount offered during the sale is 25%, Bert will pay $15 for the polo shirt. However, if the discount is different, the sale price will be different.
Maria uses 14 beads to make a bracelet. Sarah uses 6 times as many beads as Maria to make a necklace. How many beads does Sarah use? Enter your answer in the box.
I need help im stuck (image attach)
Answer:
∠e = 32°
∠d = 69°
∠f = 79°
Step-by-step explanation:
Angle e and 32° are vertical angles, so this means that ∠e = 32° because vertical angles are congruent.
Next, a straight line is equal to 180°, and angle e is equal to 32°, so we can write the following equation to solve for angle f:
69° + e + f = 180° ➜ 69° + 32° + f = 180° ➜ 101° + f = 180° ➜ f = 79°
Lastly, angle d and 69° are also vertical angles, so this means that ∠d = 69° because vertical angles are congruent.
Cables and parallel beams are used to support a bridge. The obtuse angles formed by the beams and the bridge are 105°. The acute angle formed by the cable and the bridge is 42°.
Note: Picture is not drawn to scale.
What is the measure of angle X?
Answer:
Step-by-step explanation:
A$AP BEAN$: Khan Academy can help you. Do your work and stop cheating!!!!!
BEAN$ OUT!!!!!
What is the solution, if any, to the equation 3(x-2)+4=3x+6
Answer:
if I'm correct it's 3(4x5)
BUT RECHECK JUST INCASE
Step-by-step explanation:
Jiang is building a doghouse for her new puppy. The image below is what the base of the house will look like. What is the area of the dog bed that Jiang will need to make to fit the entire base of the dog house
Answer:
Step-by-step explanation:
lets take the top part as 1 and the seconds part as 2
so for the first part
1 Area= L*B
so 8*5 =40
2 Area=L*B
so 11*4=44
44+40=84cm2
r each equation, determine whether its all symmetries that apply. (a) y^(2)-x+10=0
The equation y^(2)-x+10=0 has no symmetry.
Symmetry is when one part of an equation is the mirror image of another part. In this case, there is no symmetry because the equation cannot be mirrored on any axis.
To check for symmetry with respect to the y-axis, we can replace x with -x and see if the equation is still true. In this case, y^(2)-(-x)+10=0 is not the same as the original equation, so there is no symmetry with respect to the y-axis.
Similarly, to check for symmetry with respect to the x-axis, we can replace y with -y and see if the equation is still true. In this case, (-y)^(2)-x+10=0 is not the same as the original equation, so there is no symmetry with respect to the x-axis.
Finally, to check for symmetry with respect to the origin, we can replace x with -x and y with -y and see if the equation is still true. In this case, (-y)^(2)-(-x)+10=0 is not the same as the original equation, so there is no symmetry with respect to the origin.
Therefore, the equation y^(2)-x+10=0 has no symmetry.
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Write the function in standard form and determine the end behavior. Be sure to use limit notation. 7. f(x)=-4 x^{3}-10 x^{12}+13 x) 8. ( f(x)=8 x^{21}-5 x^{2}+13 x ) 9. ( f(x)=x^{32}-10 x^{14}
The function in standard form of f(x)=-4 x³-10 x¹²+13 is f(x)=-10 x¹²-4 x³+13 x. The function in standard form of f(x)=8 x²¹-5 x²+13 x ) is f(x)=8 x²¹-5 x²+13 x. The function in standard form of ( f(x)=x^³²-10 x¹⁴ is f(x)=8 x²¹-5 x²+13 x
7. To determine the end behavior, we look at the highest degree term, which is -10 x^{12}. As x approaches positive infinity, -10 x^{12} will approach negative infinity. As x approaches negative infinity, -10 x^{12} will approach positive infinity. Therefore, the end behavior is:
lim_{x→∞} f(x) = -∞
lim_{x→-∞} f(x) = ∞
8. The function in standard form is f(x)=8 x^{21}-5 x^{2}+13 x. To determine the end behavior, we look at the highest degree term, which is 8 x^{21}. As x approaches positive infinity, 8 x^{21} will approach positive infinity. As x approaches negative infinity, 8 x^{21} will approach negative infinity. Therefore, the end behavior is:
lim_{x→∞} f(x) = ∞
lim_{x→-∞} f(x) = -∞
9. The function in standard form is f(x)=8 x^{21}-5 x^{2}+13 x. To determine the end behavior, we look at the highest degree term, which is x^{32}. As x approaches positive infinity, x^{32} will approach positive infinity. As x approaches negative infinity, x^{32} will approach positive infinity. Therefore, the end behavior is:
limx→∞ f(x) = ∞
limx→∞f(x) = +∞
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Rewrite the Cartesian equation y = 3x^2 as a polar equation. r(θ) = 1/3 tan(θ) sec(θ)
Enter theta for θ if needed.
The Cartesian equation y = 3x^2 as a polar equation can be write as r(θ) = 3 sec(θ) / (1 + tan²(θ)).
To rewrite the Cartesian equation y = 3x² as a polar equation, we need to use the conversion formulas between Cartesian coordinates (x, y) and polar coordinates (r, θ):
x = r cos(θ)
y = r sin(θ)
Substituting these formulas into the Cartesian equation gives us:
r sin(θ) = 3(r cos(θ))²
Dividing both sides by r and rearranging terms gives us:
r = 3 cos²(θ) / sin(θ)
Using the identity 1 + tan²(θ) = sec²(θ), we can rewrite the equation in terms of tan(θ) and sec(θ):
r = 3 (1 / (1 + tan²(θ))) / (1 / sec(θ))
Simplifying gives us the final polar equation:
r(θ) = 3 sec(θ) / (1 + tan²(θ))
Therefore, the Cartesian equation y = 3x²can be rewritten as the polar equation r(θ) = 3 sec(θ) / (1 + tan²(θ)).
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Fraction division Divide. Write your answer in simplest form. (13)/(15)-:(7)/(10)
The answer to 13)/(15)-:(7)/(10) in simplest form is (26)/(21).
To divide the fractions (13)/(15) and (7)/(10), we need to multiply the first fraction by the reciprocal of the second fraction. The reciprocal of a fraction is obtained by flipping the numerator and denominator. So, the reciprocal of (7)/(10) is (10)/(7).
Now, we can multiply the two fractions as follows:
(13)/(15) * (10)/(7) = (13*10)/(15*7) = (130)/(105)
Next, we need to simplify the fraction by finding the greatest common factor (GCF) of the numerator and denominator.The GCF of 130 and 105 is 5. So, we can divide both the numerator and denominator by 5 to get the simplest form of the fraction:
(130)/(105) = (130/5)/(105/5) = (26)/(21)
Therefore, the answer in simplest form is (26)/(21).
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I need help pls pls pls pls pls.
Answer:
G
Step-by-step explanation:
Since these triangles are similar, side lengths in one must be proportional to the other. Segment AB is similar to DE, AC similar to DF, and BC similar to EF. With that in mind, G is the only possible answer because DE/AG correspond to one another, just as DF/AC correspond to one another.
Michelle reads 16 pages in ¼ of an hour. How many pages will she read in one hour?
Answer:
reads 64 pages in one hour
Step-by-step explanation:
For simplicities' sake, we'll convert 1/4 of an hour into 15 minutes (60 minutes x 1/4 = 15 mins)
Since we have a ratio, of 16 pages to 15 mins, we can create the ratio by making a fraction of 16/15. Since we want to get to 1 hour, to find how many pages she read in an hour, we can multiply the top and bottom (to equalize) by 4 (because 4 x 15 = 60).
16•4/15•4
64/60
Because of this, we can see that the ratio is now 64 pages to 60 mins or one hour. Therefore, she reads 64 pages in one hour.
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Answer:
64
Step-by-step explanation:
To do this problem you have to figure out Michelle's hourly rate for reading pages. We are given that she will read 16 pages in 1/4 of an hour. This means that in 15 minutes (60 minutes in an hour / 4 for the 1/4 an hour) she will read 16 pages. We only have 15 minutes though and need 60 minutes so 60/15 gives us 4. This means that we will need 4, 15 minute periods to get to a hour. So 4*15minutes = 60 minutes or one hour and 16pages*4 = 64 pages. So in one hour she will read 64 pages.
A faster way to do this is to just multiply both sides by the recripical.
16 pages = 1/4 hour
*4/1 *4/1
64 pages = 4/4 hour
64 pages = 1 hour
Solve the system of equation using the method of choice. Show work and steps.
10x+5y=-20
-4x+6y=-8
Answer:
Step-by-step explanation:
1
Juan is deciding between two truck rental companies. Company A charges an initial fee of $65 for the rental plus $3 per mile driven. Company B charges an initial fee of $100 for the rental plus $2 per mile driven. Let A represent the amount Company A would charge if Juan drives a miles, and let B represent the amount Company B would charge if Juan drives a miles. Write an equation for each situation in terms of x, and determine the number of miles driven, x, that would make the cost of each company the same.
Answer:
35 miles.
Step-by-step explanation:
A=3x+65
B=2x+100
3x+65=2x+100
x+65=100
x=35
The graph and the table show the high temperatures in a city over a 10-day period.
a. What was the high temperature on Day 7?
b. On which days was the high temperature 61 degrees?
c. Is the high temperature a function of the day? Explain how you know.
d. Is the day a function of the high temperature? Explain how you know.
The graph and the table show the high temperatures in a city over a 10-day period. here are the right answer:
a. 60, b. 2, 4, and 6 days
c. Yes, the high temperature is a function of the day.
Reading the graph:The following group is shows a the high temperatures in a city for 10 -days. Where height temperature is on day 9, lowest temparature in day 7. Here the high temperature is a function of the day.
Here we have
The graph and the table show the high temperatures in a city for 10-days
From the table,
The high temperature on Day 7 = 60
The high temperature 61 degrees on 2, 4, and 6 days
Since the graph shows the change in temperature as well as the change in the number of days the high temperature is a function of the day
Therefore,
The answer are
a. 60, b. 2, 4, and 6 days
c. Yes, the high temperature is a function of the day.
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95/56 and 2 5/7 correct symbol
The correct symbol in the expression is 95/56 < 2 5/7
How to determine the correct symbolFrom the question, we have the following parameters that can be used in our computation:
95/56 and 2 5/7
Convert the improper number to a mixed number
So, we have the following representation
1 39/56 and 2 5/7
By comparison;
1 39/56 is less than 2 5/7
So, the inequality symbol is a less than inequality
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Create a data set with 6 numbers that has a median of 12
Answer:
Median is a middle number.
Step-by-step explanation:
You can use any set of number with 12 in the middle
14, 18, 12, 12, 17, 18
Middle numbers are third and forth number added together and divided by 2
Find the value of x if the 8th term of the expansion of
(x^3+1)^12 is equal to 25952256
a. 2
b. 4
c. 6
d. 3
The value of x is if the 8th term of the expansion of given equation is equal to 25952256 = 2. The correct answer is option a.
In order to find the value of x, we first need to find the 8th term of the expansion of [tex](x^3 + 1)^12.[/tex] Using the binomial theorem, the general term in the expansion is given by: [tex]T(k) = C(n, k-1) * (x^3)^(n-k+1) * 1^(k-1).[/tex]where T(k) is the kth term, n is the power (in this case, 12), C(n, k-1) represents the binomial coefficient [tex](n! / [(k-1)! * (n-k+1)!]),[/tex] and x^3 is the term in the expansion.
Now, we plug in the values to find the 8th term: [tex]T(8) = C(12, 7) * (x^3)^(12-7+1) * 1^7, T(8) = 792 * (x^3)^6.[/tex] We are given that the 8th term of the expansion is equal to 25952256. So, 25952256 = 792 * (x^3)^6
Now, we need to solve for [tex]x: (25952256 / 792) = (x^3)^6, 32768 = (x^3)^6[/tex]
Taking the 6th root of both sides, we get:
[tex]x^3 = 2[/tex]
Now, taking the cube root of both sides, we find the value of x:
x = 2 (option a). The correct answer is option a.
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Y varies inversely as x. If x = 8 when y = 30, find x when y = 20
[tex]\qquad \qquad \textit{inverse proportional variation} \\\\ \textit{\underline{y} varies inversely with \underline{x}} ~\hspace{6em} \stackrel{\textit{constant of variation}}{y=\cfrac{\stackrel{\downarrow }{k}}{x}~\hfill } \\\\ \textit{\underline{x} varies inversely with }\underline{z^5} ~\hspace{5.5em} \stackrel{\textit{constant of variation}}{x=\cfrac{\stackrel{\downarrow }{k}}{z^5}~\hfill } \\\\[-0.35em] ~\dotfill[/tex]
[tex]\stackrel{\textit{"y" varies inversely with "x"}}{y = \cfrac{k}{x}}\hspace{5em}\textit{we also know that} \begin{cases} x=8\\ y=30 \end{cases} \\\\\\ 30=\cfrac{k}{8}\implies 240=k\hspace{9em}\boxed{y=\cfrac{240}{x}} \\\\\\ \textit{when y = 20, what's "x"?}\qquad 20=\cfrac{240}{x}\implies x=\cfrac{240}{20}\implies x=12[/tex]
Find the period and amplitude. \[ y=-4 \sin \left(\frac{\pi x}{5}\right) \] period amplitude Need Help?
The period of the function is 10 and the amplitude is 4.
The period and amplitude of the function \[ y=-4 \sin \left(\frac{\pi x}{5}\right) \] can be found by examining the coefficients in the equation.
The amplitude is the absolute value of the coefficient in front of the sine function, which in this case is |-4| = 4. Therefore, the amplitude of the function is 4.
The period is found by dividing 2π by the coefficient in front of the x variable inside the sine function. In this case, the coefficient is \[\frac{\pi}{5}\]. So, the period is \[\frac{2\pi}{\frac{\pi}{5}} = 10\].
Therefore, the period of the function is 10 and the amplitude is 4.
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Mr. Cortez is comparing the cost to join two different gyms. Gym A charges a $55 registration fee plus $24 per month . Gym B does not charge a registration fee but charges $35 per month. At what number of months will the cost be the same for both gyms?
Mr. Cortez is comparing the cost to join two different gyms. Gym A charges a $55 registration fee plus $24 per month. Gym B does not charge a registration fee but charges $35 per month. The cost will be the same for both gyms on the 5 months.
What is gym?
A gymnasium or gym is a place where people can exercise indoors. The word is a translation of "gymnasium," which is Greek. They are typically seen in athletic and fitness facilities, in addition to acting as activity and learning spaces in educational institutions. Gym is slang for "fitness centre," which is often a location for indoor entertainment.
Given,
Gym A charges a $55 registration fee and $24 per month
Total charges after 5 months with registration fee= (24*5+55)= $175
Gym B charges $35 per month
Total charges after 5 months= 35*5= $175
Hence the correct answer is the cost will be the same for both gyms on the 5 months.
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A calculator is not allowed on this question. If (mx+3)(nx+5)=8x^(2)+px+15 for all values of x, and m+n=6, what are the two possible values for p?
The answer of two possible values for p are 26 and 22
If (mx+3)(nx+5)=8x^(2)+px+15 for all values of x, and m+n=6, we can use the distributive property to expand the left side of the equation:
mnx^(2)+(5m+3n)x+15=8x^(2)+px+15
Next, we can compare the coefficients of each term to find the values of m, n, and p:
mn=8
5m+3n=p
m+n=6
Since m+n=6, we can solve for one variable in terms of the other:
n=6-m
Substituting this into the equation for p gives us:
5m+3(6-m)=p
5m+18-3m=p
2m+18=p
Now we can substitute this back into the equation for mn:
m(6-m)=8
6m-m^(2)=8
m^(2)-6m+8=0
Using the quadratic formula, we can find the two possible values for m:
m=(6±√(6^(2)-4(1)(8)))/(2(1))
m=(6±√(36-32))/2
m=(6±√4)/2
m=(6±2)/2
m=4 or m=2
If m=4, then n=6-4=2, and p=2(4)+18=26
If m=2, then n=6-2=4, and p=2(2)+18=22
Therefore, the two possible values for p are 26 and 22.
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Can someone help with this please?
According to the Synthetic Division, the Quotient is equal to 7x² -4x + 5 and the Remainder is equal to 8.
Synthetic Division: What is it?Synthetic division can divide two polynomials more quickly than the standard long division algorithm.
With this method, the dividend and divisor polynomials are reduced to a set of numerical numbers.
Given:
(7x³-25x² +17x -7) ÷ (x-3).
So, Synthetic Division for this is as follows:
(x-3) | (7x³-25x² +17x -7) | 7x² -4x + 5
7x³-21x²
_________
-4x² + 17x
-4x² + 12x
_______________
5x - 7
5x - 15
________
8
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Natalie has two number pyramids each labeled 1 to 4. Natalie is going to conduct an experiment by tossing both pyramids a total of 96 times. She will find the difference of each pair of numbers rolled by subtracting the lesser number from the greater number
a. The total number of possible outcomes is 16 x 96 = 1,536.
b. She should expect to see a difference of 1 approximately 12/16 x 96 = 72 times.
c. She should expect to see a difference of 0 approximately 8/16 x 96 = 48 times.
a) There are a total of 16 possible pairs of numbers that can be rolled on the two pyramids:
(1,1), (1,2), (1,3), (1,4), (2,1), (2,2), (2,3), (2,4), (3,1), (3,2), (3,3), (3,4), (4,1), (4,2), (4,3), and (4,4).
We can use a probability distribution to calculate the probability of each pair of numbers being rolled. Since each pyramid has four sides, there are 4 x 4 = 16 possible outcomes when both pyramids are rolled. Therefore, the probability of each outcome is 1/16.
Since Natalie is going to toss each pair of pyramids 96 times, the total number of possible outcomes is 16 x 96 = 1,536.
To calculate the difference between each pair of numbers, we can simply subtract the smaller number from the larger number.
Here is a table that shows the probability of each difference:
Difference Possible Pairs Probability
0 (1,1), (2,2), (3,3), (4,4) 4/16 = 1/4
1 (1,2), (1,3), (1,4), (2,3), (2,4), (3,4), 12/16 = 3/4
(2,1), (3,1), (4,1), (3,2), (4,2), (4,3)
2 None 0
3 None 0
Therefore, the probability of the difference being 0 is 1/4, the probability of the difference being 1 is 3/4, and the probability of the difference being 2 or 3 is 0.
Since Natalie is going to conduct the experiment a total of 96 times, we can use these probabilities to calculate the expected number of times each difference will occur:
Difference Probability Expected Number of Occurrences
0 1/4 96 x 1/4 = 24
1 3/4 96 x 3/4 = 72
2 0 96 x 0 = 0
3 0 96 x 0 = 0
b) we can expect the difference between each pair of numbers to be 0 about 24 times and
c) to be 1 about 72 times over the course of the experiment.
Differences 2 and 3 are not expected to occur.
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The correct question should be:
Natalie has two number pyramids each labeled 1 to 4. Natalie is going to conduct an experiment by tossing both pyramids a total of 96 times. She will find the difference of each pair of numbers rolled by subtracting the lesser number from the greater number.
a. How many possible outcomes are there? _________________
b. How many times should Natalie toss a difference of 1? ______________
c. How many times should Natalie toss a difference of 0?
Given that tangent theta = negative 1, what is the value of secant theta, for StartFraction 3 pi Over 2 EndFraction less-than theta less-than 2 pi?
Negative StartRoot 2 EndRoot
StartRoot 2 EndRoot
0
1
Answer:
Step-by-step explanation:
Since tangent is negative and is equal to the ratio of opposite over adjacent, we can assume that the angle theta lies in either the third or the fourth quadrant, where the x and y coordinates have opposite signs.
In the third quadrant, sine is positive and cosine is negative. Therefore, we have:
tangent theta = opposite/adjacent = -1
sine theta = opposite/hypotenuse > 0
cosine theta = adjacent/hypotenuse < 0
Using the Pythagorean identity, we have:
sin² theta + cos² theta = 1
=> opposite²/hypotenuse² + adjacent²/hypotenuse² = 1
=> opposite² + adjacent² = hypotenuse²
Since we are given that 3π/2 < theta < 2π, we can place the triangle in the third quadrant with a reference angle of π/2. This means that the opposite side is equal to the absolute value of the adjacent side. Let's assume that the hypotenuse is equal to 1, then we have:
opposite/adjacent = -1
=> opposite = -adjacent
opposite² + adjacent² = hypotenuse²
=> adjacent² + (-adjacent)² = 1
=> 2adjacent² = 1
=> adjacent = ±1/√2
Since cosine is negative in the third quadrant, we have:
cosine theta = adjacent/hypotenuse = -1/√2
Finally, we can use the reciprocal identity to find secant:
secant theta = 1/cosine theta = -√2
Therefore, the answer is A. Negative StartRoot 2 EndRoot.
The value of tangent theta is equal to the negative 1. At this value the value of secant theta is [tex]\sqrt{2}[/tex].
What is tangent theta?The tangent theta in a triangle is the ratio of sine theta and cos theta. It can be written as,
[tex]tan\theta=\dfrac{sin\theta}{cos\theta}[/tex]
Given information-
The value of tangent theta is equal to the negative 1.
[tex]tan\theta=-1[/tex]
The tangent theta in a triangle is the ratio of sine theta and cos theta. It can be written as,
[tex]tan\theta=\dfrac{sin\theta}{cos\theta}[/tex]
The value of tangent theta is equal to the negative 1. Thus put the value in above expression as,
[tex]-1=\dfrac{sin\theta}{cos\theta}[/tex]
Simplify it further as,
[tex]-cos\theta=sin\theta[/tex]
When the value of cosine and sine theta is equal, then the angle exist in
4th quadrant with the value of [tex]\dfrac{7\pi }{4}[/tex]. Which extent to the [tex]\sqrt{2}/2[/tex] for the cosine function.
In the trigonometry cosine theta is the reciprocal of the secant theta. Thus,
[tex]\dfrac{1}{sec\theta} =\dfrac{\sqrt{2} }{}[/tex]
[tex]sec\theta=\sqrt{}\dfrac{2}{2}[/tex]
Thus the value of secant theta is [tex]\sqrt{2}[/tex]
Learn more about the tangent theta here:
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