Answer:
See Below.
Step-by-step explanation:
We want to prove the trigonometric identity:
[tex]\displaystyle \frac{\sec^2(\theta)-1}{\sin(\theta)}=\frac{\sin(\theta)}{1-\sin^2(\theta)}[/tex]
To start, let's simplify the right side. Recall the Pythagorean Identity:
[tex]\sin^2(\theta)+\cos^2(\theta)=1[/tex]
Therefore:
[tex]\cos^2(\theta)=1-\sin^2(\theta)[/tex]
Substitute:
[tex]\displaystyle \frac{\sin(\theta)}{1-\sin^2(\theta)}=\frac{\sin(\theta)}{\cos^2(\theta)}[/tex]
Split:
[tex]\displaystyle =\frac{\sin(\theta)}{\cos(\theta)}\left(\frac{1}{\cos(\theta)}\right)=\tan(\theta)\sec(\theta)[/tex]
Therefore, our equation becomes:
[tex]\displaystyle \frac{\sec^2(\theta)-1}{\sin(\theta)}=\tan(\theta)\sec(\theta)[/tex]
From the Pythagorean Identity, we can divide both sides by cos²(θ). This yields:
[tex]\displaystyle \tan^2(\theta)+1=\sec^2(\theta)[/tex]
So:
[tex]\tan^2(\theta)=\sec^2(\theta)-1[/tex]
Substitute:
[tex]\displaystyle \frac{\tan^2(\theta)}{\sin(\theta)}=\tan(\theta)\sec(\theta)[/tex]
Rewrite:
[tex]\displaystyle (\tan(\theta))^2\left(\frac{1}{\sin(\theta)}\right)=\tan(\theta)\sec(\theta)[/tex]
Recall that tan(θ) = sin(θ)/cos(θ). So:
[tex]\displaystyle \frac{\sin^2(\theta)}{\cos^2(\theta)}\left(\frac{1}{\sin(\theta)}\right)=\tan(\theta)\sec(\theta)[/tex]
Simplify:
[tex]\displaystyle \frac{\sin(\theta)}{\cos^2(\theta)}=\tan(\theta)\sec(\theta)[/tex]
Simplify:
[tex]\tan(\theta)\sec(\theta)=\tan(\theta)\sec(\theta)}[/tex]
Hence proven.
The table shows Kayden's quiz scores for a semester. What are the mean and the standard deviation, rounded to the nearest hundredth, for the quiz data?
45 48 42 47 43
Mean =
Standard deviation =
Answer:
The mean is of 45.
The standard deviation is 2.28.
Step-by-step explanation:
The mean is the sum of all values divided by the number of values.
The standard deviation is the square root of the sum of the difference squared between each value and the mean, and divided by the number of values.
Mean:
[tex]M = \frac{45 + 48 + 42 + 47 + 43}{5} = 45[/tex]
The mean is of 45.
Standard deviation:
[tex]M = \sqrt{\frac{(45-45)^2 + (48-45)^2 + (42-45)^2 + (47-45)^2 + (43-45)^2}{5}} = 2.28[/tex]
The standard deviation is 2.28.
Help please?
51c^3 -34 c
4.3x10^3 in standard notation
Answer:
4300
Step-by-step explanation:
i think this is what you want
A card is selected at random from a standard deck of playing cards.
Compute the probability that the card is a 3. (Enter the probability as a fraction.)
Answer:
It's 4/52
Step-by-step explanation:
because there is four threes in a standard deck of playing cards which has 52 cards in all.
Answer:
4/52 = 1/13
Step-by-step explanation:
the are four 3's in standard deck of cards
I need help with this please. I need to find the volume. For number 7
Explanation is in a file
bit.[tex]^{}[/tex]ly/3a8Nt8n
Help thanks in advance
Answer:
C) 2/7, 1/2, 5/6, 4/2
Step-by-step explanation:
hey I need help with my coordinates
Ok send the picture so I can help you.
Answer:
what is the question? I can help
Step-by-step explanation:
pls help ASAP this is timing meeee!!!!
Answer:
48
Step-by-step explanation:
details will be added
In a class test (+ 5) marks are given for every correct answer and (–2) marks are given for every incorrect answer and no marks for not attempting any question.
(i) Rayan scored 30 marks. If she has got 10 correct answers, how many questions has she attempted incorrectly?
(ii) Riya scores 44 marks by attempting 13 questions. How many questions has he attempted correctly and how many has he attempted incorrectly?
OK ITS OK IM GOD CALM DOWN
Answer:
Step-by-step explanation:
(1)
If she got 10 answers right, she got 5 * 10 = 50 marks before anything was deducted.
Since she wound up with 30 marks, she lost 50 - 30 = 20 marks. If you don't feel comfortable with this, call it -20
Each incorrect answer cost her -2 marks.
Divide -20 /-2 = 10
She has answered 10 questions incorrectly.
(ii)
5x - 2y = 44
x + y = 13 Multiply this equation by 2
2x + 2y = 26
Rewrite the first and third equations
5x - 2y = 44
2x + 2y = 26 Add
7x = 70 Divide by 7
x = 70/7
x = 10
So 10 questions were answered correctly
x + y = 13
10 + y = 13
y = 3
Answer
10 were answered correctly
3 were answered incorrectly
7. What is the area of a triangle whose base is 14 cm and a height of 12 cm?
Answer:
84 cm²
Step-by-step explanation:
1/2x14x12=84
having trouble i don't understand the problem
Answer:
1
a= 25/36
b= -1
c= -8/15
2
a=positive
b=negative
c=positive
d=positive
hope it helps
If A = (-2,-4) and B = (-8,4), what is the length of AE?
A. 12 units
B. 16 units
C. 10 units
D. 14 units
Answer:
It's C
Step-by-step explanation: How far is it from -2 to -8 horizontally?
They are 6 apart, right?
How far is -4 to 4 vertically? 8.
__________________________________________________________
6^2 + 8^2 = c^2
36 + 64 = c^2
c = 100 = 10
___________________________________________________________
So in short yea...it's C UwU
Please help! I need help ASAP and the question is in the picture. I’m giving 35 points! PLEASE HELP ASAP
Answer:
36x
Step-by-step explanation:
I believe this should be the answer. I hope it helps.
I will mark branliest if correct
Answer:
The answer is: y = - |x+1|+1
Josephine has a rectangular garden with an area of 2x2 + x – 6 square feet.
Which expressions can represent the length and width of the garden?
length = x2 – 3 feet; width = 2 feet
length = 2x + 3 feet; width = x – 2 feet
length = 2x + 2 feet; width = x – 3 feet
length = 2x – 3 feet; width = x + 2 feet
Answer:
D
Step-by-step explanation:
This is for 20 points 0.4 Divided by 29
Answer:
72.500
Step-by-step explanation:
I'm not sure, but I'm 80% sure
someone plz help i will give brainliest (no links)
Answer:
for part b the answer is 90 degrees
Step-by-step explanation:
2 is adjacent from 5 so they are the same degree number
What is the domain of (-2, 5), (8, 6), (3, 12) (2, 6)
Answer:
The domain is (-2, 8, 3, 2)
Step-by-step explanation:
The domain is such because domain is the x-element or the one you see at first in the ordered pairs.
A recipe used 1/4 cup of sugar for every 3 cups of flour. How much sugar was used for each cup of flour?
Answer:
12 cups of sugar
Step-by-step explanation:
3÷0.25=12
1/4=0.25
How can you Divide the number 30 into 4 equal parts? Or can you?
Answer:
7.5
Step-by-step explanation:
thirty divided by 4 is 7.5 or 7 1/2
Answer:
That might be possible though.
Step-by-step explanation:
If you divide 30 by 7.5, you'll get 4. With any other number, you'll get 3,10,6,5,1, and 30. If you use a decimal, it might be possible it'll give you the answer you wanted.
I NEED HELP jimthompson5910
15. The price of a coat was reduced from $150 to $90. By what percentage was the price reduced?
16. Nate’s hourly pay rate was raised from $10 to $12. By what percentage was Nate’s pay raised?
17. A tree grew from a height of 8 feet to a height of 12 feet. By what percentage did the tree’s height increase?
18. This week only, the local coffee shop is reducing the price of a cup of coffee from $3 .00 to $2 .28. By what percentage is the price reduced?
19. On a sunny day, the area of a puddle decreased from 700 square inches to 560 square inches. By what percentage did the area decrease?
20. A baker increased the amount of chocolate in a recipe from 8 ounces to14 ounces. By what percentage was the amount of chocolate increased?
The price of an $8,000 car is temporarily reduced to $5,000. By what percentage is the price reduced?
22. Zoe needs to raise the reduced price of the car to its original price. By what percentage should she raise the reduced price?
I'll do the first four problems to get you started
====================================================
Problem 15
Answer: 40%-----------------
Explanation:
The price dropped by $60, since 150-90 = 60. This is the discount, which is the amount you save. Divide this over the original price ($150) and we get 60/150 = 0.40 = 40%
Therefore, the price has been reduced by 40%. If we take 40% of 150, then 0.40*150 = 60 showing that 40% of $150 is 60 dollars. This matches up with the discount value mentioned earlier. So it helps verify the answer.
====================================================
Problem 16
Answer: 20%-----------------
Explanation:
The pay raised by 12-10 = 2 dollars. So the percent increase is 2/10 = 0.20 = 20%
Note how 20% of 10 = 0.20*10 = 2 is the pay raise amount; meaning that 2 is 20% of 10.
====================================================
Problem 17
Answer: 50%-----------------
Explanation:
I'll use a formula this time, but we could follow the same idea as the previous problems. This is to give another approach.
A = old value = 8
B = new value = 12
C = percent change
C = [ (B-A)/A ] * 100%
C = [ (12-8)/8 ] * 100%
C = (4/8)*100%
C = 0.50*100%
C = 50%
50% is the same as half of the value. Note how half of 8 is 4, ie 50% of 8 = 0.50*8 = 4. So this is the amount of increase when going from 8 to 12.
Side note: The "B-A" portion in the numerator represents the change in value. We divide this change over the original amount "A" and then multiply by 100% to get the percent change we're after.
====================================================
Problem 18
Answer: 24%-----------------
Explanation:
I'll use the formula again
A = old value = 3.00
B = new value = 2.28
C = percent change
C = [ (B-A)/A ] * 100%
C = [ (2.28-3.00)/3.00 ] * 100%
C = (-0.72/3.00)*100%
C = -0.24*100
C = -24%
The negative percent change means we have a percent decrease. So we have a 24% decrease. Since your teacher is asking about how much the percent reduction is, we'll write the positive version of this percentage.
Problems 19 through 22 will follow the same ideas as shown in the previous problems.
A vehicle moves along a straight road. The vehicle's position is given by f(t) , where tt is measured in seconds since the vehicle starts moving. During the first 10 seconds of the motion, the vehicle's acceleration is proportional to the cube root of the time since the start. Which of the following differential equations describes this relationship, where k is a positive constant?
a. df/dt = kſ
b. df/dt - = kgf
c. d²f/dt = kyt dt2
d. d²f/dt = ky
Answer:
Step-by-step explanation:
a
or D
If 21 is added to a number, the result is 41 less than twice the number. Find the number.
Answer: 62
Step-by-step explanation: Set up your equation
21+x=2x-41 because the number (x) that 21 is added to is 41 less than (so subtraction) twice (so multiply the number x by 2) the number (x).Get the x's on one side by subtracting the single x21=x-41Now add 41 to both sides to completely isolate x62=xSo, your answer is x=62. The number is 62.Check your work. 62+21=83. 2(62)= 124. 124-41=83. Both equations equal 83, so the solution is correct.
What is the area of a square that has a perimeter of 16?
Step-by-step explanation:
If the figure is a square with a perimeter of 16 inches, then each side of the square is 4 inch in length. Area is found by multiplying the length x the width. For a square, the length and width are the same. A = 16 sq
pls give brainlist + thanks
When you see average rate of change what should you automatically think of
Answer:
Steady rate
Step-by-step explanation:
It is a measure of how much the function changed per unit, on average, over that interval. It is derived from the slope of the straight line connecting the interval's endpoints on the function's graph.
a.
Write the center and radius of a circle with the equation:
x2 + 8x + y2 - 24y + 144 = 0
center = (12,-4)
0 center = (-8,24)
radius=4
radius=12
b. center = (8,-24)
d. center = (-4,12)
radius=12
radius=4
Please select the best answer from the choices provided
Answer: The answer is D
Step-by-step explanation:
Edge 2021
What does congruent mean in mathematics
Answer:
same shape and size
Step-by-step explanation:
Answer:
Similarity in my opinion
A professional pool player claims that she sinks at least one ball in a pocket on 80% of her opening break shots. In a recent set of 20 games, she sunk at least one ball on her break shot in only 14 of the games. A simulation of 100 trials was conducted to see how unusual these most recent games are.
Based on the dotplot of the simulation results and the pool player’s last 20 games, which conclusion can be drawn?
The player’s true probability of sinking at least one ball on her break is 27%.
Another 100 simulated trials would show a much different outcome; therefore, we cannot draw any conclusions.
It is most likely that she would sink at least one ball during her break exactly 16 times.
There is about a 27% chance of her sinking at least one ball on her break 14 or fewer times in 20 games. This result is not unusual and there is no convincing evidence that her average is less than 80%.
Answer: B
Step-by-step explanation:
What is the unit rate for this table? Please help!!!
Three methods for teaching mathematics to first graders are to be compared. Ten students are
randomly assigned to each of the methods, and the score on a quiz is recorded. What are the
response and treatment for this experiment?
Answer: The three methods are the treatments and the response is the score achieved on the quiz.
Step-by-step explanation:
The three techniques are the treatments, and the results are the final score.
what is response variable in the experiment?
The researcher predicts that a variable will change if the controlled variable changes; this variable is known as a responsive variable. A dependent variable is another name for a responsive variable.
Solution:
Three methods are assigned for teaching mathematics.
Randomly ten students are assigned to each of the method.
There scores are then recorded on a quiz.
It is clear that the response will be the score card of the students either scoring high marks or not being able to clear it.
And the treatments are three for if the student is not able to understand any of the method.
Hence the three methods serve as treatments, and the outcomes determine the final grade.
Learn more about "Logical reasoning questions" here-
brainly.com/question/25175983
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