Pls help
Prove the trigonometric identity
csc x cos x/tan x+cot x =cos^2 x
Drag the expressions into each box to complete the proof.
Answers
Answer:
Here is the proof of the trigonometric identity
\(\frac{\csc x \cos x}{\tan x + \cot x} = \cos^2 x\)
Given:
\(\frac{\csc x \cos x}{\tan x + \cot x}\)
To prove:
\(\frac{\csc x \cos x}{\tan x + \cot x} = \cos^2x\)
Step 1: Rewrite all trigonometric functions in terms of sine and cosine.
\(\frac{\csc x \cos x}{\tan x + \cot x} \\ = \frac{\frac{1}{\sin x} \cos x}{\frac{\sin x}{\cos x} + \frac{\cos x}{\sin x}} \\ = \frac{\cos x}{\sin x \cdot \frac{\sin x}{\cos x} + \cos x \cdot \frac{\cos x}{\sin x}} \\ \\ = \frac{\cos x}{\frac{\sin^2 x + \cos^2 x}{\sin x \cos x}} \\ \\ = \frac{\cos x}{\frac{1}{\sin x \cos x}} \\ \\ = \cos x \cdot \sin x \cos x \\ \\ = \cos^2 x \)
Step 2:Simplify the expression.
\(\cos^2 x = \cos^2 x\)
Conclusion:Therefore,
\(\frac{\csc x \cos x}{\tan x + \cot x} = \cos^2 x.\)
Hence Proved:
Answer:
\(\dfrac{ \csc x \cos x}{\tan x + \cot x}=\cos^2 x\)\(\textsf{Rewrite $\csc x=\dfrac{1}{\sin x}}, \tan x=\dfrac{\sin x}{\cos x}$, and $\cot x=\dfrac{\cos x}{\sin x}$:}\)
\(\boxed{\dfrac{ \dfrac{1}{\sin x} \cdot \cos x}{\dfrac{\sin x}{\cos x} + \dfrac{\cos x}{\sin x}}}=\cos^2 x\)
\(\boxed{\dfrac{\dfrac{\cos x}{\sin x}}{\dfrac{\sin^2 x }{\sin x + \cos x}+\dfrac{\cos^2 x }{\sin x + \cos x}}}=\cos^2 x\)
\(\boxed{\dfrac{\dfrac{\cos x}{\sin x}}{\dfrac{\sin^2 x + \cos^2 x}{\sin x + \cos x}}}=\cos^2 x\)
\(\boxed{\dfrac{\dfrac{\cos x}{\sin x}}{\dfrac{1}{\sin x + \cos x}}}=\cos^2 x\)
\(\boxed{\dfrac{\cos x}{\sin x} \cdot \sin x \cos x}=\cos^2 x\)
\(\cos^2 x=\cos^2 x\)
Step-by-step explanation:
Given trigonometric identity:
\(\dfrac{ \csc x \cos x}{\tan x + \cot x}=\cos^2 x\)
\(\textsf{Use the identities\;\;$\csc x=\dfrac{1}{\sin x}$\;,\;$\tan x = \dfrac{\sin x}{\cos x}$\;\;and\;\;$\cot x=\dfrac{\cos x}{\sin x}$}:\)
\(\textsf{Rewrite $\csc x=\dfrac{1}{\sin x}}, \tan x=\dfrac{\sin x}{\cos x}$, and $\cot x=\dfrac{\cos x}{\sin x}$:}\)
\(\boxed{\dfrac{ \dfrac{1}{\sin x} \cdot \cos x}{\dfrac{\sin x}{\cos x} + \dfrac{\cos x}{\sin x}}}=\cos^2 x\)
Simplify the numerator and make the fractions in the denominator like fractions:
\(\boxed{\dfrac{\dfrac{\cos x}{\sin x}}{\dfrac{\sin^2 x }{\sin x + \cos x}+\dfrac{\cos^2 x }{\sin x + \cos x}}}=\cos^2 x\)
\(\textsf{Apply\;the\;fraction\;rule\;\;$\dfrac{a}{b}+\dfrac{c}{b}=\dfrac{a+c}{b}$\;to\;the\;denominator}:\)
\(\boxed{\dfrac{\dfrac{\cos x}{\sin x}}{\dfrac{\sin^2 x + \cos^2 x}{\sin x + \cos x}}}=\cos^2 x\)
\(\textsf{Use\;the\;identity\;\;$\sin^2x+\cos^2x=1$}:\)
\(\boxed{\dfrac{\dfrac{\cos x}{\sin x}}{\dfrac{1}{\sin x + \cos x}}}=\cos^2 x\)
\(\textsf{Apply\;the\;fraction\;rule\;\;$\dfrac{a}{\frac{b}{c}}=a \cdot \dfrac{c}{b}$}:\)
\(\boxed{\dfrac{\cos x}{\sin x} \cdot \sin x \cos x}=\cos^2 x\)
Cancel the common factor sin x, and apply the exponent rule aa = a²:
\(\cos^2 x=\cos^2 x\)
Related Questions
elementary surveying two ridges b and c forms a valley having a at the bottom part in between the ridges. elevation difference between b nad c is 111..356 m. observing at point a, the angle of elevation is
Answers
the difference of elevation between a and b.Let AB = x, tan 36 = x/111.356 ,x = 111.356 tan 36= 64.39 m and Difference in elevation = 64.39m
We are given that two ridges B and C form a valley with a point A at the bottom part with an elevation difference of 111.356m between the ridges and and angle of elevation of 36 degrees at point A. To find the difference of elevation between A and B, we use the formula of tangent which is tan θ = Opposite/Adjacent. We substitute the values and solve for x, which is the elevation difference between A and B. So the difference in elevation between A and B is 64.39m.
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3. Ditmas Middle School has 4,000 students. Montauk Middle School has
1/10 (one tenth) as many students as Ditmas Middle School. How many
students are there at Montauk Middle School.
Answers
Answer:
400 students
Step-by-step explanation:
4,000/10 = 400
Answer:
400
Step-by-step explanation:
help, please
thank you
i will brainlist
Answers
Answer:
The answer is √156
Step-by-step explanation:
I hope this helped and if it did I would appreciate it if you marked me Brainliest. Thank you and have a nice day!
Question 12(Multiple Choice Worth 5 points) (02.05 MC) Triangle ABC is transformed to obtain triangle A′B′C′: A coordinate grid is labeled from negative 12 to 0 to 12 on both x- and y-axes at increments of 1. Triangle ABC has A at ordered pair 2, 2, B at 2, 10, C at 8, 12. Triangle A prime B prime C prime has A prime at ordered pair negative 1, negative 1,B prime at negative 1, negative 5, C prime at negative 4, negative 6. Which statement is correct for Triangle ABC. and Triangle A prime B prime C prime.? Triangle ABC is similar to triangle A prime B prime C prime., because Triangle A prime B prime C prime. is obtained by dilating Triangle ABC. by a scale factor of 1 over 6. and then rotating it about the origin by 90 degrees Triangle ABC is similar to triangle A prime B prime C prime., because Triangle A prime B prime C prime. is obtained by dilating Triangle ABC. by a scale factor of 1 over 2. and then rotating it about the origin by 180 degrees Triangle ABC is similar to triangle A prime B prime C prime., because Triangle A prime B prime C prime. is obtained by dilating Triangle ABC. by a scale factor of 1 over 3. and then rotating it about the origin by 180 degrees Triangle ABC is similar to triangle A prime B prime C prime., because Triangle A prime B prime C prime. is obtained by dilating Triangle ABC. by a scale factor of 1 over 3. and then rotating it about the origin by 90 degrees
Answers
Answer:
Triangle A prime B prime C prime. is obtained by dilating Triangle ABC. by a scale factor of 1 over 2. and then rotating it about the origin by 180 degrees
hope it helps!
Step-by-step explanation:
Triangle ABC is similar to triangle A'B'C' because Triangle A'B'C' is obtained by dilating Triangle ABC by a scale factor of 1 over 3 and then rotating it about the origin by 180 degrees.
T
Radius i den store sirkelen er r.
Vis at summen av arealet til de fire hvite feltene kan skrives A = r²(2 – 2 ).
Løs oppgaven her.
Answers
Thank you :)
\(\pink{ \rule{1pt}{90000000pt}}\color{blue}{ \rule{1pt}{90000000pt}}\color{yellow}{ \rule{1pt}{90000000pt}}\purple{ \rule{1pt}{90000000pt}} \green{\rule{1pt}{90000000pt}}
\)
One rectangle is "framed" within another. Find the area of the shaded region if the "frame" is 3 units wide.
Answers
The area of the shaded region, with a frame of 3 units wide, is 264 square units. The inner rectangle is 6x4 units, and the outer rectangle is 18x16 units.
To solve the given problem, we have to find the area of the shaded region if the frame is 3 units wide. If the frame is 3 units wide, then the dimensions of the inner rectangle (the shaded region) will be (12 - 6) × (10 - 6) which simplifies to 6 × 4.
Therefore, we can say that the inner rectangle has a length of 6 units and a width of 4 units.The dimensions of the outer rectangle are (12 + 3 + 3) × (10 + 3 + 3) which simplifies to 18 × 16. Therefore, we can say that the outer rectangle has a length of 18 units and a width of 16 units.
The area of the shaded region can be obtained by subtracting the area of the inner rectangle from the area of the outer rectangle. Therefore, the Area of the outer rectangle = length × width= 18 × 16 = 288 square units
Area of the inner rectangle = length × width= 6 × 4 = 24 square units
Area of the shaded region = Area of the outer rectangle - Area of the inner rectangle = 288 - 24= 264 square units
Therefore, the area of the shaded region is 264 square units.
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Suppose you need to graph the function f(x) = x(4x + 7)(x - 1)^7. Are there going to be any repeated zeros?
Answers
Answer:
yes
Step-by-step explanation:
hello,
as we have
\((x-1)^7\)
the zero 1 is repeated 7 times
hope this helps
Answer:
yes
Step-by-step explanation:
The exponent on each factor tells you the number of times it is repeated. The zero represented by the factor (x -1)⁷ is repeated 7 times.
__
When a zero is repeated, the graph is flattened there. The more times it is repeated, the "flatter" the function is at that point. If the repeat-count is even, the graph touches the axis, but does not cross. Here, the repeat-count is odd, so the graph flattens out at the x-intercept, and crosses the x-axis.
A student answers 90% of the math questions on a math test correctly. If he answers 27 questions correctly how many questions are on the test?
Answers
So 30 questions on the test
27 ÷ 90 = 30
What is the solution of |x − 1| + 2 = 4? Enter your answers from least to greatest in the boxes.
Answers
Answer:
x=3 or -3
Step-by-step explanation:
hope it helps
The area is 20yd^2.Write and equation and solve it to find the value of T.
Answers
Given figure is a rectangle with length 4 yd and width t yd.
The area of the rectangle is
\(4t\text{ yd}\)Accordingly,
\(\begin{gathered} 4t=20 \\ t=5 \end{gathered}\)Hence, the value of t is 5.
Figure ABCD is a parallelogram. Is it also a rhombus? Why or why not?
D
5m
B
5 m
C
OA. It cannot be determined, because a parallelogram with congruent
adjacent sides may or may not be a rhombus.
B. No, because all four sides are congruent.
C. Yes, because adjacent sides are congruent.
D. No, because adjacent sides are congruent.
Answers
The true statement about the parallelogram is (c) Yes, because adjacent sides are congruent.
How to determine the true statement about the parallelogramGiven that
Figure ABCD is a parallelogram
Such that
Side lengths = 5 cm
This means that
The figure is a square
As a general rule
All squares can be classified as rhombus
This is because a rhombus is a quadrilateral with all four sides of equal length and square is a type of rhombus in which all four sides are equal in length
Hence, the true statement is (c)
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A sensor is used to monitor the performance of a nuclear reactor. The sensor accurately reflects the state of the reactor with a probability of .97. But with a probability of .02, it gives a false alarm (by reporting excessive radiation even though the reactor is performing normally), and with a probability of .01, it misses excessive radiation (by failing to report excessive radiation even though the reactor is performing abnormally).
What is the probability that a sensor will give an incorrect report, that is, either a false alarm or a miss?
To reduce costly shutdowns caused by false alarms, management introduces a second completely independent sensor, and the reactor is shut down only when both sensors report excessive radiation. (According to this perspective, solitary reports of excessive radiation should be viewed as false alarms and ignored, since both sensors provide accurate information much of the time.) What is the new probability that the reactor will be shut down because of simultaneous false alarms by both the first and second sensors?
Being more concerned about failures to detect excessive radiation, someone who lives near the nuclear reactor proposes an entirely different strategy: Shut down the reactor whenever either sensor reports excessive radiation. (According to this point of view, even a solitary report of excessive radiation should trigger a shutdown, since a failure to detect excessive radiation is potentially catastrophic.) If this policy were adopted, what is the new probability that excessive radiation will be missed simultaneously by both the first and second sensors?
Answers
1. The probability that the sensor will give an incorrect report is 0.03 or 3%.
2. The probability that the reactor will be shut down because of simultaneous false alarms by both sensors is 0.0004 or 0.04%.
3. The probability that excessive radiation will be missed simultaneously by both sensors is 0.0001 or 0.01%.
1. Probability of the sensor giving an incorrect report (false alarm or miss):The probability of a false alarm is 0.02.
The probability of a miss is 0.01.
P(false alarm or miss)
= P(false alarm) + P(miss)
= 0.02 + 0.01 = 0.03
Therefore, the probability that the sensor will give an incorrect report is 0.03 or 3%.
2. Probability of simultaneous false alarms by both sensors:
P(false alarm by both sensors)
= P(false alarm by sensor 1) * P(false alarm by sensor 2)
= 0.02 * 0.02
= 0.0004
Therefore, the probability that the reactor will be shut down because of simultaneous false alarms by both sensors is 0.0004 or 0.04%.
3. Probability of missing excessive radiation by both sensors:
P(miss by both sensors)
= P(miss by sensor 1) * P(miss by sensor 2)
= 0.01 * 0.01
= 0.0001
Therefore, the probability that excessive radiation will be missed simultaneously by both sensors is 0.0001 or 0.01%.
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A farmer needs to fence a rectangular piece of land. She wants the length of the field to be 80 feet longer than the width. If she has 1080 feet of fencing material, what should be the length and width of the field?
Answers
The width of the field is 230 feet and the length is 310 feet.
Let's denote the width of the field as "x" feet.
The length of the field is 80 feet longer than the width, so it can be represented as "x + 80" feet.
To find the total amount of fencing material needed, we sum up the lengths of all four sides of the rectangular field:
2(length) + 2(width) = perimeter
Substituting the given values:
2(x + 80) + 2(x) = 1080
2x + 160 + 2x = 1080
4x + 160 = 1080
4x = 920
x = 920/4
x = 230
Therefore, the width of the field is 230 feet.
Now we can find the length by adding 80 feet to the width:
Length = Width + 80 = 230 + 80 = 310 feet.
So, the width of the field is 230 feet and the length is 310 feet.
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Please,can you help me with this one?
Parker has 4 different piggy banks
one each for pennies, nickels,dimes, and quarters
If there are twenty coins in each of his 4 piggy banks,
what is the average amount of money contained in
each piggy back?
Answers
Answer:
Just multiply two with the value of one penny,one,nickel,one dimes,one quarters and add them you will get the ans.
Name the image of C(6,-4) under a rotation 90° about the origin. PLEASE HELP ME :)
Answers
Answer:
(4, 6)
Step-by-step explanation:
90 degree goes counterclockwise so
(6,-4) becomes( 4, 6)
(x,y) rotation 90 or -270 is (-y,x)
Multiply: 3,217 × 5,716
Answers
Answer:
Step-by-step explanation:
183,883,72
Answer:
18388372
Step-by-step explanation:
1 )Use the algorithm method.
3 2 1 7
× 5 7 1 6
1 1 1 4
1 9 3 0 2
3 2 1 7 0
2 1 1 4
2 2 5 1 9 0 0
1 1 3
1 6 0 8 5 0 0 0
1 1 1
1 8 3 8 8 3 7 2
==================step 2==================2 )Therefore, 3217 × 5716 = 18388372.
18388372
help meeeeee please
Answers
Find the equation of the line that has a slope of 2 and passes through the point (-8,-1). Write the equation in point-slope form.
Answers
The equation of the line in point-slope form is y + 1 = 2(x + 8).
What is equation of line?
An equation with terms that have variables x and y independently and a degree of one is referred to be a line equation in a two-dimensional coordinate system. A linear equation is any equation with degree 1. The relationship between the x and y coordinates of any point on a straight line is known as the equation of a straight line.
The point-slope form of the equation of a line with slope m and passing through the point (x1, y1) is given by:
y - y1 = m(x - x1)
Using the given information, we have:
m = 2
x1 = -8
y1 = -1
Substituting these values into the point-slope form, we get:
y - (-1) = 2(x - (-8))
Simplifying, we get:
y + 1 = 2(x + 8)
This is the equation of the line in point-slope form.
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Solve the given initial-value problem. dy + 2(t + 1)y2 = 0, y(0) dt 1 15 1 y(t) = 2 + 2t + 15 x Give the largest interval I on which the solution is defined. (Enter your answer using interval notation.)
Answers
The Largest interval in which the solution of the differential equation is defined is (-5, 3).
dy/dt + 2(t + 1)y² = 0 and y(0) = -1/15
Now, dy/dt + 2(t + 1)y² = 0
dy/dt = -2(t + 1)y²
By separating variables
dy/y² = -2(t + 1) dt
Integrating both sides
∫dy/y² = -∫(t + 1) dt
-1/y = -(t²+2t) + c
y(t) = /(t²+2t - c)
We know, y(0) = -1/15
So, -1/15 = 1/(0+0-c)
c = 15
∴ y(t) = 1/(t² + 2t - 15)
Now, y(t) is defined if t² + 2t - 15 ≠ 0
Let t² + 5t - 3t - 15 = 0
(t + 5)(t - 3) = 0
t = -5, 3
So, the intervals are (-∞ , -5), (-5, 3) and (3, ∞)
As t = 0 lies in between (-5, 3) hence the interval in which the solution is defined is (-5, 3)
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If 5 children share 1/3 pound of candy equally, how much candy will each child get?
Answers
Answer:
Each child will receive \(\frac{1}{15}\) of a pound
Step-by-step explanation:
\(\frac{1}{5*3}\)
the answer for these questions
Answers
The triangles are similar by SSS congruence of triangles.
What is SSS congruence of triangles?
The two triangles are congruent if the three sides of one triangle are the same as the corresponding three sides (SSS) of the other triangle.
We are given two triangle ABC and PQR.
In order to see whether these are similar or not, we will use the SSS congruence of triangles.
The ratio of sides are:
⇒AB : PQ = 8 : 6
⇒AB : PQ = 4 : 3
Similarly,
⇒BC : QR = 12 : 9
⇒BC : QR = 4 : 3
Similarly,
⇒AC : PR = 12 : 9
⇒AC : PR = 4 : 3
Since, the ratio of all the sides is same so, the triangles are similar.
Hence, the triangles are similar by SSS congruence of triangles.
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Since, there are multiple questions so, the question answered above is attached below
URGENT DUE IN FEW MINS6th grade math Calculate to TOTAL Surface Area.
Do NOT count the surface under the pyramid.
Answers
The surface area of the solid is 281 ft².
Given that, a solid figure made of a rectangular prism and a square pyramid,
We need to find its surface area,
To find the same, we will find the lateral surface area of square pyramid, and total surface area of rectangular prism,
So,
Surface area = 2×side×slant height + 2(length·width+width·height+height·length)-base area of the square pyramid,
= 2×5×5 + 2(4·12+12·5+4·5)-5²
= 50+256-25
= 281 ft²
Hence the surface area of the solid is 281 ft².
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3.) The volume of a cylinder, V(h), is given by the area of the base times the height, h. The base of the cylinder shown below is 5 inches squared.
Part A: Write the function, V(h), that represents the volume of the cylinder.
Part B: Find V(3)
Part C: What does V(3) represent?
Answers
Step-by-step explanation:
The volume of a cylinder is given by :
\(V=\pi r^2 h\) ...(1)
r is radius and h is height of the cylinder
The diameter of a cylinder = 5 inches
Radius = 5/2 = 2.5 inches
Part A :
Put r = 2.5 inch in equation (1)
\(V=\pi (2.5)^2 h\\\\=6.25\pi h\ \text{inches}^3\) .....(2)
Part B :
Put h = 3 in equation (2) as follows :
\(V(3)=6.25\pi \times 3\\\\=18.75\pi \text{inches}^3\)
Part C:
V(3) represents the volume of the cylinder when its height is 3 inches.
Hence, this is the required solution.
SAT Math scores have a bell-shaped distribution with a mean of 515 and a standard deviation of 114.
1) What percentage of SAT scores is between 287 and 857?
2) What percentage of SAT scores is greater than 629?
Please explain!
Answers
1) For our first job, PROBLEM A, it has a mean of 515 points and a standard deviation of 114 points. We're given the goal of calculating what % belongs between 401 and 629. As we can see, if we remove 515, then subtract 114 from 515, we obtain our first value of 401, then if we add our standard deviation tournament, we get our second number of 629. So, using the empirical rule, we know that in this scenario, 68 of our data points will fall inside this range.
2) Part B requires us to identify the scores that are less than 401 and larger than 629. So, in part A, we have the inclusive section of this, and now we have the exclusive section. Simply remove 60 percent from 100 percent, and we get 32 percent outside of this range.
Eric is x years old. Eric's sister is 5 years older than Eric. His mother is 3 times his sister's age. His father is x years older than his mother. How old is Eric's father if he was 8 years old? How old is Eric's mother is he was 8 years old?
Answers
Answer:
Eric´s mother is 39 years old.
Step-by-step explanation:
Eric is 8, so 8+5 = 13 then 13x3 = 39
Answer:
Eric mother was 39 years old and the father is 47 years old.
Step-by-step explanation:
I hope this helped.
What is the mean, median and mode for...45,50,55,55,55,60,60,60,65,65,70?
Answers
Answer:
Mean = 58.18
Median = 60
Modes = 55 and 60
Step-by-step explanation:
To find the mean/average, add up all the numbers and divide the sum by the # of numbers there are (ex. there are 11 numbers so divide the sum by 11).
The median is the middle number of a set. In this case, the middle number is easy to see because the # of numbers is odd, but if it was even, just find the average of the two middle numbers.
The mode is/are the most frequent number(s) in the set. In this set, the numbers that appear the most are 55 and 60.
Ms. Marcus purchased 5 pounds of grapefruit for $4.90. At this price how much will 7 pounds of grapefruit cost?
Answers
Answer: I thinnk the answer is 6.86
4.90 divided by 5=0.98
0.98 times 7=6.86
Step-by-step explanation:
The difference of two numbers is 596 The smaller number is 716 what is the other number?
Answers
Answer
1,312 is the other number
Answer:1,312
Step-by-step explanation: if you have 716 and you add 596 to create the answer of 1,312
what is x divided by one
Answers
Answer:
\( x \div 1\)
\( = x\)
Answer:
\(x\div 1=x\)
Step-by-step explanation:
When x is divided by one it is called reciprocal.
reciprocal is the inverse of a number or a value.
examples: The reciprocal of 3 is 1/3, and the reciprocal of 5 is 1/3.
OAmalOHopeO
Which ordered pair is a solution to the system of linear equations 2x 3y equals 6?
Answers
The ordered pair which is a solution to the system of linear equations is (0,2).
What is linear equation?
A linear equation is one that has a degree of 1 as its maximum value. No variable in a linear equation, thus, has an exponent greater than 1. A linear equation's graph will always be a straight line.
Given equations are
2x + 3y = 6 ....(i)
-3x + 5y = 10 ....(ii)
Multiply 3 with equation (i) and multiply 2 with equation (ii)
6x + 9y = 18 ......(iii)
-6x + 10y = 20 ......(iv)
Add equation (iii) and (iv)
6x + 9y -6x + 10y = 18 + 20
19y = 38
y = 2
Putting y = 2 in equation (i)
2x + 6 = 6
2x = 0
x = 0
The solution is (0,2).
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