Trigonometric substitution may useful when integrating the integrals root of a sum or difference of two squares.
In arithmetic, trigonometric substitution is the substitution of trigonometric features for other expressions. In calculus, trigonometric substitution is a way of comparing integrals.
Moreover, one can also use trigonometric identities to simplify sure integrals containing radical expressions.
You can use trig substitution even when you do not have square roots. particularly, if you have an integrand that looks like an expression in the square roots proven inside the above table, then you can use trig substitution.
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The angles of elevation of the top of a tower from two points at a distance of 4 m and 9 m from the base of the tower and in the same straight line with it are complementary Prove that the height of the tower is 6 m.
Kindly help!
Answer:
Proof explained below
Step-by-step explanation:
Tan trigonometric ratio
[tex]\sf \tan(\theta)=\dfrac{O}{A}[/tex]
where:
[tex]\theta[/tex] is the angleO is the side opposite the angleA is the side adjacent the angleDraw a diagram to help visualize the scenario (see attached).
If the angles are complementary, their sum is 90°.
Therefore, let one of the angles be [tex]x[/tex] and the other angle be [tex]90^{\circ}-x[/tex].
Use the tan trig ratio to create two equations with the missing side length.
Right triangle ACD
Given:
[tex]\theta = x[/tex]O = CD (tower)A = 9 mSubstituting the given values into the tan trig ratio:
[tex]\implies \tan (x)= \dfrac{\textsf{CD}}{9}[/tex]
Right triangle BCD
Given:
[tex]\theta = 90^{\circ} - x[/tex]O = CD (tower)A = 4 mSubstituting the given values into the tan trig ratio:
[tex]\implies \tan (90^{\circ} - x)= \dfrac{\textsf{CD}}{4}[/tex]
[tex]\textsf{As }\:\tan(90^{\circ} - x)= \cot (x):[/tex]
[tex]\implies \cot(x)= \dfrac{\textsf{CD}}{4}[/tex]
[tex]\implies \dfrac{1}{\tan(x)}= \dfrac{\textsf{CD}}{4}[/tex]
[tex]\implies \tan(x)= \dfrac{4}{\textsf{CD}}[/tex]
Therefore, the two equations are:
[tex]\textsf{Equation 1}: \quad \tan (x)= \dfrac{\textsf{CD}}{9}[/tex]
[tex]\textsf{Equation 2}: \quad \tan(x)= \dfrac{4}{\textsf{CD}}[/tex]
Substitute one equation into the other and solve for CD:
[tex]\implies \sf \dfrac{CD}{9}=\dfrac{4}{CD}[/tex]
[tex]\implies \sf CD \cdot CD= 4 \cdot 9[/tex]
[tex]\implies \sf CD^2=36[/tex]
[tex]\implies \sf CD=\sqrt{36}[/tex]
[tex]\implies \sf CD= \pm 6[/tex]
As distance cannot be negative, CD = 6 m only, thus proving that the height of the tower is 6 m.
You enter a room. 2 dogs, 4 horses, 1 giraffe and a duck lie on the bed. 3 hens fly over a chair. How many legs are there on the floor?.
Answer:
28 legs are on the floor once again
There are no legs on the floor.
Hence answer is zero.
According to the question,
For 2 dogs: Each dog has 4 legs.
Therefore, 2 dogs have a total of 8 legs.
For 4 horses: Each horse has 4 legs.
Therefore, 4 horses have a total of 16 legs.
For 1 giraffe: A giraffe has 4 legs.
For 1 duck: A duck has 2 legs.
For 3 hens: Each hen has 2 legs.
Therefore, 3 hens have a total of 6 legs.
To get the total number of legs on the bed,
we simply add up all the legs of the animals on the bed,
⇒ 8 legs + 16 legs + 4 legs + 2 legs + 6 legs = 36 legs.
Now, to get the number of legs on the floor,
We need to subtract the number of legs on the bed from the total number of legs of all the animals,
For 2 dogs with a total of 8 legs.
For 4 horses with a total of 16 legs.
For 1 giraffe with 4 legs.
For 1 duck with 2 legs.
For 3 hens with a total of 6 legs.
So, the total number of legs of all the animals is,
⇒ 8 legs + 16 legs + 4 legs + 2 legs + 6 legs = 36 legs.
Since all the animals are on the bed,
There are no legs on the floor.
Therefore, the answer is 0 legs on the floor.
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Solve Triangle DEF. - Math
Answer:
[tex]p = 52.942259755149776[/tex]
Step-by-step explanation:
This is basic trigonometry.
[tex] \cos(39) = \frac{e}{22} [/tex]
[tex]e = \cos(39) \times 22[/tex]
[tex] \sin(39) = \frac{d}{22} [/tex]
[tex]d = \sin(39) \times 22[/tex]
And finally perimeter (p):
[tex]p = 22 + e+ d[/tex]
how to solve this pleaseee
Answer:
if you divide 10 by 8 it should be 10/8 so a
Hope This Helps!!!
Answer: A
Step-by-step explanation: Step 3 is incorrect as it should be 10/8 (or 5/4 simplified.) When we have 8x = 10, we are looking at what number "x" multiplied by 8 gives 10. Meaning we need to have 8(?) = 10. So, to find out what x is, we just divide by 8 from both sides. Since 8x/8 cancels out and becomes just x, we get x = 10/8 since we are dividing 10 by 8. We get 5/4 simplified or 10/8 unsimplified.
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Which sequence of transformations produced the image?
a counter-clockwise rotation 90° about the origin followed by a translation 2 units to the right and 1 unit up
a translation 2 units to the right and 1 unit up followed by a counter-clockwise rotation 90° about the origin
a clockwise rotation 90° about the origin followed by a reflection across the y-axis
a reflection across the y-axis followed by a clockwise rotation 90° about the origin
The sequence of transformation is; B. A translation 2 units to the right and 1 unit up followed by a counter-clockwise rotation 90° about the origin.
What is the sequence of Transformation?From the given image in the transformation, we can see that the pre image which is the original image is on the right hand side while the image after transformation is on the left hand side.
Now it will be noticed that in the pre image, all the x and y coordinates are positive while the image after transformation has all the x-coordinates as negative with the y-coordinates as positive.
This transformation when critically examined is seen to be a translation 2 units to the right and 1 unit up followed by a counter-clockwise rotation 90° about the origin.
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Answer: its B
Step-by-step explanation:
helpp me plss! geometry, how to find x. i kinda need step by step for better understanding, if possible, too pleaseee
Answer:
Step-by-step explanation:
help me please ty ty
Answer:
A
Step-by-step explanation:
(f - g)(x)
= f(x) - g(x)
= 3x² + 4x - 6 - (6x³ - 5x² - 2) ← distribute parenthesis by - 1
= 3x² + 4x - 6 - 6x³ + 5x² + 2 ← collect like terms
= - 6x³ + 8x² + 4x - 4
A 3x2 flag is divided into six squares. Each square is to be coloured green or blue, so that every square shares at least one edge with another square of the same colour. In how many ways can this be done?
The computation shows that the number of ways that this can be done is 64.
How to illustrate the information?From the information given, it's was stated that a 3x2 flag is divided into six squares and that each square is to be coloured green or blue.
In this case, the number of ways that this can be illustrated will be:
= 2^n
= 2^6
= 64
Therefore, there are 64 ways.
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These two figures shown are congruent. Which statement is true?
Answer:
The answer is C
Step-by-step explanation:
First if you look at the problem you will notice the triangles aren't the same size so if it's neither rotation nor translation then the shapes do not
Expand and combine like terms. (5a^3 - 2)(5a^3+2)
Answer:
[tex]25a^6-4[/tex]
Step-by-step explanation:
So if you look at the equation, you'll notice it's being multiplied by it's conjugate which can be expressed as: x+a and x-a. See how only the sign changes? Well if remember the differences of squares, it looks like that, because it is. Essentially the difference of squares says that: [tex]a^2-b^2=(a-b)(a+b)[/tex]. So expanding this out will result in [tex](5a^3)^2-(2)^2[/tex] which simplifies to: [tex]25a^6-4[/tex]. If you're confused on how I got 25a^6. Try to think of a^3 as three a's rather than simply an exponent. and since it has a coefficient of 5. 5a^3 is the same as (5 * a * a * a). Now if you square this you have (5 * a * a * a) * (5 * a * a * a). So if you group like terms you'll end up with 6 a's being multiplied by each other (which can be expressed as a^6) and 5 * 5 which is 25.
The multiplication can be transformed into the difference of squares using the rule:(a−b)(a+b)=a²-b² . Gets the square of 2.
(5a³)²-4
Expand (5a³)².
5² (a³)² -4
To raise a power to another power, multiply the exponents. Multiply 3 and 2 to get 6.
5²a⁶-4
25a⁶-4 ====> Answer
{ Pisces04 }
Alyssa's favorite bowling team has 10 players on them. She has taken a picture with every single athlete individually and wants to post some of the pictures on her wall. But she only wants to post 4 pictures on her wall. How many different sets of pictures can she post on her wall
There are 84 sets of pictures she can post on her wall.
Since Alyssa has taken a picture with every single athlete individually. It means she has taken 9 pictures.
Now she wants to post 3 pictures on her wall.
We need to use Combination for arranging pictures. A combination is a mathematical technique that determines the number of possible arrangements in a collection of items where the order of the selection does not matter. In combinations, we can select the items in any order.
Therefore, The number of different sets of pictures = Selecting 3 pictures from a set of 9 pictures
9C3 = (9!)/((9-3)!*3!)
(9*8*7)(3*2*1)= 84 ways
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the intersection of a line and a triangle can be
Answer:
nothing
a point
a segment
Step-by-step explanation:
nothing
a point
a segment
The number of students who attend a school could be divided among 10, 12, or 16 buses, such that each bus transports an equal number of students. What is the minimum number of students that could attend the school
Answer:
240 students
Step-by-step explanation:
Least common multiple:To find the minimum number of students, we have to find the LCM of 10, 12, 16
10 = 2 * 5
12 = 2 * 2 * 3
16 = 2 * 2 * 2 * 2
LCM = 2 * 2 * 2 * 2 * 3 * 5
= 240
which box is in the wrong spot?
The first function isn't decreasing over the largest interval, so the box in the wrong spot is the first one.
Which box is in the wrong spot?
Here we have 3 cubic functions.
A function is decreasing when reading from left to right, the curve goes downwards.
Now, if you look at the graph, you can see that the interval where the function decreases is really small (the graph is small so I can't see the exact interval).
While if you look at g(x), you can see that, at least in the interval [-2, 2] the function is decreasing.
So the box in the wrong spot is the first box.
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Draw a graph of the line that contains the given point and
(-3,0); m =
1
8
Answer:
Step-by-step explanation:
how to reduce this 25/35 please please help explain !!!!!!
Answer:
5/7
Step-by-step explanation:
Note that the highest common factor to the numbers are 5. Therefore, to simplify will be:
= 25/35
= 5/7
check all that apply.
if secØ= 13/12, then:
Based on trigonometric formulas, we can find that the options that applies to the given trigonometric relationship are tan θ = 5/12 and cos θ = 12/13.
How to find the exact form of trigonometric functions
There six trigonometric functions, which are related by a group of expressions, which will be used in this question:
[tex]\cos \theta = \frac{1}{\sec \theta} = \frac{12}{13}[/tex]
[tex]\sin \theta = \sqrt{1 - \cos^{2}\theta}[/tex]
[tex]\sin \theta = \sqrt{1 -\left(\frac{12}{13} \right)^{2}}[/tex]
[tex]\sin \theta = \frac{5}{13}[/tex]
[tex]\csc \theta = \frac{1}{\sin \theta} = \frac{13}{5}[/tex]
[tex]\tan \theta = \sqrt{\sec^{2}\theta - 1}[/tex]
[tex]\tan \theta = \sqrt{\left(\frac{13}{12} \right)^{2}-1}[/tex]
[tex]\tan \theta = \frac{5}{12}[/tex]
Based on trigonometric formulas, we can find that the options that applies to the given trigonometric relationship are tan θ = 5/12 and cos θ = 12/13.
Remark
The statement is incomplete. Complete form is shown below:
Check all that apply:
If sec θ = 13/12, then:
a) sin θ = 12/13
b) tan θ = 5/12
c) cos θ = 12/13
d) csc θ = 12/13
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A normal score is the expected z-score of a data value, assuming the distribution of the random variable is normal. Is this statement true or false?.
The given statement about normal score that it is expected z score of a data value is true.
Given A statement that a normal z score is the expected z score of a data value, assuming the distribution of the random variable is normal.
We have to tell whether the given statement is true or false.
In z test the z score or standard score is the number of standard deviations by which the value of a raw score is above or below the mean value of what is observed or measured.
A normal distribution is arrangement of a data set in which most values cluster in the middle of the range and the rest taper off symmetrically toward either extreme.
A random variable is a variable which is identified without following any criteria.
A Z test is any statistical test for which the distribution of the test statistic under the null hypothesis ca be approximated by a normal distribution.
Hence the given statement about normal score is absolutely true.
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The cylinder below has a volume of 785 Feet cubed, and the cone has a volume of 314 Feet cubed.
A cylinder has a volume of 78.5 feet squared and a cone has a volume of 78.5 feet squared.
Which explains which figure has the greater height?
The cylinder has a greater height because the bases are the same, but the cylinder has a greater volume.
The cylinder has a greater height because the volume of the cylinder is greater than twice the volume of the cone.
The cone has a greater height because the bases are the same, but the volume of the cylinder is less than 3 times the volume of the cone.
The cone has a greater height because the bases are the same, but the volume of the cylinder is less than 4 times the volume of the cone.
The cone has a greater height because the bases are the same, but the volume of the cylinder is less than 3 times the volume of the cone.
What is the relationship between the cone and the cylinder?The volume of a cylinder is three times that of a cone.
Volume of a cylinder = πr²h
Volume of a cone = 1/3πr²h
π = 22/7r = radiush = heightIf the height of the cylinder would be greater than that of the cone, the volume of the cylinder should be at least 3 times that of the cone.
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A scuba diver is 30 ft below sea level. Which of the following best describes this situation?
1. |30|
2. |-30|
3. -30
4. ±30
Answer:
-30
Step-by-step explanation:
The scuba driver is below the sea and ground level which is 0 ft, and he dives down 30 ft so it should be negative.
Hope it helps!
The word fable derives from the Latin word fabula, which originally meant about the same as the Greek mythos. Like mythos, it came to mean a fictitious or untrue story. Myths, in contrast, are not presented as fictitious or untrue. Fables, like some myths, feature personified animals or natural objects as characters.
The information given about fables and myths is correct. Therefore, the correct option is true.
What is a fable?It should be noted that a fable simply illustrates a story that's considered false or fictional.
In this case, fable derives from the Latin word fabula, which originally meant about the same as the Greek mythos.
Myths are typically not presented as fictitious or untrue. Therefore, the correct option is true.
Complete question:
The word fable derives from the Latin word fabula, which originally meant about the same as the Greek mythos. Like mythos, it came to mean a fictitious or untrue story. Myths, in contrast, are not presented as fictitious or untrue. Fables, like some myths, feature personified animals or natural objects as characters. True or false.
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A sports company conducted a road test of a new model of a geared bike. the test rider cycled (3x − 2) miles on a flat road, (x2 − 5) miles uphill, and (2x 7) miles downhill. which simplified expression is equivalent to the total distance, in miles, for which the bike was tested? x2 − x x2 5x x2 − x 14 x2 5x 14
The simplified equation equivalent to the total distance in miles is given as,
[tex]x^{2} +5x[/tex]
What is the equation representing the total distance?
It is given that in a road test of a geared bike,
The distance covered on a flat road = (3x - 2)
The distance covered uphill = (x² - 5)
The distance covered downhill = (2x + 7)
Total distance covered in miles as equation,
(3x - 2) + (x² - 5) + (2x + 7)
= 3x - 2 + x² -5 + 2x + 7
= 5x - 7 + x² + 7
= x² + 5x
Thus, the final equation representing the total distance covered in miles by the bike is (x² + 5x)
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A diver is standing on a spring board 24 feet above the pool. She jumps from the platform with an initial upward velocity of 8feet / (sec(or)) . Use the formula d(t)=-16(2t; -3)(t+1) , where d is the height of the diver above the water and t is the time in seconds. How long will it take for her to hit the water?
It would take 1.5 seconds for the diver standing on a spring board 24 feet above the pool to hit the water.
What is an equation?An equation is an expression that shows the relationship between two or more numbers and variables.
Given that d represent the height of the diver above the water and t is the time in seconds.
d(t) = -16(2t - 3)(t + 1)
The diver hit the water at d(t) = 0, hence:
0 = -16(2t - 3)(t + 1)
2t - 3 = 0; or t + 1 = 0
t = 1.5 seconds
It would take 1.5 seconds for the diver standing on a spring board 24 feet above the pool to hit the water.
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Which part of the proof does the fourth statement and reason represent?
Given: AB ≅ BC
BC ≅ EF
Prove: EG = AB + FG
A line contains point A, B, C.
A 2-column table has 6 rows. Column 1 is labeled Statements with entries line segment A B is-congruent-to line segment B C, line segment B C is congruent to line segment E F, line segment A B is-congruent-to line segment E F, A B = E F, E G = E F + F G, E G = A B + F G. Column 2 is labeled reasons with entries given, given, transitive property, definition of congruent segments, segment addition, and substitution.
It is part of the given information.
It is part of the argument.
It is the conclusion.
It is an assumption.
Mark this and return
The proof does the fourth statement and reason represent It is an argument.
AB≅ BC and BC≅EF
What is the transitive property?a = b and b = c then, a =c
ABC is the side of the triangle
⇒AB≅EF
AB = EF [def of segment]
By Segment Addition Postulate states that given two points A and C, and a third point B lies on the line segment AC if and only if the distances between the points satisfy the equation
AB + BC = AC.
Since, the line segment EG, F lies on the line segment;
then, by segment addition postulates we have;
EG = EF + FG
By substitution AB=EF
⇒ EG = AB + FG hence proved!
In the fourth statement, the reason is: It is an argument.
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The diagram below shows the partial construction of which shape?
Answer:
im gonna say circle or even a hour glass
Step-by-step explanation:
URGENT: statistics: A researcher wishes to estimate the percentage of adults who support abolishing the penny what size sample should be obtained if he wishes the estimate to be within three percentage points with a 95% confidence if he:
A: uses a previous estimate of 26%?
B: he does not use any prior estimates?
Using the margin of error for the z-distribution, the sample sizes are given as follows:
a) 822.
b) 1068.
What is a confidence interval of proportions?A confidence interval of proportions is given by:
[tex]\pi \pm z\sqrt{\frac{\pi(1-\pi)}{n}}[/tex]
The margin of error is given by:
[tex]M = z\sqrt{\frac{\pi(1-\pi)}{n}}[/tex]
In which:
[tex]\pi[/tex] is the sample proportion.z is the critical value.n is the sample size.We have a 95% confidence level, hence[tex]\alpha = 0.95[/tex], z is the value of Z that has a p-value of [tex]\frac{1+0.95}{2} = 0.975[/tex], so the critical value is z = 1.96.
Item a:
The estimate is of [tex]\pi = 0.26[/tex], hence we solve for n when M = 0.03.
[tex]M = z\sqrt{\frac{\pi(1-\pi)}{n}}[/tex]
[tex]0.03 = 1.96\sqrt{\frac{0.26(0.74)}{n}}[/tex]
[tex]0.03\sqrt{n} = 1.96\sqrt{0.26(0.74)}[/tex]
[tex]\sqrt{n} = \frac{1.96\sqrt{0.26(0.74)}}{0.03}[/tex]
[tex](\sqrt{n})^2 = \left(\frac{1.96\sqrt{0.26(0.74)}}{0.03}\right)^2[/tex]
n = 821.2
A sample of 822 is needed.
Item b:
No prior estimate, hence [tex]\pi = 0.5[/tex].
[tex]M = z\sqrt{\frac{\pi(1-\pi)}{n}}[/tex]
[tex]0.03 = 1.96\sqrt{\frac{0.5(0.5)}{n}}[/tex]
[tex]0.03\sqrt{n} = 1.96\sqrt{0.5(0.5)}[/tex]
[tex]\sqrt{n} = \frac{1.96\sqrt{0.5(0.5)}}{0.03}[/tex]
[tex](\sqrt{n})^2 = \left(\frac{1.96\sqrt{0.5(0.5)}}{0.03}\right)^2[/tex]
n = 1067.11
A sample of 1068 is needed.
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Which answer describes the type of sequence?
3, 4, 7, 11,
• geometric
•
arithmetic
neither arithmetic nor geometric
Hello and Good Morning/Afternoon!
To determine whether a sequence is arithmetic or geometric:
⇒ must consider their definitions
Arithmetic Sequence: sequences that go up by adding or subtracting a numberGeometric Sequences: sequences that go up by multiplying or dividing a numberLet's test out this sequence to see what kind of sequence it is:
Testing for arithemetic sequence3 ⇒ 4, 3 + 1
4 ⇒ 7, 4 + 3
7 ⇒ 11, 7 + 4
NOT AN ARITHMETIC SEQUENCE
Testing for geometric sequence3 ⇒ 4, 3 * (4/3)
4 ⇒ 7, 4 * (7/4)
7 ⇒ 11, 7 * (11/7)
NOT AN GEOMETRIC SEQUENCE
Answer: Neither arithetic nor geometric
Hopefully that helps!
Find the value of x.
A) 71
B) 72
C) 73
D) 75
Answer:
The answer is c) 73
Step-by-step explanation:
The angle 146 and (-39 + x) lie on the same line, so that means their sum is 180 degrees because a straight line has 180 degrees.
180 = 149 - 39 + x
180 = 107 + x
180 - 107 = x
73 = x
Answer:
73
Step-by-step explanation:
The two angles form a straight line and a straight line is 180 degrees
-39+x + 146 = 180
Combine like terms
107 +x = 180
Subtract 107 from each side
107+x-107 = 180-107
x =73
Does anyone know what the answer to this
Answer:
60 degrees
Step-by-step explanation:
Since all three sides are the same length, all three angles must be the same size. This means the triangle is an equilateral triangle
The angles are 60 degrees
Answer:
Hello! The answer is 60°.
Step-by-step explanation:
As marked by the dashes on each side, the sides are congruent. If the sides are congruent, this means that the angle measures are equal.
Having three equal sides is an equilateral triangle, and the total angle measure sum is 180.
To find the measure of angle y, you will have to divide the total angle measure by the number of angles:
180 ÷ 3 = 60
The measure of angle y = 60°
-6 + 6 how do you solve it
Answer:
0.
Step-by-step explanation:
-6+6=0 because any negative number plus the same positive number cancels each other out.