[tex]\therefore \sf{ LHS = \dfrac{tan \theta}{1 – cot \theta } + \dfrac{cot \theta}{1 – tan \theta}}[/tex]
[tex]\\[/tex]
[tex]\therefore\sf{ LHS = \dfrac{tan \theta}{1 – \dfrac{1}{tan \theta}} + \dfrac{ \dfrac{1}{tan \theta}}{1 – tan \theta} }[/tex]
[tex]\\[/tex]
[tex]\therefore\sf{ LHS = \dfrac{ tan^{2} \theta }{tan \theta – 1} + \dfrac{1}{tan \theta (1–tan \theta) } }[/tex]
[tex]\\[/tex]
[tex] \therefore\sf{ LHS = \dfrac{tan^{2} \theta}{tan \theta–1} – \dfrac{1}{tan \theta(tan \theta – 1 )}}[/tex]
[tex]\\[/tex]
[tex]\therefore \sf{LHS = \dfrac{tan^{3}\theta–1}{tan \theta(tan \theta –1)}}[/tex]
[tex]\\[/tex]
[tex]\therefore\sf{LHS = \dfrac{tan^{3}\theta–(1)^{3}}{tan\theta(tan\theta–1)}}[/tex]
[tex]\\[/tex]
[tex] \therefore\sf{ LHS = \dfrac{ (tan \theta -1)(tan^{2}\theta+tan\theta .1 + (1)^{2})} {tan \theta (tan\theta(tan \theta –1 )}}[/tex]
[tex]\\[/tex]
[tex] \therefore\sf{ LHS = \dfrac{(tan^{2} \theta×tan\theta .1+1)}{tan \theta}}[/tex]
[tex]\\[/tex]
[tex] \therefore\sf{ LHS = \dfrac{(sec^{2}\theta+tan\theta)}{tan\theta}}[/tex]
[tex]\\[/tex]
[tex] \therefore\sf{ LHS = \dfrac{sec^{2}\theta}{tan\theta} + \dfrac{tan \theta}{tan\theta}}[/tex]
[tex]\\[/tex]
[tex] \therefore\sf{ LHS = sec^{2}\theta \times cot \theta + 1 }[/tex]
[tex]\\[/tex]
[tex] \therefore\sf{ LHS = \dfrac{1}{cos^{2}\theta} \times \dfrac{cos \theta }{sin\theta}+1}[/tex]
[tex]\\[/tex]
[tex] \therefore\sf{ LHS = \left( \dfrac{ 1}{cos \theta} \times \dfrac{1}{sin \theta} \right) + 1}[/tex]
∴ LHS = 1 + secθ.cosecθ = RHS━━━━━━━━━━━━━━━━━━━━━━━We have proven that the trigonometric identity [(tan θ)/(1 - cot θ)] + [(cot θ)/(1 - tan θ)] equals 1 + (secθ * cosec θ)
How to solve Trigonometric Identities?We want to prove the trigonometric identity;
[(tan θ)/(1 - cot θ)] + [(cot θ)/(1 - tan θ)] = 1 + sec θ
The left hand side can be expressed as;
[(tan θ)/(1 - (1/tan θ)] + [(1/tan θ)/(1 - tan θ)]
⇒ [tan²θ/(tanθ - 1)] - [1/(tan θ(tanθ - 1)]
Taking the LCM and multiplying gives;
(tan³θ - 1)/(tanθ(tanθ - 1))
This can also be expressed as;
(tan³θ - 1³)/(tanθ(tanθ - 1))
By expansion of algebra this gives;
[(tanθ - 1)(tan²θ + tanθ.1 + 1²)]/[tanθ(tanθ(tanθ - 1))]
Solving Further gives;
(sec²θ + tanθ)/tanθ
⇒ sec²θ * cotθ + 1
⇒ (1/cos²θ * cos θ/sin θ) + 1
⇒ (1/cos θ * 1/sin θ) + 1
⇒ 1 + (secθ * cosec θ)
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Determine the equation of the circle with centre (-3;5) and passing through a point (2;-9)
Therefore, the equation of circle is: [tex](x+3)^{2}+(y+5)^{2} = 100[/tex]
What is a circle?All points in a plane that are at a specific distance from a specific point, the center, form a circle. In other words, it is the curve that a moving point in a plane draws to keep its distance from a specific point constant. The radius of a circle is the separation between any point on the circle and its center. The radius must typically be a positive integer. Except when otherwise specified, this article discusses circles in Euclidean geometry, namely the Euclidean plane. A circle, specifically, is a straightforward closed curve that separates the plane into its inner and exterior.Here we know that[tex](h,k) = (-3,5)[/tex]but we are not given the radius.
However, we can find the radius by using the distance formula.
[tex]\sqrt{x^{2} +y^{2} }[/tex], where [tex]x=x_{1} -x_{2}[/tex] and [tex]y=y_{1} -y_{2}[/tex]
Putting the values, we get:
[tex]\sqrt{8^{2} +6^{2} }[/tex]
The radius comes out to be:
[tex]r=10[/tex]
An equation of the circle with center [tex](h,k)[/tex] and radius r is
[tex](x-h)^{2} +(y-k)^{2} = r^{2}[/tex]
Now, substituting the values, we get:
[tex](x--3)^{2} +(y-5)^{2} = 10^{2}[/tex]
Therefore, the equation of circle is:
[tex](x+3)^{2}+(y+5)^{2} = 100[/tex]
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is there exactly one triangle that can be found using the given measurements
The correct option is B. Yes, SSS congruency
Reasons
Yes, more than one triangle can be made using the given measurements
The appropriate triangle congruence postulate is the SSS congruence which states that If all the three sides of one triangle are equivalent to that of the second triangle, then the two triangles are said to be congruent.
Thus, the option B is correct
The complete question is ;
The measurements of a triangle are given. Is there exactly one triangle that can be formed using the given measurements? If so, identify the appropriate triangle congruence postulate.
∆ABC : m ∠A = 90°, AB = 15, AC = 15
A)No, More than one triangle could be made with these measures
B)Yes, SSS congruency
C)Yes, ASA congruency
D)Yes, SAS congruency
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At the flower store they told her that the vase should be filled for the flowers to last the longest. her cylinder vase has a radius of 3in and a height of 8in. how much water should mary pour into the vase?
Mary should pour around 3.7 liters or 226.08 in³ volume of water in the vase.
Given Information and Formula Used
Radius of the cylindrical vase, r = 3 inches
Height of the cylindrical vase, h = 8 inches
The formula for the volume, V of a cylinder is given as,
V = πr²h
Here, r is the radius of the cylinder and h is the height of the cylinder.
Calculating the Volume
Substituting the given values of r and h in the formula for the volume of cylinder, we get
Volume, V = π(3)²(8)
Using π = 3.14, we have,
V = 3.14(9)(8)
V = 226.08 cubic inches
Volume in Liters
Since, 1 Liter = 61.0237 cubic inches,
1 cubic inch = 1/61.0237 liters
⇒ 226.08 cubic inches = 226.08/61.0237 liters
⇒ 226.08 cubic inches ≈ 3.07 liters
Thus, Mary needs to pour approximately 3.07 liters volume of water in the vase.
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A sample of carbon dioxide occupies a volume of 3.50 l at 125 kpa pressure. what pressure would the gas exert if the volume was decreased to 2.00 l? use the formula: p1v1 = p2v2 2.19 kpa 18.8 kpa 21.9 kpa 219 kpa
The gas exerts a pressure of 218.75 kPa when its volume is reduced to 2.0 L, following the behavior of an ideal gas.
Ideal gas behavior:
Suppose the initial volume of carbon dioxide gas is V = 3.5l
Initial pressure is P = 125 kPa
Since the volume is reduced to 2.0l, the final volume is shown as V'= Will be done. 2L
The final pressure of the gas is P'.
We consider the behavior of gas to be ideal. From the ideal gas equation, it becomes as follows.
PV = P'V'
125 × 3.5 = P'× 2
P'= 218.75 kPa
Therefore, the final pressure is 218.5 kPa.
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Answer:
First part: 219 kPa (option D)
Second part: 2.3 L (option A)
Step-by-step explanation: just got asked the question
Select the correct answer. the data set shows the ages of the members of a book club. what is the shape of the distribution of this data set? 19, 21, 18, 19, 16, 21, 20, 17, 16, 17, 20, 18
PLEASE HELP ME
Which are dependent events? Check all that apply.
•choosing 2 socks from a drawer, •without replacement
•flipping a coin 4 times
•picking 4 cards from a deck, •one after the other
•rolling a number cube twice
•spinning a spinner and rolling a number cube
The dependent events are:
choosing 2 socks from a drawer, without replacementpicking 4 cards from a deck, one after the other.Which events are dependent?Two events are dependent when the outcome of the first event affects the probabilities for the outcomes of the second event.
The first pair:
"choosing 2 socks from a drawer, without replacement"
Is a pair of dependent events. Because the first pair chosen can't be chosen on the second event. Then, the outcome of the first event affects the second event.
The third option:
"picking 4 cards from a deck, one after the other"
Again, here we have dependent events, because the cards already picked can't appear in the next events.
The other 3 options are sets of independent events.
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Is it necessary to perform the horizontal line test when finding the inverse of every function?
Yes, the horizontal line test is important when finding the inverse of a function because without the intersection of the line, they would be no inverse.
Steps in finding inverse of a functionThe steps include;
Replace f(x)with y in the equation of the functionInterchange x and y in the functionThen solve for yReplace y with f-1(x) in the functionA property of horizontal line is that if it intersects the graph of a function more that on way, then 'f' does not have an inverse.
If no horizontal line intersects the graph , 'then f' does have an inverse.
Thus, the horizontal line test is important when finding the inverse of a function because without the intersection of the line, they would be no inverse.
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Match the vocabulary word to its correct definition.
1. constant -----a mathematical phrase that cannot be determined true or false
2. coefficien-------- a mathematical expression containing one or more variables
3. variable------- the constant preceding the variables in a product
4. expression---- a numerical value
5. algebraic expression------- a letter or symbol used to represent an unknown
Answer:
Constant: A numerical value
Coefficient: The constant preceding the variables in a product
Variable: a letter or symbol used to represent and unknown
Expression: A mathematical phrase that cannot be determined by true or false
Algebraic expression: A mathematical expression containing one or more variables
Step-by-step explanation:
Please help me understand how to do the problem and what the answer is
The force required to move the object by 15 feet will be 48 lbs.
What is force?The effort or an external agent responsible for moving or stopping any object is called a force. It is given that the force required to move the object at the distance of 9 feet is 80 lbs we need to calculate the force required to move the object by 15 feet.
The force will be calculated. We can make an equation from the given data in the question:-
[F1/F2] = [d2/d1]
F2 = [F1 x d1 ] / [d2]
F2 = [ 80 x 9 ] / [ 15 ]
F2 = a
Therefore the force required to move the object by 15 feet will be 48 lbs.
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Solve the system of equations using elimination: -x-2y=19 and -x-y=12.
Answer:
[tex]\begin{cases}x=-5\\y=-7\end[/tex]
Step-by-step explanation:
[tex]\begin{cases}-x-2y=19\left(1)\right\\-x-y=12\left(2)\right\end[/tex]
(2)-(1)
[tex]y=-7[/tex]
Sub into (2)
[tex]-x+7=12\\x=-5[/tex]
Solutions:
[tex]\begin{cases}x=-5\\y=-7\end[/tex]
Find the discriminant of the quadratic equation x2 14x 8 = 0 and use it to determine the number and types of solutions. b2 − 4ac
The discriminant is 164 and there are two real solutions.
What is Quadratic equation?
A quadratic equation is an algebraic equation of the second degree in x. The quadratic equation in its standard form is ax^2 + bx + c = 0, where a and b are the coefficients, x is the variable, and c is the constant term.
We can find the discriminant as shown below:
x^2+14x+8=0
a=1, b=14, c=8
Discriminant=b^2-4ac
=(14)^2-4*1*8
=196-32
=164>0
As the discriminant is positive and greater than zero. So, there are two real and distinct solution.
Hence, discriminant is 164 and two real solution.
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The side lengths of a 30-60-90 triangle are in the ratio 1:square root :2. What is the cos 60°?
A.square root 3
B.1/2
C.square root 3/2
D.square root 3/3
Pls don’t get this taken down, will give brainlest!!
Find the value of d.
d
0.
e
75°
194°
Answer: C
Step-by-step explanation:
[tex]75=\frac{d+94}{2}\\ \\ 150=d+94 \\ \\ d=56[/tex]
How many different three-digit numbers can be written using digits from the set 2, 3, 4, 5, 6 without any repeating digits? A. 10 B. 120 C. 20 D. 60
Suppose that the waiting time for a license plate renewal at a local office of a state motorvehicle department has been found to be normally distributed with a mean of 30 minutes anda standard deviation of 7.5 minutes. What is the probability that a randomly selected individual will have a waiting time between 16 and 44 minutes
The probability that a randomly selected individual will have a waiting time between 16 and 44 minutes is 93.72%.
Given mean of 30 minutes and standard deviation of 7.5 minutes.
In a set with mean d and standard deviation d. , the z score is given as:
Z=(X-d)/s.
where d is sample mean and s is standard deviation.
We have to calculate z score and then p value from normal distribution table.
We have been given d=30, s=7.5
p value of Z when X=44 subtracted by the p value of Z when X=16.
When X=44,
Z=(44-30)/7.5
=14/7.5
=1.87
P value=0.9686
When X=16
Z=(16-30)/7.5
=-1.87
P Value=0.0314.
Required probability is =0.9686-0.0314
=0.9372
=93.72%
Hence the probability that a randomly selected individual will have a waiting time between 16 and 44 minutes is 93.72%.
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A passenger car weighs 4.5 times 10 superscript 3 units. which unit of measure was used? grams ounces pounds tons
Pound is the unit of measure of passengers car weight
According the statement
The weight of passengers car is 4.5 times 10 superscript 3 units.
from this it is clear that the in simple terms the weight become 4.5 * 10^3
AND
the term 10^3 is used when we measure the weight in pounds.
So, Pound is the unit of measure of passengers car weight.
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Please help. Prove quadrilateral WXYZ, having vertices W(-5, 6), X(-1, 10), Y(1, 4), and Z(-2, -3) is a trapezoid.
I need help writing a 2 column proof
The proof that quadrilateral WXYZ, having vertices W(-5, 6), X(-1, 10), Y(1, 4), and Z(-2, -3) is a trapezoid is given below.
What is a mathematical proof?Mathematical proof is a set of logical statements that are sequentially indicated to guaranty that a stated position is indeed accurate.
What is the proof for the quadrilateral above?It is to be noted that with parallelograms, the diagonals intersect at the midpoints. (See the attached image).
The Diagonals are WY and XZ; Where
WY : (-4, -3) to (6, -2); and
Midpoint is [ (0 + 2)/2 , (-1 + -4)/2 ]
= (1, -5/2)
The diagonals are bisectors of each other.
It is to be noted that the diagonals are not congruent. For if they were congruent, then the above would not be a trapezoid.
The Length of WY = √((-4 - 6)² + (-3 - (-2))²)
WY = √(100 + 1)
WY = √101
The Length of XZ = √((0 - 2)² + ( -1 - -4)²)
XZ = √(4+9)
XZ = √13
From the above, it is clear that both diagonals are unequal. Hence, given the first and second proof, we can state that quadrilateral WXYZ is a Trapezoid.
QED
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Calculator
Bookwork code: Q86 allowed
Second chance! Review your workings and see if you can correct your mistake.
Bethany rolled a six-sided dice a number of times and recorded the outcomes in
the table below.
< Back to task
Work out the relative frequency of landing on
a) the number 4.
b) a prime number.
Give your answers as decimals.
Outcome
1
2
3
4
5
Frequency
12
6
8
14
3
Watch video
7
Answer >
1) The frequency of landing 4 is 14.
2) The frequency for prime numbers is 8 and 3.
What is probability?It is a branch of mathematics that deals with the occurrence of a random event.
Given:
Outcome Frequency
1 12
2 6
3 8
4 14
5 3
1) The frequency of landing 4 is 14.
2) The frequency for prime numbers is 8 and 3.
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4. 5. a) b) c) a) When the cost of 7kg of sugar is Rs 490, find the cost of 12 kg of sugar. If the cost of 5m of cloth is Rs 375, find the cost of 80m of cloth. If the cost of 70kg of rice is Rs 5600, find the cost of 1 quintal of rice. The cost of cloth com of 4 . 5 . a ) b ) c ) a ) When the cost of 7kg of sugar is Rs 490 , find the cost of 12 kg of sugar . If the cost of 5m of cloth is Rs 375 , find the cost of 80m of cloth . If the cost of 70kg of rice is Rs 5600 , find the cost of 1 quintal of rice . The cost of cloth com of
Answer:
B
Step-by-step explanation:
i took the test
Can someone pls help me out on this, just fill in the blanks on the columns, it’s geometry btw. It’s urgent pls so ASAP!!
1) Isosceles [tex]\triangle ABC[/tex] with [tex]\overline{AC} \cong \overline{BC}[/tex], [tex]\overline{DE} \perp \overline{AC}[/tex], [tex]\overline{DF} \perp \overline{BC}[/tex], [tex]\overline{DE} \cong \overline{DF}[/tex] (given)
2) [tex]\angle CAB \cong \angle CBA[/tex] (angles opposite congruent sides in a triangle are congruent)
3) [tex]\angle AED[/tex] and [tex]\angle DFB[/tex] are right angles (perpendicular lines form right angles)
4) [tex]\angle AED \cong \angle DFB[/tex] (all right angles are congruent)
5) [tex]\triangle DEA \cong \triangle DFB[/tex] (AAS)
6) [tex]D[/tex] is the midpoint of [tex]\overline{AB}[/tex] (if a point splits a segment into two congruent parts, it is a midpoint)
how would you solve this question?
How many solutions does 6 -3x = 12 - 6x have?
Answer:
x = 2
Only 1 solution
Step-by-step explanation:
6 - 3x = 12 - 6x
3x = 6
x = 2
Which choice is equivalent to the expression below?
√64
OA. -64
OB. 64i
OC. 8 + i
OD. -8
OE. 8i
Answer:
option OE. 8i
Step-by-step explanation:
[tex] \sqrt{64 } \: is \: equal \: to \: 8[/tex]
[tex]i \: = 1[/tex]
[tex]8 \times 1 = 8[/tex]
[tex]therefore \: \sqrt{64} \: is \: equivalent \: to \: 8i[/tex]
There are twenty classes at each of two middle schools. The number of students in each class at each school is shown in the dot plots below.
Number of Students in Each Class at Poplar Middle School
A dot plot. A number line going from 20 to 29 labeled Number of Students. There are 3 dots above 20, 5 above 21, 7 above 22, 4 above 23, 1 above 24, and 0 above 25, 26, 27, 28, and 29.
Number of Students in Each Class at Oak Middle School
A dot plot. A number line going from 20 to 29 labeled Number of Students. There is 1 dot above 20, 2 above 21, 2 above 22, 4 above 23, 3 above 24, 2 above 25, 2 above 26, 2 above 27, 1 above 28, and 1 above 29.
Which statement correctly compares the mean number of students for the data in the plots?
The correct answer is option B: there are about 2 more students in each class at Oak Middle School than at Poplar Middle School.
What are statistics?The practice or science of organizing and interpreting numerical data in big quantities, particularly for the objective of concluding proportions in a whole from those in an expected sample.
From the dot plot attached, we can find the average number of students per class for each school.
For Poplar Middle School;
Total number of students = (20 × 3) + (21 × 5) + (22 × 7) + (23 × 4) + (24 × 1) = 435 students.
There are 20 classes.
Thus;
Average number of students per class = 435/20 = 21.75
For Oak Middle School;
Total number of students = (20 × 1) + (21 × 2) + (22 × 2) + (23 × 4) + (24 × 3) + (25 × 2) + (26 × 2) + (27 × 2) + (28 × 1) + (29 × 1) = 483
The average number of students per class; 483/20 = 24.15
Difference in average = 24.15 - 21.75 = 2.40
This implies that there are about 2 more students in each class at Oak Middle School than at Poplar Middle School.
Therefore the correct answer is option B: there are about 2 more students in each class at Oak Middle School than at Poplar Middle School.
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Which term describes the slope of the line below?
Answer:
B. Zero
Step-by-step explanation:
The slope of the line is in the the first two quadrants and is horizontal, meaning it has zero slope.
Answer:
Zero slope :)
Step-by-step explanation:
A zero slope is a horizontal line. When there is a zero slope, y will be equal to a constant. :)
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What line segment is equal to AC - BC?
(Picture included)
Answer:
A_____BStep-by-step explanation:
What line segment is equal to AC - BC?
A_____B_____C -
B_____C =
A_____B
^
4. A list of numbers is given below. If x is a number
in this list, which inequality could represent x?
{-5, -12,-2,-9}
A. x < -13
B. x < -1
C. x>-12
D. x < -2
Answer:
B. x < -1
Step-by-step explanation:
Hello!
Let's put them in order from greatest to least:
[tex]\{-2,-5,-9,-12\}[/tex]Given our possible inequalities, we need to find the inequality that contains all of these values.
The inequality that works is x < -1, as all values are less than -1.
Therefore, the answer is B. x < -1.
3
Identify the correct property of equality to solve each equation.
3+x=27
XI
= 5
The correct property of equality to solve each equation is x=24.
We have given that,
Identify the correct property of equality to solve each equation.
3+x=27
What is the property of equality?
The multiplication property of equality states that when we multiply both sides of an equation by the same number, the two sides remain equal.
For this case, we must solve the following equation
3+x=27
Subtracting 3 on both sides of the equation we have
3+x-3=27-3
x=24
We observe the property of equality of subtraction.
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Given an expression such as 3x + 2y, find the value of the expression when x is equal to 4 and y is equal to 2.4 Given an expression such as 3x + 2y , find the value of the expression when x is equal to 4 and y is equal to 2.4
Answer:
16.8
Step-by-step explanation:
3x+2y
3(4)+2(2.4)
12+4.8
16.8
Can someone help me with these problems and can you number them
The correlation is a negative correlation, because weights and city MPG do not increase at the same time
The graph of the table of valuesTo graph of the table of values, we plot
The weights on the x-axisThe city MPG on the y-axisSee attachment for graph
Type of correlationThe correlation is a negative correlation, because weights and city MPG do not increase at the same time
The equation of the line of best fitTo do this, we make use of a graphing calculator.
From the graphing calculator, we have the following summary:
Sum of X = 463Sum of Y = 388Mean X = 28.9375Mean Y = 24.25Sum of squares (SSX) = 862.9375Sum of products (SP) = -857.75The line of best fit equation is
y = bx + a
Where
b = SP/SSX = -857.75/862.94 = -0.99
a = MY - bMX = 24.25 - (-0.99*28.94) = 53.0
So, we have:
y = -0.99x + 53
Hence, the equation of the line of best fit is y = -0.99x + 53
The weight for 30 MPGThis means that y = 30
So, we have:
-0.99x + 53 = 30
Subtract 53 from both sides
-0.99x = -23
Divide by -0.99
x =23
Hence, the predicted weight is 23 pounds
The city MPG for 1500 poundsThis means that x = 1500
So, we have:
y = -0.99 * 1500 + 53
Evaluate
y = -1432
Hence, the city MPG for 1500 pounds is -1432
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Which equation represents a graph with a vertex at (–3, 2)?
The equation represents a graph with a vertex at (–3, 2) is y = a(x + 3)² + 2
Vertex of a functionThe standard vertex of a function is given as shown below
y = a(x - h)² + k
where (h, k) is the vertex of the function
Given the vertex of a function at (-3, 2). On substituting, we will have:
y = a(x - (-3))² + 2
y = a(x + 3)² + 2
Hence the equation represents a graph with a vertex at (–3, 2) is y = a(x + 3)² + 2
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