Please answer this question, i request
If cot θ = 7/8 , evaluate :-
(1 + sin θ)(1 – sin θ)/(1 + cos θ)(1 - cos θ)
Answers
[tex]{\large{\textsf{\textbf{\underline{\underline{Given :}}}}}}[/tex]
[tex] \star \: \tt \cot \theta = \dfrac{7}{8} [/tex]
[tex] {\large{\textsf{\textbf{\underline{\underline{To \: Evaluate :}}}}}}[/tex]
[tex] \star \: \tt \dfrac{(1 + \sin \theta)(1 - \sin \theta) }{(1 + \cos \theta) (1 - \cos \theta) }[/tex]
[tex]{\large{\textsf{\textbf{\underline{\underline{Solution :}}}}}}[/tex]
Consider a [tex]\triangle[/tex] ABC right angled at C and [tex]\sf \angle \: B = \theta [/tex]
Then,
‣ Base [B] = BC
‣ Perpendicular [P] = AC
‣ Hypotenuse [H] = AB
[tex] \therefore \tt \cot \theta = \dfrac{Base}{ Perpendicular} = \dfrac{BC}{AC} = \dfrac{7}{8}[/tex]
Let,
Base = 7k and Perpendicular = 8k, where k is any positive integer
In [tex]\triangle[/tex] ABC, H² = B² + P² by Pythagoras theorem
[tex] \longrightarrow \tt {AB}^{2} = {BC}^{2} + {AC}^{2} [/tex]
[tex] \longrightarrow \tt {AB}^{2} = {(7k)}^{2} + {(8k)}^{2} [/tex]
[tex]\longrightarrow \tt {AB}^{2} = 49{k}^{2} + 64{k}^{2} [/tex]
[tex]\longrightarrow \tt {AB}^{2} = 113{k}^{2} [/tex]
[tex]\longrightarrow \tt AB = \sqrt{113 {k}^{2} } [/tex]
[tex]\longrightarrow \tt AB = \red{ \sqrt{113} \: k}[/tex]
Calculating Sin [tex]\sf \theta [/tex]
[tex] \longrightarrow \tt \sin \theta = \dfrac{Perpendicular}{Hypotenuse}[/tex]
[tex] \longrightarrow \tt \sin \theta = \dfrac{AC}{AB}[/tex]
[tex]\longrightarrow \tt \sin \theta = \dfrac{8 \cancel{k}}{ \sqrt{113} \: \cancel{ k } }[/tex]
[tex]\longrightarrow \tt \sin \theta = \purple{ \dfrac{8}{ \sqrt{113} } }[/tex]
Calculating Cos [tex]\sf \theta [/tex]
[tex] \longrightarrow \tt \cos \theta = \dfrac{Base}{Hypotenuse}[/tex]
[tex] \longrightarrow \tt \cos \theta = \dfrac{BC}{ AB} [/tex]
[tex] \longrightarrow \tt \cos \theta = \dfrac{7 \cancel{k}}{ \sqrt{113} \: \cancel{k } }[/tex]
[tex]\longrightarrow \tt \cos \theta = \purple{ \dfrac{7}{ \sqrt{113} } }[/tex]
Solving the given expression :-
[tex] \longrightarrow \: \tt \dfrac{(1 + \sin \theta)(1 - \sin \theta) }{(1 + \cos \theta) (1 - \cos \theta) } [/tex]
Putting,
• Sin [tex]\sf \theta [/tex] = [tex]\dfrac{8}{ \sqrt{113} }[/tex]
• Cos [tex]\sf \theta [/tex] = [tex]\dfrac{7}{ \sqrt{113} }[/tex]
[tex] \longrightarrow \: \tt \dfrac{ \bigg(1 + \dfrac{8}{ \sqrt{133}} \bigg) \bigg(1 - \dfrac{8}{ \sqrt{133}} \bigg) }{\bigg(1 + \dfrac{7}{ \sqrt{133}} \bigg) \bigg(1 - \dfrac{7}{ \sqrt{133}} \bigg)} [/tex]
Using (a + b ) (a - b ) = a² - b²
[tex]\longrightarrow \: \tt \dfrac{ { \bigg(1 \bigg)}^{2} - { \bigg( \dfrac{8}{ \sqrt{133} } \bigg)}^{2} }{ { \bigg(1 \bigg)}^{2} - { \bigg( \dfrac{7}{ \sqrt{133} } \bigg)}^{2} } [/tex]
[tex]\longrightarrow \: \tt \dfrac{1 - \dfrac{64}{113} }{ 1 - \dfrac{49}{113} } [/tex]
[tex]\longrightarrow \: \tt \dfrac{ \dfrac{113 - 64}{113} }{ \dfrac{113 - 49}{113} } [/tex]
[tex]\longrightarrow \: \tt { \dfrac { \dfrac{49}{113} }{ \dfrac{64}{113} } }[/tex]
[tex]\longrightarrow \: \tt { \dfrac{49}{113} }÷{ \dfrac{64}{113} }[/tex]
[tex]\longrightarrow \: \tt \dfrac{49}{ \cancel{113}} \times \dfrac{ \cancel{113}}{64} [/tex]
[tex]\longrightarrow \: \tt \dfrac{49}{64} [/tex]
[tex]\qquad \: \therefore \: \tt \dfrac{(1 + \sin \theta)(1 - \sin \theta) }{(1 + \cos \theta) (1 - \cos \theta) } = \pink{\dfrac{49}{64} }[/tex]
[tex]\begin{gathered} {\underline{\rule{300pt}{4pt}}} \end{gathered} [/tex]
[tex] {\large{\textsf{\textbf{\underline{\underline{We \: know :}}}}}}[/tex]
✧ Basic Formulas of Trigonometry is given by :-
[tex]\begin{gathered}\begin{gathered}\boxed { \begin{array}{c c} \\ \bigstar \: \sf{ In \:a \:Right \:Angled \: Triangle :} \\ \\ \sf {\star Sin \theta = \dfrac{Perpendicular}{Hypotenuse}} \\\\ \sf{ \star \cos \theta = \dfrac{ Base }{Hypotenuse}}\\\\ \sf{\star \tan \theta = \dfrac{Perpendicular}{Base}}\\\\ \sf{\star \cosec \theta = \dfrac{Hypotenuse}{Perpendicular}} \\\\ \sf{\star \sec \theta = \dfrac{Hypotenuse}{Base}}\\\\ \sf{\star \cot \theta = \dfrac{Base}{Perpendicular}} \end{array}}\\\end{gathered} \end{gathered}[/tex]
[tex]{\large{\textsf{\textbf{\underline{\underline{Note :}}}}}}[/tex]
✧ Figure in attachment
[tex]\begin{gathered} {\underline{\rule{200pt}{1pt}}} \end{gathered} [/tex]
Related Questions
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Answers
Answer:
E. 18.6
Step-by-step explanation:
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__
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Step-by-step explanation:
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Step-by-step explanation:
Kian borrowed £1900 at a simple interest rate of 8% per year.
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Answers
[tex]~~~~~~ \textit{Simple Interest Earned Amount} \\\\ A=P(1+rt)\qquad \begin{cases} A=\textit{accumulated amount}\dotfill & \pounds 2812\\ P=\textit{original amount deposited}\dotfill & \$1900\\ r=rate\to 8\%\to \frac{8}{100}\dotfill &0.08\\ t=years \end{cases} \\\\\\ 2812=1900[1+(0.08)(t)]\implies \cfrac{2812}{1900}=1+0.08t\implies \cfrac{37}{25}=1+0.08t \\\\\\ \cfrac{37}{25}-1=0.08t\implies \cfrac{12}{25}=0.08t\implies \cfrac{ ~~ \frac{12}{25} ~~ }{0.08}=t\implies 6=t[/tex]
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(-2, 1). write the new function.
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A local public library decides to track the number of hours that a certain computer is being used. The table represents the number of hours, y, which is dependent on the number of days, x.
x
y
3
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What is the linear equation that represents this situation?
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y = 2 x minus 32
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That statistic is based on a pinterest quote, the actual statistic is closer to 8-10%
You and your family are going on a
family cruise this summer! WOO HOO!
Your parents had to put down a $250
deposit. Then, the cruise costs $500
per person on top of that. There are 5
people in your family. How much will
the entire cruise cost? Okay
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Answers
Answer: x=12
Step-by-step explanation:
Answer:
x=1
Step-by-step explanation:
The equation can be simplified and solved using the rules of exponents and the usual inverse operations.
__
Interpreted according to the Order of Operations, your equation is ...
((2^2)x)^3 = 64
(4x)^3 = 64x^3 = 64 . . . . . . use (ab)^c = (a^c)(b^c)
x^3 = 1 . . . . . . . . . . . . . . divide by 64
x = ∛1 = 1
__
Perhaps you want x in the exponent:
(2^(2x))^3 = 64
2^(6x) = 2^6 . . . . . . . . . use (a^b)^c = a^(bc)
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x = 1 . . . . . . divide by 6
_____
Additional comment
Folks are often not careful about identifying exponents. It is helpful if you write the exponent in a way that leaves no ambiguity.
Read the statement.
Doubling the dimensions of a rectangle increases the area by a factor of 4.
If p represents doubling the dimensions of a rectangle and q represents the area increasing by a factor of 4, which are
true? Select two options.
Op q represents the original conditional statement.
~p~q represents the inverse of the original conditional statement.
q- p represents the original conditional statement.
-q- ~p represents the converse of the original conditional statement.
Op q represents the contrapositive of the original conditional statement.
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Mark this and return
Submit
Next
Answers
The options that are true as a result of doubling the rectangle dimensions are;
p → q represents the original conditional statement.
p → ~q represents the contrapositive of the original conditional statement.
What are conditional statements?We are told that;
Doubling the dimensions of a rectangle increases the area by a factor of 4.
Now, p represents doubling the dimensions of a rectangle and q represents the area increasing by a factor of 4.
Thus, using conditional statements notation, we can say that;
p → q represents the original conditional statement.
p → ~q represents the contrapositive of the original conditional statement.
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Without using the formula, work out the gradient of the graph shown.
Answers
Answer:
1/3
Step-by-step explanation:
if you do 12 minutes of a 45 minute workout how many calories is it? (the 45 minutes is 1000 calories in total)
help!
Answers
Answer:
266[tex]\frac{2}{3}[/tex]
Step-by-step explanation:
if a 45 minute workout is 1000 calories, at the same rate, a 12 minute workout would be 266 calories and 2 thirds. [tex]\frac{12}{45}[/tex] x 1000 = 266[tex]\frac{2}{3}[/tex]
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Answers
Answer:
not equivalent
Step-by-step explanation:
solve for x
4-5x=24
-5x=20
x = - 4
5x=20
x = 4
no they are not equivalent
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Select each correct answer.
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Answers
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Answers
Answer:
-34+x/4
Step-by-step explanation:
Answer:
x=17
Step-by-step explanation:
Since you did not specify, we will assume that your equation is equal to 0.
1. (-34+2x)/(-8) = 0 (multiply both sides by -8)
2. (-34+2x) = 0 (add 34 to both sides)
3. 2x = 34 (divide both sides by 2)
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Answers
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Given that the subscript n in [tex]u_n[/tex] represents number of years and [tex]u_n[/tex] represents the amount left to be paid after n years.
We can conclude that:
u₁₂ represents the amount of loan remaining after 12 years
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(3x-2)(3x-2)(3x-2)(2x+1)
Answers
What is f(x) + f(x) + f(x)?
✔ 3f(x)
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Evaluate 3f(2) =
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Answers
Answer:
12
Step-by-step explanation:
The problem statement tells us the definition of 3f(x). In order to find the value of 3f(2), we substitute 2 for x and do the arithmetic.
__
definition3f(x) = 3x² . . . . given
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The value of 3f(2) is 12.
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Answers
Answer:
-3ab
Step-by-step explanation:
-27/9= 3
(a^2)/a=a
(b^2)/b=b
Solve system of equation using elimination by addition.
Will give brainliest!!
Answers
Answer:
x = -8
y = 0
Step-by-step explanation:
The two shown equations can be added together:
x + 4y = -8
+ ( x - 4y = -8 )
The two x's would combine to make two x, the 4y and -4y cancel out to zero, and -8 added to -8 is -16:
2x = -16
Therefore, x = -8. We can plug this into the first equation to get:
-8 + 4y = -8
Subtracting 8 on both sides yields:
4y = 0
Now divide by 4 on both sides to get
y = 0
We can check these values by plugging them into the second equation:
(-8) - 4(0) = -8
-8 - 0 = -8
-8 = -8
Since -8 always equals -8, the answer is correct.
our cell phone costs $50 a month plus $0.25 for each text message, t. How much would your monthly bill be if you sent 40 text messages? our cell phone costs $50 a month plus $0.25 for each text message, t. How much would your monthly bill be if you sent 40 text messages? our cell phone costs $50 a month plus $0.25 for each text message, t. How much would your monthly bill be if you sent 40 text messages? our cell phone costs $50 a month plus $0.25 for each text message, t. How much would your monthly bill be if you sent 40 text messages? our cell phone costs $50 a month plus $0.25 for each text message, t. How much would your monthly bill be if you sent 40 text messages? our cell phone costs $50 a month plus $0.25 for each text message, t. How much would your monthly bill be if you sent 40 text messages? our cell phone costs $50 a month plus $0.25 for each text message, t. How much would your monthly bill be if you sent 40 text messages?
Answers
Answer:
$60
Step-by-step explanation:
cost = $50 + $0.25x
Since x = 40
$(50 + 10) = $60
Answer:
$60/month
Step-by-step explanation:
So the equation would be 50 + (0.25 * 40), solve parenthesis 50 + 10, so answer would be $60.
a rectangle mural measure 234 inches by 245 inches, rihanna created a mural 33 inches longer
Answers
Rhiannna needs 1024 inches longer bedspread if Rhiannna gets a new bedspread that is 33 in length.
What is a rectangle?It is defined as the two-dimensional geometry in which the angle between the adjacent sides are 90 degrees. It is a type of quadrilateral.
The question is incomplete.
The complete question is:
Rhiannna has a rectangular bedspread that measures 234 in. By 245 in. She wants to sew a fringe around the edge of the bedspread. What length of fringe does Rhiannna need to purchase. Rhiannna gets a new bedspread that is 33 in length. What length of the fringe does Rhiannna need for the longer bedspread?
We know perimeter of the rectangle = 2(length + width)
= 2(234 + 245 )
= 958 inches
Rihannon gets a new bedspread that is 33 in length:
New perimeter = 2(33) + 958 = 1024 inches
Thus, Rhiannna needs 1024 inches longer bedspread if Rhiannna gets a new bedspread that is 33 in length.
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♡♡♡ ▲ ▲ ▲ ▲ Ratio of hearts to triangles in simplest form.
Answers
3 hearts
4 triangle
H:T
3:4
So, ratio of hearts to triangles in simplest form is 3:4
(we can't simplify any further - HCF of 3 & 4 is 1)
Hope this helps!
What is the answer (6√2)(-3√5)
Answers
Solution:
[tex] - 18 \sqrt{10} [/tex]
Step-by-step explanation:1. Apply rule : a(-b) = -ab
[tex] = - 6 \sqrt{2} \: \: • \: \: 3 \sqrt{5} [/tex]
2. Apply radical rule : [tex] \sqrt{a} \sqrt{b} = \sqrt{ab} [/tex] , [tex]a \geqslant 0[/tex], [tex]b[/tex]
[tex] = 6 \sqrt{2 \: • \: 5} \: • \: 3[/tex]
3.Multiply the numbers : 2 • 5 = 10=[tex]6 \sqrt{10} •3[/tex]
4.Multiply the numbers : - 6 • 3= - 18[tex] = - 18 \sqrt{10} [/tex]
LL
F
C
9
5
D
2x+3
E
What is the value of x and the length of segment DE?
1 5
9
9
2x + 3
2. 10x+15=9(9)
Length of DE=
units
Answers
Answer:
[tex]x=\boxed{6.6}[/tex]
[tex]\overline{\sf DE}=\boxed{13.2}\:\:\sf units[/tex]
Step-by-step explanation:
Geometric Mean Theorem - Altitude Rule
The altitude drawn from the vertex of the right angle perpendicular to the hypotenuse separates the hypotenuse into two segments. The ratio of one segment to the altitude is equal to the ratio of the altitude to the other segment:
[tex]\sf \dfrac{segment\:1}{altitude}=\dfrac{altitude}{segment\:2}[/tex]
From inspection of the given diagram:
altitude = FD = 9segment 1 = CD = 5segment 2 = DE = [tex]2x+3[/tex][tex]\begin{aligned}\sf \dfrac{segment\:1}{altitude} & = \sf \dfrac{altitude}{segment\:2}\\\\\implies \dfrac{5}{9} & = \dfrac{9}{2x+3}\\\\5(2x+3) & = 81\\\\10x+15 & = 81\\\\10x & = 66\\\\ \implies x & = 6.6\end{aligned}[/tex]
Substitute the found value of x into the expression for DE:
[tex]\begin{aligned}\sf \overline{DE} & = 2x+3\\\implies \sf \overline{DE} & = 2(6.6)+3\\& = 13.2+3\\& =16.2\:\: \sf units\end{aligned}[/tex]
If f(x) = x² - 2x, find:
f(5) = [?]
Answers
Answer:
[tex]15[/tex]
Step-by-step explanation:
if [tex]f(x) = x^2 -2x[/tex], then that means that [tex]f(5) = (5)^2 - 2(5)[/tex] which equals [tex]15[/tex].
Step-by-step explanation:
A function f is defined by f (x) = 2x - 5. Write down the values of In mathematics, a function means a correspondence from one value x of the first set to another value y of the second set. NCERT Solutions Class 11 Math's Chapter 2 Exercise 2.3 Question 3 A function f is defined by f (x) = 2x - 5.
You used $3000 of your budget for training your employees. each hour of training costs $40 per employee plus a one-time $200 fee. a. how many hours of training can your purchase? b. how many hours of training will each of your employees receive? remember how many employees do you have?
Answers
A. The number of hours of training that you can purchase with a $3,000 training budget at $40 per employee per hour and a one-time $200 fee is 70-employee-training-hours.
B. Each employee will get 7 hours of training, assuming your organization has 10 employees.
What is the cost of training employees?The training cost of employees varies from one organization to another.
The training cost also depends on the training provider and the number of hours of training being offered.
Data and Calculations:Training budget = $3,000
One-time fee = $200
Remainder after the one-time fee = $2,800 ($3,000 - $200)
Training costs per employer per hour = $40
Hours of training to purchase = 70 ($2,800/$40)
Thus, an organization with 10 employees can obtain 70 hours of training with the training budget.
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Consider 8x-4 = 8^10
Answers
A country has two political parties, the Reds and the Blues. The 100-member senate has 44 Reds and 56 Blues. How many ways are there to pick a 10 member committee of senators with the same number of Reds as Blues
Answers
There are 4.15E12 ways to select the members of the committee to have the same number of Reds as Blues
How to determine the number of ways?The given parameters are:
Reds = 44
Blues = 56
Committee member = 10
To have the same number of Reds as Blues, the committee must have 5 reds and 5 blues.
So, the number of ways is;
Ways = 44C5 * 56C5
Evaluate the combination expressions:
Ways = 1.086008 E+6 * 3.819816 E+6
Evaluate the product
Ways = 4.15E12
Hence, there are 4.15E12 ways to select the members of the committee
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n triangle ABC (see sketch), AD is the angle bisector of the angle A and BH is the height from side AC. The obtuse angle between BH and AD is four times the side of DAB. How big is the angle CAB?
A. 30 B.45 C.60 D.75 E.90
Answers
Answer: C
Step-by-step explanation:
We know that [tex]\angle AHB=90^{\circ}[/tex], and thus by the exterior angle theorem,
[tex]90^{\circ}+\alpha=4\alpha\\\\90=3\alpha\\\\\alpha=30^{\circ}[/tex]
Thus, [tex]\angle CAB=2\theta=\boxed{60^{\circ}}[/tex]