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]
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Step-by-step explanation:
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Answers
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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.
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Data and Calculations:Training budget = $3,000
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Read the statement.
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What are conditional statements?We are told that;
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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;
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Answers
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Step-by-step explanation:
Answer:
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Step-by-step explanation:
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We can conclude that:
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The question is incomplete.
The complete question is:
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Answer:
266[tex]\frac{2}{3}[/tex]
Step-by-step explanation:
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l
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Answers
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Answers
Answer:
not equivalent
Step-by-step explanation:
solve for x
4-5x=24
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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:
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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
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Hence, there are 4.15E12 ways to select the members of the committee
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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.
After a certain number of years, he owes a total of £2812 on this loan.
How many years have passed since he took out the loan?
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[tex]15[/tex]
Step-by-step explanation:
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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.
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.
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3
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What is the linear equation that represents this situation?
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y = 8 x minus 3
y = 2 x + 16
y = 2 x minus 32
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-34 + 2x divided by -8
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)
4. x = 17
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
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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.
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]
Consider 8x-4 = 8^10