Correct Question :-
If sec[tex]\theta[/tex] + tan[tex]\theta[/tex]= x , then prove that ,
[tex]\implies\sf sin\theta =\dfrac{x^2-1}{x^2+1}[/tex]
Proof :-
Here we are given that ,
[tex]\longrightarrow \sec\theta + tan\theta = x[/tex]
Firstly write everything in terms of sine and cosine .
[tex]\longrightarrow \dfrac{1}{\cos\theta}+\dfrac{\sin\theta}{\cos\theta}=x [/tex]
Add ,
[tex]\longrightarrow \dfrac{1+\sin\theta}{\cos\theta}=x [/tex]
On squaring both sides , we have ;
[tex]\longrightarrow \dfrac{(\sin\theta+1)^2}{(\cos\theta)^2}=x^2[/tex]
Simplify using identity sin²x + cos²x = 1 ,
[tex]\longrightarrow \dfrac{(1+\sin\theta)^2}{1-\sin^2\theta}=x^2 [/tex]
Simplify using identity (a+b)(a-b)=a²-b² ,
[tex]\longrightarrow \dfrac{(1+\sin\theta)^2}{(1+\sin\theta)(1-\sin\theta)}=x^2 [/tex]
Simplify,
[tex]\longrightarrow \dfrac{1+\sin\theta}{1-\sin\theta}=x^2 [/tex]
On using Componendo and Dividendo , we have ;
[tex]\longrightarrow \dfrac{1+\sin\theta+1-\sin\theta}{1+\sin\theta-1+\sin\theta}=\dfrac{x^2+1}{x^2-1}[/tex]
[tex]\longrightarrow \dfrac{2}{2\sin\theta}=\dfrac{x^2+1}{x^2-1}[/tex]
Simplify,
[tex]\longrightarrow \dfrac{1}{\sin\theta}=\dfrac{x^2+1}{x^2-1}\\[/tex]
Divide both the sides by 1 ,
[tex]\longrightarrow \underline{\underline{\sin\theta =\dfrac{x^2-1}{x^2+1}}} [/tex]
Hence proved .
And we are done !
[tex]\rule{200}4[/tex]
More to Know :-1) Trigonometric table :-
[tex]\small{ \begin{tabular}{|c|c|c|c|c|c|} \cline{1-6} \theta & \sf 0^{\circ} & \sf 30^{\circ} & \sf 45^{\circ} & \sf 60^{\circ} & \sf 90^{\circ} \\ \cline{1-6} $ \sin $ & 0 & $\dfrac{1}{2 }$ & $\dfrac{1}{ \sqrt{2} }$ & $\dfrac{ \sqrt{3}}{2}$ & 1 \\ \cline{1-6} $ \cos $ & 1 & $ \dfrac{ \sqrt{ 3 }}{2} } $ & $ \dfrac{1}{ \sqrt{2} } $ & $ \dfrac{ 1 }{ 2 } $ & 0 \\ \cline{1-6} $ \tan $ & 0 & $ \dfrac{1}{ \sqrt{3} } $ & 1 & $ \sqrt{3} $ & $ \infty $ \\ \cline{1-6} \cot & $ \infty $ &$ \sqrt{3} $ & 1 & $ \dfrac{1}{ \sqrt{3} } $ &0 \\ \cline{1 - 6} \sec & 1 & $ \dfrac{2}{ \sqrt{3}} $ & $ \sqrt{2} $ & 2 & $ \infty $ \\ \cline{1-6} \csc & $ \infty $ & 2 & $ \sqrt{2 } $ & $ \dfrac{ 2 }{ \sqrt{ 3 } } $ & 1 \\ \cline{1 - 6}\end{tabular}}[/tex]
[tex]\rule{200}4[/tex]
2) Important identities :-
[tex]\boxed{\begin{minipage}{6cm} Important Trigonometric identities :- \\ \\ $\: \: 1)\:\sin^2\theta+\cos^2\theta=1 \\ \\ 2)\:\sin^2\theta= 1-\cos^2\theta \\ \\ 3)\:\cos^2\theta=1-\sin^2\theta \\ \\ 4)\:1+\cot^2\theta=\text{cosec}^2 \, \theta \\ \\5)\: \text{cosec}^2 \, \theta-\cot^2\theta =1 \\ \\ 6)\:\text{cosec}^2 \, \theta= 1+\cot^2\theta \\\ \\ 7)\:\sec^2\theta=1+\tan^2\theta \\ \\ 8)\:\sec^2\theta-\tan^2\theta=1 \\ \\ 9)\:\tan^2\theta=\sec^2\theta-1$\end{minipage}}[/tex]
[tex]\rule{200}4[/tex]
If sec theta + tan theta equals to x then prove that:-
[tex]\longrightarrow \: sin \theta = \frac{ {x}^{2} - { \bold{1}} }{ {x}^{2} + 1 } [/tex]
Solution:Given that:
[tex]\longrightarrow \: sec \: \theta + tan \: \theta =x \: \: ..(i) [/tex]
We have to prove:
[tex]\longrightarrow \: sin \: \theta = \frac{ {x}^{2} - 1 }{ {x}^{2} + 1} [/tex]
We know that:
[tex]\longrightarrow \: { \sec}^{2} \theta \: - \: { \tan}^{2} \theta = 1[/tex]
[tex] \longrightarrow ( \sec \: \theta \: + \: \tan \: \theta)( \sec \: \theta - \tan \: \theta )[/tex]
[tex] \longrightarrow x( \sec \: \theta - \tan \: \theta) \: = 1[/tex]
[tex] \longrightarrow \: \sec \theta - \tan \: \theta = \frac{1}{x} \ ..(ii)[/tex]
Adding equation (i) and (ii), we get:
[tex] \longrightarrow \: 2\sec \theta = x + \frac{1}{x} [/tex]
[tex] \longrightarrow \: 2 \sec \: \theta = \frac{ {x}^{2} + 1}{x} [/tex]
[tex] \longrightarrow \: \sec \: \theta = \frac{ {x}^{2} + 1}{2x} [/tex]
[tex] \longrightarrow \: \cos \: \theta = \frac{2x}{ {x}^{2} + 1 } [/tex]
Now, we know that:
[tex] \longrightarrow \: { \sin}^{2} \theta + { \cos }^{2} \: \theta = 1[/tex]
Therefore,
[tex] \longrightarrow \: \sin \theta \: = \sqrt{1 - { \cos }^{2} } \: \theta[/tex]
[tex] \longrightarrow \: \sin \: \theta = {\sqrt{1 - ( \frac{2x}{ {x}^{2} + 1} } )}^{2} [/tex]
[tex] \longrightarrow \: \sin \: \theta = \sqrt{ \frac{ {({x}^{2} + 1)}^{2} - {(2x)}^{2} }{( {x}^{2} + 1)} } [/tex]
[tex] \longrightarrow \: \sin \: \theta = \sqrt{ \frac{ {x}^{4} \: + 2 {x}^{2} + 1 - \: 4 {x}^{2} }{( {x}^{2} { + 1)}^{2} } } [/tex]
[tex] \longrightarrow \: \sin \: \theta = \sqrt{ \frac{ {x}^{4} - {2x}^{2} + 1 }{( {x}^{2} { + 1)}^{2} } } [/tex]
[tex] \longrightarrow \: \sin \: \theta = \sqrt{ \frac{( {x}^{2} { - 1)}^{2} }{ {(x}^{2} + {1)}^{2} } } [/tex]
[tex] \longrightarrow \: \sin \: \theta = \frac{ {x}^{2} - 1 }{ {x}^{2} + 1} [/tex]
Hence Proved..!![tex] \: [/tex]
Learn More:1. Relationship between sides and T-Ratios.sin θ = Height/Hypotenusecos θ = Base/Hypotenusetan θ = Height/Basecot θ = Base/Heightsec θ = Hypotenuse/Basecosec θ = Hypotenuse/Height2. Square formulae.sin²θ + cos²θ = 1cosec²θ - cot²θ = 1sec²θ - tan²θ = 13. Reciprocal Relationship.sin θ = 1/cosec θcos θ = 1/sec θ tan θ = 1/cot θcosec θ = 1/sin θsec θ = 1/cos θtan θ = 1/cot θ4. Cofunction identities.sin(90° - θ) = cos θcos(90° - θ) = sin θcosec(90° - θ) = sec θsec(90° - θ) = cosec θtan(90° - θ) = cot θcot(90° - θ) = tan θ5. Even odd identities.sin -θ = -sin θcos -θ = cos θtan -θ = -tan θfind the value of -7 + 3(-12) ÷ (-3)
A. -16
B. 5
C. -40
D. -19
1. What's the name of the line segment in the diagram below?
U
O A. UT
O B. TU
O c.T
TŮ
O D. fŮ
Answer:
C
Step-by-step explanation:
What is 1 + 1? I forgot
Answer:
2
Step-by-step explanation:
You’d like to see how many baseball and soccer games you can attend this spring. Travel time for basketball games is 2 hours and soccer games is 1 hour. You would like to spend no more than 15 hours traveling to the games. In total, you would like to attend at least 8 games.
We have to attend 7 basketball games and 1 soccer game.
Given, we had two types of game namely basketball and soccer.
Travel time for basketball games is 2 hours and soccer games is 1 hour.
Let no. of basketball games be x and no of soccer games be y.
According to the question we would like to attend at least 8 games, so we have the inequality [tex]x+y\geq 8[/tex]
And we would spend no more than 15 hours traveling to the game, so [tex]2x+1y\leq 15[/tex].
[tex]x+y=8\\[/tex]
[tex]2x+1y=15\\[/tex]
On solving we get [tex]x=7[/tex] and [tex]y= 1[/tex].
Hence we have to attend 7 basketball games and 1 soccer game.
For more details on Equation follow the link:
https://brainly.com/question/10824691
What type of slope does (-7,8) and (-7,0) have?
Answer:
Undefined slope
Step-by-step explanation:
any number divided by 0 is undefined
-8/0
Answer: m=12/7
Step-by-step explanation:
I looked it up for you bro
Sammy bought 15.3 Liters of milk. She poured the milk equally into 5 bottles. There was 0.35 liters of milk left over. Calculate the capacity of milk in 1 bottle.
please show work.
Answer:
The capacity of milk in 1 bottle is 2.99 liters
Step-by-step explanation:
Let us solve the question
∵ Sammy bought 15.3 liters of milk
∵ She poured the milk equally into 5 bottles
∵ There were 0.35 liters of milk leftover
→ At first, subtract the leftover from the quantity she bought to
find the capacity of the five bottles
∴ The capacity of the 5 bottles = 15.3 - 0.35
∴ The capacity of the 5 bottles = 14.95 liters
→ To find the capacity of each bottle divide the total capacity by 5
∴ The capacity in one bottle = 14.95 ÷ 5
∴ The capacity in one bottle = 2.99 liters
∴ The capacity of milk in 1 bottle is 2.99 liters
Can someone help me out please ?
Answer:
i think its option 1
Step-by-step explanation:
2) A square and a rectangle have the same perimeter. The square has a side length of 6 x units. The
rectangle has a length of (5x + 8) and a width of 6 units. For what value of x, will the square and
rectangle have the same perimeter?
30x usb
a. 2
30 x 54 = 30x 54
b. 4
C. 5
d. 3
Answer:
The value of X should be 8
3) 8x = 72
It’ll help
Answer:
9
Step-by-step explanation:
×=72/8
×=9
Hope it helps
Answer: 9
Step-by-step explanation: First, look for the value of x. A number with a variable in front of it means that you’ll be multiplying, but you can use division which makes it easier to find x. SO, DIVIDE 72 by 8 (82 ÷ 7), which equals to 9. (8 x 9 = 72)
Cuales son las 5 propiedades de la adición
Answer:
En realidad supongo que solo hay 4 no 5.
Step-by-step explanation:
Combine Like Terms
9x + 5 – 6 – 8x + 2
-6s + 7 + 10x – 5
Distribute
-5(2x + 7)
4(-3x + 6)
Equations: Solve for the variable (show all work – circle answer)
2y = -8
x7
= 6
2x – 6 = 10
Answer:
9x+5-6-8x+2
=(9x-8x)+(5-6+2)
=(x)+(+1)
=x+1
Find the equation of a line that passes through the points (2, 16) and (-1, 7)
Answer:
y=3x+10
Step-by-step explanation:
7-16=-9
-1-2=-3
-9/-3=3
m=3
Maria rewrote 45.59 – 7.9 as 45.59 - 7.90. Is the value of 79 changed by annexing a zero after 7.9? Why or why not?
Answer:
No
Step-by-step explanation:
because it is 7.9 and all you are doing is adding a place holder with is the zero
A granola bite contains 27 calories. Most of the calories come from grams of carbohydrates. The rest come from other ingredients. One gram of carbohydrate contains 4 calories.The equation represents the relationship between these quantities. What is the prolom asking you to find? * 4c + 5 = 27
Answer:
Quantity of carbohydrate in gram = 5.5 gram
Step-by-step explanation:
Given:
Amount of calories contain = 27
1 gram carbohydrate = 4 calories
Equation : 4c + 5 = 27
Find:
What is c
Computation:
Equation asking for quantity of carbohydrate in gram
So,
4c + 5 = 27
4c = 22
c = 5.5
Quantity of carbohydrate in gram = 5.5 gram
A drone costs $200, and you receive a 15% discount due a coupon. What is
the total amount to be paid?
Answer:
the total would be 170 dollars
Step-by-step explanation:
200x0.15
Answer:
$170
Step-by-step explanation:
15% of 200 is 30, so subtracting 30 from 200 gets you 170.
If you're having trouble with finding 15% of 200, just flip it around, and find 200% of 15. 15% of 200 and 200% of 15 are both 30. This works no matter what numbers you have. For example, 8% of 100 and 100% of 8 are equivalent, 70% of 200 and 200 and 200% of 70 are equivalent as well.
4. Which number is irrational?
A. (1.5)?
C. 149
D. (15)
A student has passed 60 percent of the 20 quizzes he has written so far successfully. If the student writes 50 quizzes during the year, and passes 80 percent of the remaining quizzes successfully, what would his average percentile be?
Answer:
A student has passed 60 percent of the 20 quizzes he has written so far successfully mean:
successfully quizzes = 60 % of 20 quizzes = ( 60 / 100 ) * 20 = 60 * 20 / 100 = 1200 / 100 = 12
The remaining quizzes = 50 - 20 = 30 quizzes
80 percent of the remaining quizzes successfully mean:
The remaining successfully quizzes = 80 % of 30 = ( 80 / 100 ) * 30 = 80 * 30 / 100 = 2400 / 100 = 24
Total successfully quizzes = 12 + 24 = 36
Average = ( 36 / 50 ) * 100 % = 0.72 * 100 % = 72 %
Pls rate Brainliest!
Answer:73%
Step-by-step explanation:A student has passed 60 percent of the 20 quizzes he has written so far successfully mean:
successfully quizzes = 60 % of 20 quizzes = ( 60 / 100 ) * 20 = 60 * 20 / 100 = 1200 / 100 = 12
The remaining quizzes = 50 - 20 = 30 quizzes
80 percent of the remaining quizzes successfully mean:
The remaining successfully quizzes = 80 % of 30 = ( 80 / 100 ) * 30 = 80 * 30 / 100 = 2400 / 100 = 24
Total successfully quizzes = 12 + 24 = 36
Average = ( 36 / 50 ) * 100 % = 0.72 * 100 % = 72 %
PLEASE HELP! I WILL MAKE YOU BRAINLIST!
Answer: A.
Step-by-step explanation: yes
I NEED HELP ASAP WE ARE GIVING BRAILIEST AND POINT HURRY I NEED
HELP!!!Q.1) The total distance an ant crawls varies directly with the time in seconds. The ant crawls a total distance of 95 centimeters in 5 seconds.
What is the time in seconds the ant crawls when the total distance is 133 centimeters?
A.6 s
B.7 s
C.19 s
D.27 s
Answer: C
Step-by-step explanation:
not A or d
B or C
All I did was divide 95 and 5 and I got 19!
hope this helps!:)
Answer:
B.) 7 seconds
Step-by-step explanation:
We are given that the ant can crawl 95 centimeters in 5 seconds, which means that it can crawl 19 centimeters in 1 second (95 centimeters/5 seconds). Now you need to divide 19 from 133 (133 centimeters/19 centimeters per second), and you get 7 seconds.
PLEASE HELP,I AM BEING TIMED!!
Measure the angle of 6
Options: 100
120
30
60
30 degrees
Measure of the angle of 6 is 30 degrees
Answer:
30 degrees
Step-by-step explanation:
because 60 degrees is already there and 120 is angled the opposite way, and 100 degrees is a little farther than 90.
6+3{[4-2(4-7)]}=
I need answer this whit steps.
pls.
Step-by-step explanation:
6+3((4-2(4-7))
1. 6+3((2(4-7))
2. 6+3((2(*-3)
3. 6+3(-6)
4. 6+(3*-6)
5. 6+-18
6. -12
Diego pays $4.50 for 3 pounds of bananas and 2 pounds of oranges. one pound of oranges cost $0.75 more than one pound of bananas let b represent the price per pound of bananas. Which equation represents the situation, and what is the price per pound of each fruit? 3b + 2b = 4.50; $0.90 per bananas, $0.90 per pound of oranges 3b + 2(0.75) = 4.50; $1.00 per bananas, $1.75 per pound of oranges 3b + 2(b + 0.75) = 4.50; $0.60 per bananas, $1.35 per pound of oranges 3b + 2b = 4.50; $1.35 per bananas, $0.60 per pound of oranges
Given:
Diego pays $4.50 for 3 pounds of bananas and 2 pounds of oranges.
One pound of oranges cost $0.75 more than one pound of bananas.
To find:
The equation which represents the situation, and what is the price per pound of each fruit.
Solution:
Let b represent the price per pound of bananas.
Cost of 3 pound of banana = $3b
One pound of oranges cost $0.75 more than one pound of bananas.
So, cost of one pound of oranges = $(b+0.75)
Cost of 2 pound of oranges = $2(b+0.75)
Diego pays $4.50 for 3 pounds of bananas and 2 pounds of oranges.
[tex]3b+2(b+0.75)=4.50[/tex]
Therefore, the required equation is [tex]3b+2(b+0.75)=4.50[/tex].
On solving the above equation, we get
[tex]3b+2b+1.50=4.50[/tex]
[tex]5b=4.50-1.50[/tex]
[tex]5b=3[/tex]
Divide both sides by 5.
[tex]b=\dfrac{3}{5}[/tex]
[tex]b=0.60[/tex]
Now,
[tex]b+0.75=0.60+0.75=1.35[/tex]
Therefore, the price of fruits are $0.60 per bananas, $1.35 per pound of oranges.
Hence, the correct option is C.
During a basketball season cathy sored 40% of her shots If she scored 140 goals how many shots did she make
Answer:
Step-by-step explanation:
Set up the following ratio. Let the total number of shots = x
40/100 = 140/x Cross multiply
40x = 100* 140 Combine the right
40x = 14000 Divide by 40
x = 14000/40
x = 350
She took 350 shots in total.
Look at this graph:
y
100
90
80
70
60
50
40
30
20
10
X
0
10
20 30 40 50 60 70 80 90 100
What is the slope?
Find the x- and y-intercepts of the graph of each ... CCSS REASONING The equation 5x + 10y = 60 ... 20. SOLUTION: The x-intercept is the point at which the y-coordinate is 0, or the line ... charges $50 for admission before 6 p.m. and $20 for ... (90, 70). 100 f (100) = 160 – (100). 60. (100, 60). 130 f (130) = 160 – (130). 30.
Leigh is renting a moving van The function C (d) = 0.12d + 25 is used to relate the cost, C, in dollars of the rental after Leigh
uses the van for d miles. What is the interpretation of the slope for this function? (1 point)
The initial cost of the moving van is $0.12
The additional cost is $25 for each additional mile the van is driven
The initial cost of the moving van is $25.
The additional cost is $0.12 for each additional mile the van is driven
Answer:
The additional cost is $0.12 for each additional mile the van is driven ⇒ D
Step-by-step explanation:
The form of the linear function is
f(x) = m x + b, where
m is the rate
b is the initial value
In our question
The function C(d) = 0.12 d + 25 represents the cost C in dollars of the rental van uses for d miles
→ Compare it with the form of the linear function above
∴ m = 0.12
∴ b = 25
∵ The cost C depends on the number of miles d
∴ $0.12 is the cost per mile ⇒ additional cost for each additional mile
∴ $25 is the initial cost ⇒ initial cost of the moving the van
The interpretation of the slope for this function is:
The additional cost is $0.12 for each additional mile the van is driven
Answer:
+0.12
Step-by-step explanation:
The additional cost is 0.12 cents for each additional mile the van is being driven.
pls help me...
A. y is all real numbers
B. y>0
C.-6</=x </=6
D.0 </=y </=8
♥️♥️♥️♥️♥️♥️♥️♥️♥️♥️♥️♥️♥️♥️
[tex]0 \leqslant y \leqslant 8[/tex]
♥️♥️♥️♥️♥️♥️♥️♥️♥️♥️♥️♥️♥️♥️
Thus the correct answer is (( D )) .
♥️♥️♥️♥️♥️♥️♥️♥️♥️♥️♥️♥️♥️♥️
What is the variable expression that represents the word phrase.
The product of 2 and x is increased by 7
Answer:
2x + 7
Step-by-step explanation:
The product of 2 and x
Product means multiplication
The product of 2 and x means multiplication of 2 and x
Which is equivalent to
2 * x = 2x
The product of 2 and x is increased by 7
That is, 7 is added to the product of 2 and x
Here,
2x + 7
The variable expression that represent the word phrase "The product of 2 and x is increased by 7" is 2x + 7
The y-intercept is...
Hey love, if the question itself is what the y-intercept stands for then it's the point when the line passes the y axis. if that wasn't your question give me some more context and i'll try my best to help!
What is 2.8 divides by 0.4? Please Show your work.
what is the distance of LM
Answer
(-4,3)
Step-by-step explanation: