Answer: (a), (b), (c), and (d)
Step-by-step explanation:
Check the options
[tex](a)\\\Rightarrow [\sin x+\cos x]^2=\sin ^2x+\cos ^2x+2\sin x\cos x\\\Rightarrow [\sin x+\cos x]^2=1+2\sin x\cos x\\\Rightarrow \Rightarrow [\sin x+\cos x]^2=1+\sin 2x[/tex]
[tex](b)\\\Rightarrow \sin (6x)=\sin 2(3x)\\\Rightarrow \sin 2(3x)=2\sin (3x)\cos (3x)[/tex]
[tex](c)\\\Rightarrow \dfrac{\sin 3x}{\sin x\cos x}=\dfrac{3\sin x-4\sin ^3x}{\sin x\cos x}\\\\\Rightarrow 3\sec x-4\sin ^2x\sec x\\\Rightarrow 3\sec x-4[1-\cos ^2x]\sec x\\\Rightarrow 3\sec x-4\sec x+4\cos x\\\Rightarrow 4\cos x-\sec x[/tex]
[tex](d)\\\Rightarrow \dfrac{\sin 3x-\sin x}{\cos 3x+\cos x}=\dfrac{2\cos [\frac{3x+x}{2}] \sin [\frac{3x-x}{2}]}{2\cos [\frac{3x+x}{2}]\cos [\frac{3x-x}{2}]}\\\\\Rightarrow \dfrac{2\cos 2x\sin x}{2\cos 2x\cos x}=\dfrac{\sin x}{\cos x}\\\\\Rightarrow \tan x[/tex]
Thus, all the identities are correct.
A. Not an identity
B. An identity
C. Not an identity
D. An identity
To check whether each expression is an identity, we need to verify if the equation holds true for all values of the variable x. If it is true for all values of x, then it is an identity. Let's check each option:
A. [tex]\((\sin x + \cos x)^2 = 1 + \sin 2x\)[/tex]
To check if this is an identity, let's expand the left-hand side (LHS):
[tex]\((\sin x + \cos x)^2 = \sin^2 x + 2\sin x \cos x + \cos^2 x\)[/tex]
Now, we can use the trigonometric identity [tex]\(\sin^2 x + \cos^2 x = 1\)[/tex] to simplify the LHS:
[tex]\(\sin^2 x + 2\sin x \cos x + \cos^2 x = 1 + 2\sin x \cos x\)[/tex]
The simplified LHS is not equal to the right-hand side (RHS) 1 + sin 2x since it is missing the sin 2x term. Therefore, option A is not an identity.
B. [tex]\(\sin 6x = 2 \sin 3x \cos 3x\)[/tex]
To check if this is an identity, we can use the double-angle identity for sine:[tex]\(\sin 2\theta = 2\sin \theta \cos \theta\)[/tex]
Let [tex]\(2\theta = 6x\)[/tex], which means [tex]\(\theta = 3x\):[/tex]
[tex]\(\sin 6x = 2 \sin 3x \cos 3x\)[/tex]
The equation holds true with the double-angle identity, so option B is an identity.
C. [tex]\(\frac{\sin 3x}{\sin x \cos x} = 4\cos x - \sec x\)[/tex]
To check if this is an identity, we can simplify the right-hand side (RHS) using trigonometric identities.
Recall that [tex]\(\sec x = \frac{1}{\cos x}\):[/tex]
[tex]\(4\cos x - \sec x = 4\cos x - \frac{1}{\cos x} = \frac{4\cos^2 x - 1}{\cos x}\)[/tex]
Now, using the double-angle identity for sine, [tex]\(\sin 2\theta = 2\sin \theta \cos \theta\),[/tex] let [tex]\(\theta = x\):[/tex]
[tex]\(\sin 2x = 2\sin x \cos x\)[/tex]
Multiply both sides by 2: [tex]\(2\sin x \cos x = \sin 2x\)[/tex]
Now, the left-hand side (LHS) becomes:
[tex]\(\frac{\sin 3x}{\sin x \cos x} = \frac{\sin 2x}{\sin x \cos x}\)[/tex]
Using the double-angle identity for sine again, let [tex]\(2\theta = 2x\):[/tex]
[tex]\(\frac{\sin 2x}{\sin x \cos x} = \frac{2\sin x \cos x}{\sin x \cos x} = 2\)[/tex]
So, the LHS is 2, which is not equal to the RHS [tex]\(\frac{4\cos^2 x - 1}{\cos x}\)[/tex]. Therefore, option C is not an identity.
D. [tex]\(\frac{\sin 3x - \sin x}{\cos 3x + \cos x} = \tan x\)[/tex]
To check if this is an identity, we can use the sum-to-product trigonometric identities:
[tex]\(\sin A - \sin B = 2\sin \frac{A-B}{2} \cos \frac{A+B}{2}\)\(\cos A + \cos B = 2\cos \frac{A+B}{2} \cos \frac{A-B}{2}\)[/tex]
Let A = 3x and B = x:
[tex]\(\sin 3x - \sin x = 2\sin x \cos 2x\)\(\cos 3x + \cos x = 2\cos 2x \cos x\)[/tex]
Now, we can rewrite the expression:
[tex]\(\frac{\sin 3x - \sin x}{\cos 3x + \cos x} = \frac{2\sin x \cos 2x}{2\cos 2x \cos x} = \frac{\sin x}{\cos x} = \tan x\)[/tex]
The equation holds true, so option D is an identity.
To know more about identity:
https://brainly.com/question/28974915
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The closed form sum of
$$12 \left[ 1^2 \cdot 2 + 2^2 \cdot 3 + \ldots + n^2 (n+1) \right]$$
for $n \geq 1$ is $n(n+1)(n+2)(an+b).$ Find $an + b.$
Perhaps you know that
[tex]S_2 = \displaystyle\sum_{k=1}^n k^2 = \frac{n(n+1)(2n+1)}6[/tex]
and
[tex]S_3 = \displaystyle\sum_{k=1}^n k^3 = \frac{n^2(n+1)^2}4[/tex]
Then the problem is trivial, since
[tex]\displaystyle\sum_{k=1}^n k^2(k+1) = S_2 + S_3 \\\\ = \frac{2n(n+1)(2n+1)+3n^2(n+1)^2}{12} \\\\ = \frac{n(n+1)\big((2(2n+1)+3n(n+1)\big)}{12} \\\\ = \frac{n(n+1)\big(4n+2+3n^2+3n\big)}{12} \\\\ = \frac{n(n+1)(3n^2+7n+2)}{12} \\\\ = \frac{n(n+1)(3n+1)(n+2)}{12}[/tex]
Then
[tex]12\bigg(1^2\cdot2+2^2\cdot3+3^2\cdot4+\cdots+n^2(n+1)\bigg) = n(n+1)(n+2)(3n+1)[/tex]
so that a = 3 and b = 1.
josh earns $415 a week plus 3% commision on any sales he makes. in a particular week, he gets paid $600. how much did he make in sales?
Answer:
$185 in commisions
Step-by-step explanation:
600 - 415 = $185
Answer:
6,166.666$ in total sales
Step-by-step explanation:
I. 600$, If 415$ is his earnings then 185 is commision [600 - 415 = 185]
II. Find 3% so 3/100 = 185/x
3x = 18,500
x = 6,166.666
Hope it help :)
the ratio of boys and girls in a 4:3.If there are 20 boys, find the number of girls.
Answer:
There are 15 girls.
Step-by-step explanation:
The ratio is 4:3, meaning there are 7 parts.
4 parts= 20 (Boys)
This means 1 part is 20 divided by 4.
1 part= 5
3 parts= Amount of girls.
3 x 5 = 15.
There are 15 girls.
Answer:
15
Step-by-step explanation:
Add:
boys : girls
4 : 3 = 7
20: ? = ?
Divide:
20 / 4 = 5
5 × 3 = 15
20: 15
So, there are 15 girls.
Find a 2-digit number smaller than 50, the sum of whose digits does not change after being multiplied by a number greater than 1
The only 2-digit number that is lesser than 50 and the sum of its digits remain unaffected despite being multiplied by a number < 1 would be '18.'
To prove, we will look at some situations:
If we add up the two digits of 18. We get,
[tex]1 + 8 = 9[/tex]
And we multiply 18 by 2 which is greater than 1. We get,
[tex]18[/tex] × [tex]2 = 36[/tex]
The sum remains the same i.e. [tex]3 + 6 = 9[/tex]
Similarly,
If 18 is multiplied to 3(greater than 1), the sum of the two digits comprising the number still remains the same;
[tex]18[/tex] × [tex]3 = 54[/tex]
where (5 + 4 = 9)
Once more,
Even if 18 is multiplied to 4 or 5(greater than 1), the sum of its digits will still be 9.
[tex]18[/tex] × [tex]4 = 72[/tex]
[tex](7 + 2 = 9)[/tex]
[tex]18[/tex] × [tex]5 = 90[/tex]
[tex](9 + 0 = 9)[/tex]
Thus, 18 is the answer.
Learn more about 'numbers' here: brainly.com/question/1624562
the unknown side of a trapezium.
Answer:
11 mm
Step-by-step explanation:
121=area of a rectangle and area of triangle
121=x*8+(0.5)*x*(14-8), 11x=121, x=11
Answer:
[tex]11mm[/tex]
Step-by-step explanation:
Let the unknown side be x
[tex]Area \: \: \: of \: \: the \: \: trapezium = \frac{1}{2} \times (sum \: \: of \: \: \: the \: \: parallel \: sides) \times height \\ 121= \frac{1}{2} \times (8 + 14) \times x \\ 121 \times 2 = 22x \\ 242 = 22x \\ \frac{242}{22} = \frac{22x}{22} \\ 11 = x[/tex]
HELP!!!!! PLEASE!!!!
Mrs Dlamini bought a 10kg bag of rice. She cooks 500g of rice per day. How many days will the bag of rice last her family
Answer:
20 days
Step-by-step explanation:
10kg= 10,000g
10,000g/500g= 20 days
Answer:
20 days
Step-by-step explanation:
1 kg=1000 grams
10 kg=10000 grams
10000÷500
=20 days
A net is a two-dimensional pattern that can be folded to form a three-dimensional surface. T/F
Answer: True
Explanation:
Consider a 3D cardboard box that you can unfold. It would unfold into a 2D net that you can then re-fold back into its 3D form. The 2D net is useful to help visualize and calculate the surface area. The surface area is simply the total area of all the external faces.
In the PQRS triangle PQ=QR, QR side extended to S Show that PQ+RS=QS. -S Q R
pls explain too
Answer:
Step-by-step explanation:
from the picture:
QP = QR
and
QR = RS
so
PQ + RS = QS
Can some some help me with this one?
Answer: 7.5 x 10^3 is the answer.
Step-by-step explanation: First, 7.5 x 10^9 is 7500000000. Next, if you convert it to kilometers you would get 7500. And, 7.5 x 10^3 = 7500. Thus, that is the answer.
Man spends 1/5 of his salary on shoes and 2/3 on school fess what fraction of his salary is left
Find 3 solutions to the equation y = −4x −1
Answer:
(0,-1)
(1,-5)
(2,-9)
Step-by-step explanation:
y = −4x −1
Let x=0
y = 0-1 = -1 (0,-1)
Let x = 1
y = -4(1) -1 = -4-1 =-5 (1,-5)
Let x = 2
y = -4(2) -1 = -8-1 =-9 (2,-9)
Find the missing part.
Step-by-step explanation:
8²+10²= x²
64+100 =x²
164=x²
√164=√x²
square and squre root do cancel each other
x=√164
hope this helps
Answer:
x = [tex]\sqrt{164[/tex]
Step-by-step explanation:
In order to find the hypotenuse of a right triangle, you have to add,
[tex]8^{2} + 10^{2}[/tex]
which simplifies to,
64 + 100 =
[tex]\sqrt164[/tex]
Hope this helps! :)
Help plz!! Trig question
Answer:
5 meters tall.
Step-by-step explanation:
We know that angle A (the bird) is 55. The opposite side from this angle is 4.5 meters, since that is the distance she is from the tree. From this, we can set up an equation tan(55)=4.5/x and simplify it to x= 4.5/tan(55). This would be height x, but it is not accounting for the 1.5 meters the apple is off the ground, so you need to add 1.5 to the result. The final answer would be 4.65 but you need to round to the nearest meter so it would be 5.
abc or d please help me!
Answer:
A
Step-by-step explanation:
Cuz that’s the only graph that goes with the statement.
on this graph it’s saying the cost of 3 hair accessories is $10.
X = 3
Y = 10
Which of the following is a rational number?
square root 97square root 98, square root 99, square root 100
square root 97
square root 98
square root 99
square root 100
Part A: Find a rational number that is between 5.2 and 5.5. Explain why it is rational. (2 points)
Part B: Find an irrational number that is between 5.2 and 5.5. Explain why it is irrational. Include the decimal approximation of the irrational number to the nearest hundredth. (3 points)
Answer:
Square root of 100.
Step-by-step explanation:
Step-by-step explanation:
We can write 10 as a fraction 10/1, therefore,[tex]\sqrt{}[/tex]100 is a rational number.
Part A : A rational no. between 5.2 and 5.5 is 5.3.
It is rational because it can be expressed in the form
p/q where p and q are integers and q is not equal to 0, which is 53/10
Part B: A rational no. between 5.2 and 5.5 is 5.29150262213
An irrational number between 5.2 and 5.5 is 5.29150262213. It is irrational because there is no pattern that repeats and it cant be written as a fraction of two whole numbers.
Answer:
Step-by-step explanation:
Square root of all prime numbers is irrational.
97 is an prime number. So √97 is an irrational number.
√98 = [tex]\sqrt{2*7*7}=7\sqrt{2}[/tex] is an irrational number.
[tex]\sqrt{99}=\sqrt{3*3*11} =3\sqrt{11}[/tex] is an irrational number
√100 = 10 is a rational number.
Part A:
5.3 is a rational number.
Rational can be written in p/q form and when divided the result will either terminating decimal or non terminating repeating decimal
Part B:
5.3020050......
Irrational numbers are non terminating non repeating numbers
Consider the following points. (−4, −1) and (4, 2) Let Y'O' be the image of YO after a reflection across line . Suppose that ′ is located at (1, 4) and ′ is located at (−2, −4). Which of the following is true about line ?
The line was reflected about the line y = -x.
-----------------------------------------
This question is solved using reflection concepts.
There are various kinds of reflections, 90º clockwise about the origin, 90º counterclockwise, among other, and each of them has a rule.
-----------------------------------------
In this question:
Point (-4,-1) became (1,4).Point (4,2) became (-2,-4).From this, we have the following rule: (x,y) -> (-y,-x)
Checking the rules, it can be said that the line was reflected about the line y = -x.
A similar question can be found at https://brainly.com/question/16130908
15 × u= 135
---------------
Answer:
15u = 135
Divide both sides by 15
u = 135/15 = 9
U is equal to 9
Hope this helps!
[tex]15 \times 4 = 135 \\ divide \: both \: sides \: of \: equcation \\ by \: 15 \\ \frac{15u}{15} = \frac{135}{15} \\ u = \frac{135}{15} \\ u = 9[/tex]
a man is now 9 times as old as his daughter. in four years time he'll be five times as old as his daughter. what are their present ages
Answer: The father is 36 right now and the son is 4
Step-by-step explanation: 9x4 is 36 and 36/9=4.
The figure is not drawn to scale.
Answer:
93°, 89°, 93°
Step-by-step explanation:
∠1 and ∠4 are supplementary angles:
m∠1 + m∠4 = 180°; m∠1 = 93°
∠2 and ∠4 are vertical angles:
m∠2 = m∠4 = 87°
∠3 and ∠4 are supplementary angles:
m∠3 + m∠4 = 180°; m∠3 = 93°
this is geometry. hi i need help pls.
Answer:
Step-by-step explanation:
Answer:
B is the answer
Step-by-step explanation:
what is an equation in point-slope form of the line that passes through (-5,-3) and has a slope of negative 1
Answer:
Y= -x-8
Step-by-step explanation:
Hope this helps
Answer:
y + 3 = - (x + 5)
Step-by-step explanation:
The equation of a line in point- slope form is
y - b = m(x - a)
where m is the slope and (a, b) a point on the line
Here m = - 1 and (a, b ) = (- 5, - 3 ) , then
y - (- 3) = - 1 (x - (- 5) ) , that is
y + 3 = - (x + 5)
Ok ong i'm struggling and need help i've been up all night- Ill give brainlyest!!
Solve the following problems below. Find the variable and show your work
1.) 7(6x-1)+x=36
2.) 11-2(8+3p)=7²
3.) 1/4(5b+11)=19
4.) 2/7(4m-18)=12
Answer:
1- 91315Step-by-step explanation
- 7(6x-1)+x=36
=> 42x - 7 + x = 36=> 43x - 7 = 36=> 43x = 36 + 6 = 43=> x = 43 / 43 = 1- 11-2(8+3p)=7²
=> 11 - 16 - 6p = 7^2=> 11 - 16 - 6p = 49=> -5 - 6p = 49=> -6p = 49 + 5 = 54=> p = (54) / (-6) = - 9- 1/4(5b+11)=19
=> (5/4)b + (11/4) = 19=> (5/4)b = 19 - (11/4) = 65 /4=> b = (65/4) / (5/4) => (65/4) * (4/5) = 13- 2/7(4m-18)=12
=> (8/7)m - (36/7) = 12=> (8/7)m = 12 + (36/7) = 120/7=> m = (120/7) / (8/7) = 120 /8 = 15
√2+1
b) Rationalize the denominator:
√2+1
Answer:
there is not a complete question here ...
[tex]\sqrt{2} +1[/tex] is not in a denominator to rationalize
if it were you would multiply the rational expression (the fraction)
by [tex]\frac{\sqrt{2} -1}{\sqrt{2} -1}[/tex] the result (in the denominator would end up being "2 -1"
which is 1
Step-by-step explanation:
e. What order will result in a final output of 36 when the initial input is 1/7?
Answer:
5
Step-by-step explanation:
7x5=35
the remainder is 1
=5.1
the nearest number is 5
Rewrite the expression in the form y^n. (y^-1/2)^4
Answer:
y^ -2
Step-by-step explanation:
We know that a^b^c = a^(b*c)
(y^-1/2)^4
y^(-1/2*4)
y^ -2
Answer:
Step-by-step explanation:
Write an expression in simplified form for the area of each rectangle. Width: 11 Length: 3x+2
Answer:
33x+22
Step-by-step explanation:
The area of a rectangle is
A = l*w where l is the length and w is the width
A = 11(3x+2)
Distribute
= 33x+22
Area of rectangle = Length * Width
Length is 3x+2Width is 11⇛Area = 11(3x+2)
⇛Area = 33x + 22
Describe how (2 cubed) (2 superscript negative 4) can be simplified.
Answer:
1/2
Step-by-step explanation:
Given:
(2 cubed) (2 superscript negative 4)
= (2³)(2^-4)
= (2³) (1 / 2⁴)
= (2³ * 1) / 2⁴
= 2³ / 2⁴
Both numerator and denominator has the same base. Thus, pick one of the bases
Also, in indices, division sign can be translated to subtraction
Therefore,
2³ / 2⁴
= 2^3-4
= 2^-1
= 1/2¹
= 1/2
(2³)(2^-4) = 1/2
Answer:
D
Step-by-step explanation:
bc i said so
Please help out explanation need it
Answer:
156m^2
Step-by-step explanation:
I dont know what you mean by "each figure," but if you need the surface area for the rectangular prism, then the answer is 156, because the top square has a surface area of 3*6, then it equals 18, and the bottom square is the same, so it is also 18, 18*2=36
The 4 rectangle on the side are all the same, and they are all 10x3 which is 30, so 30*4 because there are 4 rectangles, equals 120.
120+36=156. The surface area of the rectangle is 156m^2
In the figure below, triangle ABC is similar to triangle PQR:
A right triangle ABC with right angle at B and base BC is drawn. Length of AB is 4, length of BC is 2. A similar right triangle; triangle PQR, which is triangle ABC enlarged and reflected across a horizontal line, is drawn near it. The right angle is at Q. Angle A is congruent to angle P and angle C is congruent to angle R. The length of QR is 12.
What is the length of side PQ?
48
24
40
18
Answer:
The length of PQ is 24
Step-by-step explanation:
If you flip triangle PQR upside down the you can see which side corresponds to which. BC corresponds to QR and AB with PQ, the scalar is 6 because of BC (2) times 6 = QR (12). So you just have to do 4 times 6 which is 24.
