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Genrish500 [490]
3 years ago
12

For 0 less than or equal to theta less than 2(pi), what are thebsolutions to sin↑2(theta)=2(sin↑2)(theta/2)? ​

Mathematics
1 answer:
qaws [65]3 years ago
3 0

I assume the up arrows are supposed to indicate exponents, so that the equation is

sin²(<em>θ</em>) = 2 sin²(<em>θ</em>/2)

Recall the half-angle identity for sine,

sin²(<em>θ</em>/2) = (1 - cos(<em>θ</em>))/2,

as well as the Pythagorean identity,

sin²(<em>θ</em>) + cos²(<em>θ</em>) = 1

Rewrite the equation in terms of cosine and solve:

1 - cos²(<em>θ</em>) = 1 - cos(<em>θ</em>)

cos²(<em>θ</em>) - cos(<em>θ</em>) = 0

cos(<em>θ</em>) (cos(<em>θ</em>) - 1) = 0

cos(<em>θ</em>) = 0   <u>or</u>   cos(<em>θ</em>) - 1 = 0

cos(<em>θ</em>) = 0   <u>or</u>   cos(<em>θ</em>) = 1

[<em>θ</em> = arccos(0) + 2<em>nπ</em>   <u>or</u>   <em>θ</em> = arccos(0) - <em>π</em> + 2<em>nπ</em>]   <u>or</u>

… … … [<em>θ</em> = arccos(1) + 2<em>nπ</em>]

(where <em>n</em> is any integer)

[<em>θ</em> = <em>π</em>/2 + 2<em>nπ</em>   <u>or</u>   <em>θ</em> = -<em>π</em>/2 + 2<em>nπ</em>]   <u>or</u>   [<em>θ</em> = 2<em>nπ</em>]

In the interval 0 ≤ <em>θ</em> < 2<em>π</em>, we get the solutions <em>θ</em> = 0, <em>π</em>/2, and 3<em>π</em>/2.

(That is, for <em>n</em> = 0 in the first and third solution families, and <em>n</em> = 1 in the second family.)

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Answer:

y=-2x+3

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the slope is -2 so that equals m

(0,3) is the y-intercept

y=b

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7 0
3 years ago
There is another way to rank the difficulty of test questions using Networks: Each student gives scores for each problem, then w
Vlad [161]

Answer:

Step-by-step explanation:

Truly, on a single student level, the Massey method can be applied i.e. problems are ranked according to their level of difficulty for a single student.

Let us say there are 4 problems P1, P2, P3, P4 set in an exam where a student has to solve any two.

Assuming student 1 solves and scores more in P1 than in P2, r1>r2

student 2  more in P1 than P4 r1>r4

student 3 more in P2 than P4 r2>r4

student 4 more in P4 than P3 r4>r3

student 5 more in P2 than P3 r2>r3

and such many more cases.

then the Massey's model can be written as

P1 P2 P3  P4

\begin{bmatrix} 1 &-1 &0 &0\\ 1 &-1 &0 &0 \\ 0 &1 & 0 &-1 \\ 0 & 0 &-1 & 1 \\ 0 & 1 & -1 & 0 \end{bmatrix}.

\times \begin{bmatrix}r1 \\r2 \\r3 \\r4 \end{bmatrix}  =

\begin{bmatrix}y_{1} \\ y_{2} \\ y_{3} \\ y_{4}\\ y_{5}\end{bmatrix}.            

If Pi gives more score than Pj, the entry below Pi shall be 1 and below Pj shall be -1.

The rest of the entries shall be 0.

r1,r2,r3,r4,r5 are ranks .  

There may be more rows depending upon more combination of 2 problems.

Should in case more than one student solve the same combination and get scores in the same order, for instance, 12 students solve P1 and P2, for 7 of them r1>r2, then their scores can be averaged.

Normally, such systems are overdetermined, i.e., more rows than the number of unknowns, then it is solved by the Least square method ., as there shall be no a solution with least error is found out.

The overall ranking, in reverse order, shall give the ranking according to increasing difficulty.

7 0
3 years ago
This is adding and subtracting of like and unlike terms <br> ab+bc=
NemiM [27]

Answer:

b(a+c) this is answer okay

5 0
3 years ago
Read 2 more answers
Drag the amounts to order them from greatest to least.
vladimir2022 [97]
So the first thing to do is get all amounts into one common unit. I will convert them all to ft.

1.8ft.
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So from that, You would have 56cm -> 1.8ft -> 20in. for Greatest to Least.
3 0
3 years ago
What is the 174th prime number?
34kurt
It is 1033. I think they dont have a way to find a prime number based on its position :p but the answer is everywhere on the Internet.
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