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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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Solve for the values of x and y in the diagram
qaws [65]

Answer: y = 0.5, x = 50

Step-by-step explanation: Both triangles in the picture are isosceles, telling us that the 2 angles at the bottom are congruent. With this, we can find y by doing the following:

a triangle has 180 degrees so we subtract the given 50 which gives us 130

2(2y + 64) = 130

4y + 128 = 130

y = .5

This means that the bottom 2 angles are both 65. Since the top angle of the second triangle is supplementary to the bottom angle of the first one, the top angle of the second triangle is 115. So, we find x by:

2(45 - x/4) = 65

x = 50

This means the bottom 2 angles of the second triangle are both 32.5.

7 0
3 years ago
Read 2 more answers
Use (&lt;,&gt;,=) to compare 2/3 and 3/4
lina2011 [118]
2/3 < 3/4. Hope that helps
3 0
3 years ago
If f(x) = 2x^2 + 2, find f(5)
zlopas [31]

Well, following the order of PEMDAS, I got choice B. 52

For instance, when you plug in 5 for x, you get F(5)=2(5)^2+2.

Moreover, following PEMDAS, you're supposed to solve what's inside the parenthesis, but since there is no operation going on inside the parenthesis, then you simple move on to the exponent.

In this case, you square the number 5, which gives you F(5)=2(25)+2

After that, you Multiply (letter M in PEMDAS). This results in F(5)=50+2.

Finally, you add them, which results in F=52.


By the way, I noticed a mistake in your work. When multiplying 2 by 5, the answer is 10, not 20.

Anyway, hope this helped! :-)  

8 0
3 years ago
Hi can someone pls pls plsssss help me with this sum ​
Dafna1 [17]

do 12.5/5 then you will get ur answer

6 0
3 years ago
Write the equation of a circle that has a center at the origin and a radius with a length of 4 inches
VMariaS [17]

Answer:

x^2+y^2=16

Step-by-step explanation:

x^2+y^2=r^2 is the mother function of the equation of a circle centered at r with radius r

since r is defined to be 4 and 4^2 is 16 therefore, the equation would then be

x^2+y^2=16

4 0
3 years ago
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