1answer.
Ask question
Login Signup
Ask question
All categories
  • English
  • Mathematics
  • Social Studies
  • Business
  • History
  • Health
  • Geography
  • Biology
  • Physics
  • Chemistry
  • Computers and Technology
  • Arts
  • World Languages
  • Spanish
  • French
  • German
  • Advanced Placement (AP)
  • SAT
  • Medicine
  • Law
  • Engineering
nikdorinn [45]
3 years ago
6

For 0 ≤ θ < 2 π what are the solutions to sin^2(θ) =2sin^2(θ/2)

Mathematics
1 answer:
gregori [183]3 years ago
6 0

Recall the half-angle identity for sine,

sin²(x/2) = (1 - cos(x))/2

Then the given equation is identical to

sin²(θ) = 1 - cos(θ)

Also recall the Pythagorean identity,

sin²(θ) + cos²(θ) = 1

Then we rewrite the equation as

1 - cos²(θ) = 1 - cos(θ)

Factoring the left side, we have

(1 - cos(θ)) (1 + cos(θ)) = 1 - cos(θ)

and so

(1 - cos(θ)) (1 + cos(θ)) - (1 - cos(θ)) = 0

and we factor this further as

(1 - cos(θ)) (1 + cos(θ) - 1) = 0

which gives

cos(θ) (1 - cos(θ)) = 0

Then either

cos(θ) = 0   or   1 - cos(θ) = 0

cos(θ) = 0   or   cos(θ) = 1

[θ = arccos(0) + 2nπ   or   θ = -arccos(0) + 2nπ]

…   or   [θ = arccos(1) + 2nπ   or   θ = -arccos(1) + 2nπ]

(where n is any integer)

[θ = π/2 + 2nπ   or   θ = -π/2 + 2nπ]   or   [θ = 0 + 2nπ]

In the interval 0 ≤ θ < 2π, we get three solutions:

• first solution set with n = 0   ⇒   θ = π/2

• second solution set with n = 1   ⇒   θ = 3π/2

• third solution set with n = 0   ⇒   θ = 0

So, the first choice is correct.

You might be interested in
Your family goes to a southern style restaurant for dinner. There are 6 people in your family. Some order the chicken dinner for
Len [333]
Ok. You can use equations to find out what number of each meal was eaten. 

x*14+y*17=99

If you solve for x and y (I used guess and check- there's only a few possible combinations) and got one chicken dinner and five steak dinners.
7 0
3 years ago
Read 2 more answers
Pi/24 is the solution for 4cos2 (4x) - 3 = 0<br><br> True or false
inessss [21]

Answer:

True.

Step-by-step explanation:

Given,

4cos2 (4x) - 3 = 0

Now, 4π/24 = π/6

Or, cos π/6 = √3/2

Or, cos^2 π/6 = 3/4

Or, 4 cos^2 π/6 = 3

Now, Left Hand Side= 4cos2 (4x) - 3

                                  = 3-3 =0 = Right Hand Side (Proved)

Here, Left Hand Side= Right Hanad Side. So, 4cos2 (4x) - 3=0 is true.

5 0
3 years ago
Read 2 more answers
The graph of a quadratic function contains (4,-64) and it's vertex is at the origin.write an equation for this function.
lesya692 [45]

Answer:

Step-by-step explanation:

Not sure what form you need this in, but it really doesn't matter, as you'll see in the final equation.  I used the vertex form and solved for a:

y=a(x-h)^2+k

We are given the vertex (h, k) as the origin (0, 0), and we have a point that the graph goes through as (4, -64).  That's our x and y.  Plugging in what we have:

-64=a(4-0)^2+0 gives us

-64 = 16a and

a = -4.  That means that the quadratic equation is

y=-4x^2  which is both vertex form and standard form here, no difference.

5 0
3 years ago
What is the value of z in the equation 2(4z − 3 − 1) = 166 − 46?
ZanzabumX [31]
In this equation z=16
7 0
3 years ago
PLEASE HELP LOTS OF POINTS!!!!!!!! TWO QUESTIONS!!!!!!!!!
zubka84 [21]

Answer: Number one would be 3x+2y+-14 and x+y=-4, second one is x=-3, y=7

Step-by-step explanation:

The first one is solved by inputting the x and y in each one and finding which one comes out true, the second on is solved by substitution. to find x you would subtract x in the first equation and make it y=4-x then input that equation in the y in the second equation.

6 0
3 years ago
Other questions:
  • The weight of softballs is uniformly distributed over the interval of 8 to 12 ounces, inclusively. what is the probability that
    12·1 answer
  • Susan is buying black and green olives from the Olive bar for her party. She buys 4 pounds of olives. Black olives cost three do
    11·1 answer
  • If the overnight temperature at the Arctic Circle was -14 degrees, but the temperature rose to 9 degrees during the day, what wa
    13·1 answer
  • What weight of wheat flour did pasha use in the test batch?
    5·2 answers
  • ILL GIVE BRAINLIEST please please help (only help if u know the right answer)
    12·1 answer
  • ABCD. If AB=5x and CD=3x + 10, find CD<br> A. 25<br> B. 2<br> C. 5<br> D. 20
    14·2 answers
  • Express 6^1/4 b^3/4 c^1/4 using a radical
    5·1 answer
  • I need help please help
    6·2 answers
  • The factors of p2-2pq+q are​
    11·2 answers
  • Which expression gives the area of the triangle shown below?
    5·1 answer
Add answer
Login
Not registered? Fast signup
Signup
Login Signup
Ask question!