2)Find the solution of sin2 theta = cos theta if 0 -< theta < 180
A)30 degrees and 90 degrees
B)30 degrees and 150 degrees
C)30 degrees, 90 degrees, 150 degrees
D)0 degrees, 90 degrees, and 150 degrees
sin2 theta = cos theta
sin2 theta - 1 = 0
(sin theta + 1)(sin theta - 1)= 0
sin theta = -1 or sin theta = 1 which is B
I used the zero property
For your earlier post for this same question at 3:17 am, I said
"for #2 I too, like Bob, was puzzled
First of all is it
sin 2θ = cos θ or sin2 θ = cos θ ?
for the first one
2sinθcosθ - cosθ = 0
cosθ(1sinθ - 1) = 0
cosθ=0 or sinθ=1/2
so θ = 90º or θ = 30º or 150º for your domain, which is choice C "
how did you get your second line???
To find the solution of the equation sin2 theta = cos theta, we need to manipulate the equation and use the zero property of multiplication.
1. Begin by subtracting cos theta from both sides of the equation:
sin2 theta - cos theta = 0
2. Rewrite sin2 theta as (sin theta)^2:
(sin theta)^2 - cos theta = 0
3. Factor the equation:
(sin theta + 1)(sin theta - 1) = 0
4. Now we have two possible cases to consider:
Case 1: sin theta + 1 = 0
If sin theta + 1 = 0, then sin theta = -1. To find the value of theta, we can use the inverse sine function (also known as arcsin or sin^-1). In this case, theta will be 270 degrees or 3π/2 radians. However, since the given range is 0 -< theta < 180 degrees, this solution is not within the given range.
Case 2: sin theta - 1 = 0
If sin theta - 1 = 0, then sin theta = 1. Again, we can use the inverse sine function to find the value of theta. In this case, theta will be 90 degrees or π/2 radians, which is within the given range.
Since case 1 is not within the given range, the only valid solution within the given range is theta = 90 degrees or π/2 radians. Therefore, the correct answer is option C) 30 degrees, 90 degrees, and 150 degrees.
line: cosθ(1sinθ - 1) = 0
should have said
cosθ(2sinθ - 1) = 0
Please use ^ in front of exponents. Otherwise, we cannot tell sin (2theta) from (sin theta)^2.
Your first step is wrong, since cos theta is not necessarly equal to 1. Also, if sin theta were 1 or -1, why would the answer be B?
I will assume you meant to write sin (2theta) and will let x be theta, to simplify typing.
2 sin x cos x = cos x
2 sin x = 1
sin x = 1/2
x = 30 or 150 degrees.
You seem to have treated sin2theta as sin^2 theta, in your next-to-last step, but if you do that, the answer is completely different
my question is sin 2è = cos è NOT sin2 è = cos è. I'm sorry, I didn't understand your question earlier.
In that case the solution I just gave you above will be the correct one.
Notice that all 3 answers work in your original equation.