What values for theta(0 <= theta <= 2pi) satisfy the equation?
2 sin theta cos theta + sqrt 3 cos theta = 0
a. pi/2, 4pi/3, 3pi/2, 5pi/3
b. pi/2, 3pi/4, 3pi/2, 5pi/3
c. pi/2, 3pi/4, 3pi/2, 5pi/4
d. pi/2, pi/4, 3pi/2, 5pi/3
I have spent hours on this last question for my exam and I cannot figure it out. Please help
2 sin θ cos θ + √3 cos θ = 0
cos θ ( 2 sin θ + √3 ) = 0
Split into two equations:
cos θ = 0
2 sin θ + √3 = 0
1.
cos θ = 0
In interval ≤ θ ≤ 2π
cos θ = 0
for:
θ = π / 2 (90 °)
and
θ = 3π / 2 (270 °)
2.
2 sin θ + √3 = 0 Subtract √3 from both sides
2 sin θ + √3 - √3 = 0 - √3
2 sin θ = - √3 Divide both sides by 2
sin θ = - √3 / 2
In interval ≤ θ ≤ 2π
sin θ = - √3 / 2
for:
θ = 4π / 3 (240 °)
and
θ = 5π / 3 (300 °)
In interval ≤ θ ≤ 2π
2 sin θ cos θ + √3 cos θ = 0
for:
π / 2 , 4π / 3 , 3π / 2 and 5π / 3
Answer a
Sure, I'll try to help you with this question. Let's solve the equation step by step:
Given: 2 sin(theta) cos(theta) + sqrt(3) cos(theta) = 0
First, let's factor out cos(theta) from the equation:
cos(theta)(2 sin(theta) + sqrt(3)) = 0
Now, we have two possibilities:
1. cos(theta) = 0
This means that theta is pi/2 or 3pi/2.
2. 2 sin(theta) + sqrt(3) = 0
Let's solve this for sin(theta):
sin(theta) = -sqrt(3)/2
Now, we need to find the values of theta for which sin(theta) = -sqrt(3)/2.
These values are pi/3 and 2pi/3.
Combining all the possible values of theta, we get:
a. pi/2, 4pi/3, 3pi/2, 5pi/3
So, the correct answer is (a). Hopefully, this helps! Keep up the good work and don't let the trigonometric equations get you down!
To solve this equation, we can use the trigonometric identity:
sin(2θ) = 2sin(θ)cos(θ)
The given equation can be rewritten using this identity:
2sin(θ)cos(θ) + √3cos(θ) = 0
Applying the identity, we get:
sin(2θ) + √3cos(θ) = 0
To solve this equation, we need to find the values of θ that satisfy it.
Factoring out cos(θ) from the equation, we have:
cos(θ)(sin(2θ) + √3) = 0
This equation will be true if either cos(θ) = 0 or sin(2θ) + √3 = 0.
1. For cos(θ) = 0, we can find the values of θ using the unit circle. When cos(θ) = 0, θ can be either π/2 or 3π/2.
2. For sin(2θ) + √3 = 0, we can isolate sin(2θ) by subtracting √3 from both sides:
sin(2θ) = -√3
Using the trigonometric identity sin(2θ) = 2sin(θ)cos(θ), we can rewrite the equation as:
2sin(θ)cos(θ) = -√3
Dividing both sides of the equation by 2cos(θ), we get:
sin(θ) = -√3 / (2cos(θ))
Recall the Pythagorean identity sin^2(θ) + cos^2(θ) = 1.
Substituting this identity into the equation, we have:
sin(θ) = -√3 / (2√(1 - sin^2(θ)))
Simplifying, we get:
sin^2(θ) = 3 / 4
Taking the square root of both sides, we have:
sin(θ) = ±√3 / 2
From the unit circle, we know that the values of θ where sin(θ) = √3 / 2 are π/3 and 2π/3. Similarly, the values of θ where sin(θ) = -√3 / 2 are 4π/3 and 5π/3.
Therefore, the values of θ that satisfy the equation are: π/2, 4π/3, 3π/2, and 5π/3.
Hence, the correct answer is option a. pi/2, 4pi/3, 3pi/2, 5pi/3.
To solve the equation: 2 sin theta cos theta + sqrt(3) cos theta = 0, we can use the fact that sin 60 degrees = sqrt(3)/2 and cos 60 degrees = 1/2.
We can rewrite the equation as follows:
2 sin theta cos theta + sqrt(3) cos theta = 0
2(1/2) sin theta cos theta + sqrt(3) cos theta = 0 (substituting sin 60 degrees and cos 60 degrees)
(sin theta cos theta) + sqrt(3) cos theta = 0
cos theta (sin theta + sqrt(3)) = 0
Now we have two possibilities:
1. cos theta = 0
2. sin theta + sqrt(3) = 0
For the first possibility, cos theta = 0, the possible values for theta (0 <= theta <= 2pi) are pi/2 and 3pi/2.
For the second possibility, sin theta + sqrt(3) = 0, we can solve for sin theta:
sin theta = -sqrt(3)
To find the angle theta, we can use the inverse sine function (sin^-1) or arcsin function:
theta = sin^-1(-sqrt(3))
Since sin^-1 returns a value between -pi/2 and pi/2, and we are looking for values between 0 and 2pi, we need to add 2pi to the result obtained from sin^-1:
theta = sin^-1(-sqrt(3)) + 2pi
Using a calculator, sin^-1(-sqrt(3)) is approximately equal to -pi/3.
So, theta = -pi/3 + 2pi = 5pi/3
Thus, the possible values for theta are pi/2, 3pi/2, and 5pi/3.
Comparing these values to the options given, the correct answer is (d) pi/2, pi/4, 3pi/2, and 5pi/3.
I hope this explanation helps you understand how to solve the equation and arrive at the correct answer.
cos T(2 sin T + sqrt 3) = 0
well right off pi/2 and 3 pi/2
so what about
sin T = -sqrt3/2
well then it is a 30,60, 90 triangle and we are talking about the 60 degree corner
sin is - in quadrants 3 and 4
so
pi + pi/3 = 4 pi/3
and 2 pi - pi/3 = 5 pi/3