For the equation 4 cos theta - square root 12 = 0, where 0 < theta < (or equal to) 2 pi, determine the measure of theta.
The answers to choose from are:
A) pi/ 3, 5 pi/ 3
B) 2 pi/ 3, 4 pi/ 3
C) 5 pi/ 6, 7 pi/ 6
D) pi/ 6, 11 pi/ 6
Please explain and show work, because I don't understand how to do this question, or get answers like that.
Ah, the joy of solving trigonometric equations! Let's break it down step by step, and I'll throw in a joke or two along the way to keep things lively.
Given the equation: 4 cos(theta) - sqrt(12) = 0
First, let's isolate the cosine term by moving the square root of 12 to the other side of the equation:
4 cos(theta) = sqrt(12)
Now, divide both sides by 4 to get the cosine term by itself:
cos(theta) = sqrt(12)/4
Now, simplify the right side:
cos(theta) = sqrt(3)
Looks like we've ended up with a nice, clean cosine value. Now, we just need to find the angles that have this cosine value.
Remember that the cosine function is positive in quadrants 1 and 4. So, we need to find the angles in quadrant 1 and quadrant 4 that have a cosine value of sqrt(3).
In quadrant 1, we find the reference angle by taking the inverse cosine (or arc cos) of sqrt(3). Let's calculate that:
theta_1 = arccos(sqrt(3))
Now, for the sake of simplicity, let's put this in terms of pi. Since the reference angle is in quadrant 1, it will be less than pi:
theta_1 = arccos(sqrt(3)) ≈ pi/6
So, we have one possible value of theta: pi/6.
But wait, there's more! In quadrant 4, the angle will be the same as in quadrant 1, but with a negative sign since cosine is negative in quadrant 4. So, we have:
theta_4 = -pi/6
Hence, the measure of theta would be the two values that we found in quadrant 1 and quadrant 4:
Theta = {pi/6, -pi/6}
Now, let's match these values with the given answer choices:
A) pi/3, 5pi/3
B) 2pi/3, 4pi/3
C) 5pi/6, 7pi/6
D) pi/6, 11pi/6
Looking at the options, we see that the values we found for theta match with option D: pi/6, 11pi/6.
So, the answer is D) pi/6, 11pi/6. And remember, solving equations may be serious business, but adding a little humor (or a clown) makes it all the more enjoyable!
To determine the measure of theta for the equation 4 cos theta - √12 = 0, we will solve for theta step-by-step.
1. Begin by isolating the cosine term by adding √12 to both sides of the equation:
4 cos theta = √12
2. Next, divide both sides of the equation by 4 to isolate the cosine term:
cos theta = √12 / 4
3. Simplify the right-hand side of the equation:
cos theta = √3
4. Now, recall the definition of cosine in terms of the unit circle. The cosine of an angle is equal to the x-coordinate of the point on the unit circle corresponding to the angle.
The value of cos theta is equal to √3 when the x-coordinate on the unit circle is equal to √3/2.
5. In the given answer choices, we need to find values of theta where the x-coordinate on the unit circle is equal to √3/2.
The options that satisfy this condition are A) π/3 and 5π/3, and B) 2π/3 and 4π/3.
6. To determine which of these options are valid solutions, we need to consider the given range of 0 < theta ≤ 2π.
From options A) and B), only A) satisfies this range since 5π/3 falls outside the given range.
7. Therefore, the measure of theta for the equation 4 cos theta - √12 = 0 in the range 0 < theta ≤ 2π is π/3.
Hence, the correct answer is A) π/3, 5π/3.
To find the measure of theta for the given equation, let's solve it step by step:
1. Start with the equation: 4cos(theta) - sqrt(12) = 0
2. Add sqrt(12) to both sides of the equation to isolate the cosine term:
4cos(theta) = sqrt(12)
3. Divide both sides of the equation by 4 to solve for cos(theta):
cos(theta) = sqrt(12) / 4
cos(theta) = sqrt(3) / 2
4. Recall that cosine is positive in the first and fourth quadrants. So, we need to find the angles in the interval (0, 2pi) where cos(theta) = sqrt(3) / 2.
5. Use the unit circle or reference angles to find the angles where cosine equals the given value. Here, the reference angle is pi/6, which has a cosine equal to sqrt(3) / 2.
6. For the first quadrant, theta = pi/6.
For the fourth quadrant, theta = 2pi - pi/6 = 11pi/6.
7. Now, compare these values with the options provided:
A) pi/3, 5pi/3
B) 2pi/3, 4pi/3
C) 5pi/6, 7pi/6
D) pi/6, 11pi/6
The values pi/3 and 5pi/3 are not equal to pi/6 or 11pi/6.
The values 2pi/3 and 4pi/3 are not equal to pi/6 or 11pi/6.
The values 5pi/6 and 7pi/6 match pi/6 and 11pi/6.
8. Therefore, the answer is option C) 5pi/6, 7pi/6.
By understanding the unit circle and the properties of trigonometric functions, we were able to solve for theta and determine the correct answer.