Math / Trigonometry

Question $\cos ((1 / 2)(x))=1 / 2$ where $x$ is a value between 0 and 2pi radians

Studdy Solution

STEP 1

What is this asking? Find the angles $x$ between $0$ and $2\pi$ radians where the cosine of half the angle, $\frac{x}{2}$ , equals $\frac{1}{2}$ . Watch out! We're dealing with half the angle, $x$ , not $x$ itself!
Also, make sure your answer is in radians, not degrees.

STEP 2

1. Find solutions for half the angle
2. Find solutions for the full angle
3. Check the range

STEP 3

We're given $\cos\left(\frac{x}{2}\right) = \frac{1}{2}$ .
So, we need to figure out what $\frac{x}{2}$ could be!

STEP 4

Think about the unit circle!
Where is the cosine equal to $\frac{1}{2}$ ?
That happens at $\frac{\pi}{3}$ and also at $\frac{5\pi}{3}$ .

STEP 5

Remember, we can keep going around the unit circle!
So, the general solutions for $\frac{x}{2}$ are $\frac{\pi}{3} + 2\pi k$ and $\frac{5\pi}{3} + 2\pi k$ , where $k$ is any integer.
This represents all the possible angles.

STEP 6

Now, let's find $x$ .
If $\frac{x}{2} = \frac{\pi}{3} + 2\pi k$ , then multiplying both sides by $2$ gives us $x = \frac{2\pi}{3} + 4\pi k$ .
Similarly, if $\frac{x}{2} = \frac{5\pi}{3} + 2\pi k$ , then $x = \frac{10\pi}{3} + 4\pi k$ .

STEP 7

Let's try different values of $k$ to see which ones give us solutions between $0$ and $2\pi$ .

STEP 8

If $k = 0$ , we get $x = \frac{2\pi}{3}$ and $x = \frac{10\pi}{3}$ . $\frac{2\pi}{3}$ is definitely between $0$ and $2\pi$ .
But $\frac{10\pi}{3}$ is bigger than $2\pi$ , so it's outside our desired range.

STEP 9

What if $k$ is negative?
If $k = -1$ , we'd get negative values for $x$ , which are also outside our range.

STEP 10

So, the only solution between $0$ and $2\pi$ is $x = \frac{2\pi}{3}$ .
It's within the range, and we've considered all possibilities!

STEP 11

The solution to $\cos\left(\frac{x}{2}\right) = \frac{1}{2}$ for $0 \le x \le 2\pi$ is $x = \frac{2\pi}{3}$ .

Was this helpful?

Get Studdy

Scan QR code with your device to get Studdy on iOS or Android

QR code

Studdy solves anything!

Download the app

Start learning now

Download Studdy AI Tutor now. Learn with ease and get all help you need to be successful at school.

Parents Influencer program Contact Policy Terms