If Cos ⁡ X = 1 4 \cos X = \frac{1}{4} Cos X = 4 1 ​ , What Is The Positive Value Of Sin ⁡ X 2 \sin \frac{x}{2} Sin 2 X ​ In Simplest Radical Form With A Rational Denominator?

If $\cos x = \frac{1}{4}$, what is the positive value of $\sin \frac{x}{2}$ in simplest radical form with a rational denominator?

In trigonometry, the relationship between the sine and cosine functions is crucial for solving various problems. Given the value of $\cos x$, we can find the value of $\sin \frac{x}{2}$ using the half-angle formula. In this article, we will explore how to find the positive value of $\sin \frac{x}{2}$ in simplest radical form with a rational denominator when $\cos x = \frac{1}{4}$.

The half-angle formula for sine is given by:

$$\sin \frac{x}{2} = \pm \sqrt{\frac{1 - \cos x}{2}}$$

This formula allows us to find the value of $\sin \frac{x}{2}$ when we know the value of $\cos x$. We will use this formula to find the positive value of $\sin \frac{x}{2}$ when $\cos x = \frac{1}{4}$.

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To find the positive value of $\sin \frac{x}{2}$, we will substitute $\cos x = \frac{1}{4}$ into the half-angle formula:

$$\sin \frac{x}{2} = \pm \sqrt{\frac{1 - \frac{1}{4}}{2}}$$

Simplifying the expression, we get:

$$\sin \frac{x}{2} = \pm \sqrt{\frac{\frac{3}{4}}{2}}$$

$$\sin \frac{x}{2} = \pm \sqrt{\frac{3}{8}}$$

Since we want the positive value of $\sin \frac{x}{2}$, we will take the positive square root:

$$\sin \frac{x}{2} = \sqrt{\frac{3}{8}}$$

To simplify the radical, we can express $\frac{3}{8}$ as a product of two numbers that have a difference of 1:

$$\frac{3}{8} = \frac{1}{2} \cdot \frac{3}{4}$$

Now, we can rewrite the radical as:

$$\sin \frac{x}{2} = \sqrt{\frac{1}{2} \cdot \frac{3}{4}}$$

Using the property of radicals, we can rewrite this as:

$$\sin \frac{x}{2} = \sqrt{\frac{1}{2}} \cdot \sqrt{\frac{3}{4}}$$

Simplifying further, we get:

$$\sin \frac{x}{2} = \frac{1}{\sqrt{2}} \cdot \frac{\sqrt{3}}{2}$$

Rationalizing the denominator, we get:

$$\sin \frac{x}{2} = \frac{\sqrt{3}}{2\sqrt{2}} \cdot \frac{\sqrt{2}}{\sqrt{2}}$$

Simplifying, we get:

$$\sin \frac{x}{2} = \frac{\sqrt{6}}{4}$$

In this article, we used the half-angle formula to find the positive value of $\sin \frac{x}{2}$ when $\cos x = \frac{1}{4}$. We simplified the radical expression to get the final answer in simplest radical form with a rational denominator. The positive value of $\sin \frac{x}{2}$ is $\frac{\sqrt{6}}{4}$.

The final answer is: $\boxed{\frac{\sqrt{6}}{4}}$
Q&A: If $\cos x = \frac{1}{4}$, what is the positive value of $\sin \frac{x}{2}$ in simplest radical form with a rational denominator?

In our previous article, we explored how to find the positive value of $\sin \frac{x}{2}$ in simplest radical form with a rational denominator when $\cos x = \frac{1}{4}$. We used the half-angle formula to find the value of $\sin \frac{x}{2}$ and simplified the radical expression to get the final answer. In this article, we will answer some frequently asked questions related to this topic.

A: The half-angle formula for sine is given by:

$$\sin \frac{x}{2} = \pm \sqrt{\frac{1 - \cos x}{2}}$$

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A: To find the positive value of $\sin \frac{x}{2}$, you can substitute $\cos x = \frac{1}{4}$ into the half-angle formula:

$$\sin \frac{x}{2} = \pm \sqrt{\frac{1 - \frac{1}{4}}{2}}$$

Simplifying the expression, you get:

$$\sin \frac{x}{2} = \pm \sqrt{\frac{\frac{3}{4}}{2}}$$

$$\sin \frac{x}{2} = \pm \sqrt{\frac{3}{8}}$$

Since you want the positive value of $\sin \frac{x}{2}$, you will take the positive square root:

$$\sin \frac{x}{2} = \sqrt{\frac{3}{8}}$$

A: To simplify the radical expression, you can express $\frac{3}{8}$ as a product of two numbers that have a difference of 1:

$$\frac{3}{8} = \frac{1}{2} \cdot \frac{3}{4}$$

Now, you can rewrite the radical as:

$$\sin \frac{x}{2} = \sqrt{\frac{1}{2} \cdot \frac{3}{4}}$$

Using the property of radicals, you can rewrite this as:

$$\sin \frac{x}{2} = \sqrt{\frac{1}{2}} \cdot \sqrt{\frac{3}{4}}$$

Simplifying further, you get:

$$\sin \frac{x}{2} = \frac{1}{\sqrt{2}} \cdot \frac{\sqrt{3}}{2}$$

Rationalizing the denominator, you get:

$$\sin \frac{x}{2} = \frac{\sqrt{3}}{2\sqrt{2}} \cdot \frac{\sqrt{2}}{\sqrt{2}}$$

Simplifying, you get:

$$\sin \frac{x}{2} = \frac{\sqrt{6}}{4}$$

A: The final answer is $\boxed{\frac{\sqrt{6}}{4}}$.

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A: Yes, you can use this method to find the value of $\sin \frac{x}{2}$ for any value of $\cos x$. Simply substitute the value of $\cos x$ into the half-angle formula and simplify the radical expression.

In this article, we answered some frequently asked questions related to finding the positive value of $\sin \frac{x}{2}$ in simplest radical form with a rational denominator when $\cos x = \frac{1}{4}$. We used the half-angle formula and simplified the radical expression to get the final answer. We hope this article has been helpful in understanding this topic.