Evaluating cos((1/2)arccos(x))

• MHB
• Elissa89
In summary, the problem at hand involves finding the cosine of a half angle, which can be solved using the half-angle identity for cosine. The given angle must first be within the range of 0 to pi, and then the identity can be applied to find the cosine of the half angle.

Elissa89

I don't know how to solve this, we didn't really cover any problems like this in class

cos(1/2*cos^-1*x)

This is due tonight online and would like help please.

Last edited:
Re: Need help, due tonight

The first thing I would consider is:

$$\displaystyle 0\le\arccos(x)\le\pi$$

Hence:

$$\displaystyle 0\le\frac{1}{2}\arccos(x)\le\frac{\pi}{2}$$

This means the cosine of the given angle will be non-negative. Next, consider the half-angle identity for cosine:

$$\displaystyle \cos^2\left(\frac{\theta}{2}\right)=\frac{1+\cos(\theta)}{2}$$

Given that the cosine function will be non-negative, we may write:

$$\displaystyle \cos\left(\frac{\theta}{2}\right)=\sqrt{\frac{1+\cos(\theta)}{2}}$$

Can you proceed?

1. What does the expression cos((1/2)arccos(x)) represent?

The expression cos((1/2)arccos(x)) represents the cosine of half of the angle whose cosine is x. It is also known as the half-angle identity for cosine.

2. How do you evaluate cos((1/2)arccos(x))?

To evaluate cos((1/2)arccos(x)), first find the value of arccos(x) using a calculator or trigonometric table. Then, divide this value by 2 and take the cosine of the result. This will give you the final value of cos((1/2)arccos(x)).

3. What is the domain and range of cos((1/2)arccos(x))?

The domain of cos((1/2)arccos(x)) is all real numbers between -1 and 1, since the arccosine function is only defined for values between -1 and 1. The range of this expression is also between -1 and 1, as the cosine function has a range of -1 to 1.

4. Can cos((1/2)arccos(x)) be simplified?

Yes, cos((1/2)arccos(x)) can be simplified using the half-angle identity for cosine. This expression can be rewritten as √((1 + x)/2), where x represents the value inside the parentheses.

5. How is cos((1/2)arccos(x)) used in mathematics?

Cos((1/2)arccos(x)) is used in various mathematical calculations and equations involving trigonometric functions. It is also commonly used in geometry and physics to find unknown angles or to simplify complex expressions involving cosines.