To determine the exact value of cos(cos^-1(19Ï€)), we can use the concept of inverse trigonometric functions.
First, let's focus on the expression cos(cos^-1(19Ï€)). The inverse cosine function, cos^-1, or arccosine, returns the angle whose cosine is the given value. So, in this case, cos^-1(19Ï€) returns an angle whose cosine is 19Ï€.
Next, we can use the identity cos(cosθ) = θ to simplify the expression. Since we have cos(cos^-1(19π)), we can conclude that cos^-1(19π) is the angle itself.
So, the exact value of cos(cos^-1(19Ï€)) is just 19Ï€.
Regarding the cosine addition formula, cos(a + b) = cos(a)cos(b) - sin(a)sin(b), it is not applicable in this case since we are not dealing with a sum of angles. Therefore, we cannot use that formula here.
When using a calculator to evaluate cos(cos^-1(19Ï€)), you need to use the functions correctly. Most calculators have the inverse cosine function labeled as "cos^-1" or "arccos." To plug in the expression, you would enter "cos(arccos(19Ï€))." This will return the exact value of 19Ï€ in radians.