Determine exact value of cos(cos^-1(19 pi)).
is this the cos (a+b)= cos a cos b- sina sin b? or is it something different. When plugging it in the calculator, do we enter it with cos and then the (cos^-1(19 pi)).
To determine the exact value of cos(cos^-1(19π)), we can use the concept of inverse trigonometric functions.
First, let's focus on the expression cos(cos^-1(19π)). The inverse cosine function, cos^-1, or arccosine, returns the angle whose cosine is the given value. So, in this case, cos^-1(19π) returns an angle whose cosine is 19π.
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(19π)) is just 19π.
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(19π)), 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(19π))." This will return the exact value of 19π in radians.
To determine the exact value of cos(cos^-1(19 pi)), we can use the identity:
cos(cos^(-1)(x)) = x
In this case, x = 19 pi.
Plugging in the value, we get:
cos(cos^(-1)(19 pi)) = 19 pi
So, the exact value of cos(cos^-1(19 pi)) is 19 pi.
Regarding your second question, when entering it into a calculator, you should first calculate the inverse cosine of 19 pi (cos^(-1)(19 pi)) and then find the cosine of that result. The specific sequence of buttons may vary depending on the calculator you are using.
an even number of pi is all the way around,360 degrees
and odd number of pi leaves you at 180, the negative x axis
the cos there at (-1,0) on theta = 180 degrees is -1