# Solving for sin & cos of -1 ∞: Answers & Explanation

• intenzxboi
In summary, the inverse tangent function has a limit of pi/2 as x approaches infinity, but the inverse sine and cosine functions do not have limits as x approaches infinity. The domain for inverse sine is typically [-pi/2, pi/2], and the domain for inverse cosine is typically [0, pi]. These intervals are chosen to make the functions one-to-one. The symbol \infty can be used as shorthand for infinity, but it also has more rigorous meanings in geometric and analytic contexts.
intenzxboi
this isn't homework just wanted to know what the values are.

tan -1 (infinity) = pi/2
tan-1 (0) = 0

what is sin -1 infinty and cos -1 infinity?

Looks like LaTeX isn't working again.

intenzxboi said:
this isn't homework just wanted to know what the values are.

tan -1 (infinity) = pi/2
tan-1 (0) = 0
The second one is correct, but not the first one. What you can say, though, is that
lim(x -->infinity) tan-1(x) = pi/2

The domain of the inverse tangent function is all real numbers, but neither -infinity nor infinity is included in that set.
intenzxboi said:
what is sin -1 infinty and cos -1 infinity?
The domain for sin-1(x) is usually taken as [-pi/2, pi/2], and the domain for cos-1(x) is usually taken as [0, pi]. These intervals are chosen to make these function one-to-one, which a function has to be in order for it to have an inverse.

Unline tan-1(x), neither the inverse sine nor inverse cosine have limits as x approaches infinity, so the answer to your last questions is that they aren't anything.

o ok thanks so only tan-1 (x) as x goes to infinty is pi/2

That is the simplest way to think of it. Using the symbol $\infty$ is often useful shorthand for the same thing. It can also be put on a sound rigorous footing geometrically (projective space) or analytically (Riemann sphere), but that requires using spaces strictly larger than the real (or complex) numbers.

## 1. What is the value of sin and cos of -1 ∞?

The value of sin and cos of -1 ∞ is undefined as infinity is not a specific value and cannot be plugged into the sine and cosine functions.

## 2. Can we solve for sin and cos of -1 ∞?

No, it is not possible to solve for the values of sin and cos of -1 ∞ as infinity is not a real number and cannot be used in mathematical equations.

## 3. Why is the value of sin and cos of -1 ∞ undefined?

The value of sin and cos of -1 ∞ is undefined because infinity is not a specific value and cannot be plugged into the sine and cosine functions, which require a real number as input.

## 4. Is -1 ∞ a valid input for the sine and cosine functions?

No, -1 ∞ is not a valid input for the sine and cosine functions as infinity is not a real number and cannot be used in mathematical equations.

## 5. What would happen if we tried to calculate sin and cos of -1 ∞?

If we tried to calculate sin and cos of -1 ∞, we would get an error as infinity is not a specific value and cannot be used in mathematical equations. The result would be undefined.

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