Evaluate. Express your answer in radians and in terms of  .

cos−1 2=  

a) sin−1 3= b)
2 2  

c) tan−1(1)=

d) sin−1− 3= e) cos−1−1= f) tan−1− 3= 

2 2 3  

Let's evaluate the inverse trigonometric functions given in the question and express the answers in radians and in terms of \(\pi\).

---

### a) \(\sin^{-1}\left(\frac{3}{2}\right)\)

- The sine function only outputs values in the range \([-1, 1]\). Since \(\frac{3}{2}\) is greater than 1, this expression is **undefined**.

\[
\boxed{\text{Undefined}}
\]

---

### b) \(\cos^{-1}\left(\frac{2}{2}\right)\)

- \(\frac{2}{2} = 1\).
- The cosine of 0 radians (or 0 degrees) is 1. Therefore:

\[
\cos^{-1}(1) = 0
\]

So,

\[
\boxed{0}
\]

---

### c) \(\tan^{-1}(1)\)

- The tangent of \(\frac{\pi}{4}\) radians (or 45 degrees) is 1. Therefore:

\[
\tan^{-1}(1) = \frac{\pi}{4}
\]

So,

\[
\boxed{\frac{\pi}{4}}
\]

---

### d) \(\sin^{-1}\left(-\frac{\sqrt{3}}{2}\right)\)

- The sine of \(-\frac{\pi}{3}\) radians (or \(-60\) degrees) is \(-\frac{\sqrt{3}}{2}\). Therefore:

\[
\sin^{-1}\left(-\frac{\sqrt{3}}{2}\right) = -\frac{\pi}{3}
\]

So,

\[
\boxed{-\frac{\pi}{3}}
\]

---

### e) \(\cos^{-1}(-1)\)

- The cosine of \(\pi\) radians (or 180 degrees) is \(-1\). Therefore:

\[
\cos^{-1}(-1) = \pi
\]

So,

\[
\boxed{\pi}
\]

---

### f) \(\tan^{-1}\left(-\frac{\sqrt{3}}{3}\right)\)

- The tangent of \(-\frac{\pi}{6}\) radians (or \(-30\) degrees) is \(-\frac{\sqrt{3}}{3}\). Therefore:

\[
\tan^{-1}\left(-\frac{\sqrt{3}}{3}\right) = -\frac{\pi}{6}
\]

So,

\[
\boxed{-\frac{\pi}{6}}
\]

---

Do you have any questions or would you like to go over the details of any of these solutions?

Here are five related questions you might find helpful:

1. What is \(\sin^{-1}(0)\)?
2. What is the range of the \(\cos^{-1}(x)\) function?
3. How do you evaluate \(\tan^{-1}(\infty)\)?
4. What is the relationship between \(\sin^{-1}(x)\) and \(\cos^{-1}(x)\)?
5. Can you calculate \(\cos^{-1}\left(\frac{\sqrt{3}}{2}\right)\)?

**Tip**: Remember that the domain and range of inverse trigonometric functions are key to determining valid outputs.

Explore how to evaluate inverse trigonometric functions such as sin^-1(3/2), cos^-1(1), tan^-1(1), sin^-1(-sqrt(3)/2), cos^-1(-1), and tan^-1(-sqrt(3)/3) in radians and Pi representation. Understand the domain and range constraints of these functions with detailed solutions.

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