Find the exact value of the trigonometric function. (If an answer is undefined, enter UNDEFINED.)

csc(−450°)

To find the exact value of \( \csc(-450^\circ) \), we first need to simplify the angle by finding its coterminal angle.

### Step 1: Simplify the angle
We can simplify angles by adding or subtracting multiples of \(360^\circ\) until we get an angle between \(0^\circ\) and \(360^\circ\).

\[
-450^\circ + 360^\circ = -90^\circ
\]

Since \(-90^\circ\) is within the standard range, we can now use it for further calculation.

### Step 2: Use the trigonometric identity
The cosecant function is defined as the reciprocal of the sine function:
\[
\csc(\theta) = \frac{1}{\sin(\theta)}
\]
Thus, we need to calculate \( \csc(-90^\circ) \).

### Step 3: Find \( \sin(-90^\circ) \)
Using the known value of \( \sin(-90^\circ) \), we have:
\[
\sin(-90^\circ) = -1
\]

### Step 4: Calculate \( \csc(-90^\circ) \)
Now, calculate the reciprocal of \( \sin(-90^\circ) \):
\[
\csc(-90^\circ) = \frac{1}{\sin(-90^\circ)} = \frac{1}{-1} = -1
\]

Thus, the exact value of \( \csc(-450^\circ) \) is:

\[
\boxed{-1}
\]

Would you like more details or have any other questions?

Here are 5 related questions to further explore this topic:
1. How do you find coterminal angles for any given angle?
2. What is the sine function's behavior in different quadrants?
3. How do you find the cosecant of an angle in radians?
4. Why is the sine of \(-90^\circ\) equal to \(-1\)?
5. How do you solve trigonometric functions of angles beyond \(360^\circ\)?

**Tip:** When working with negative angles, remember that they represent clockwise rotations and can often be simplified using coterminal angles.

Find the exact value of the trigonometric function. (If an answer is undefined, enter UNDEFINED.) csc(−450°)

Learn how to find the exact value of csc(-450°) by simplifying the angle using coterminal angles and applying trigonometric identities. This detailed guide explores the relationship between sine and cosecant functions. Ideal for students in Grades 10-12.

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