Find the exact value of cosecant left parenthesis negative StartFraction 9 pi Over 4 EndFraction right parenthesis
.

To find the exact value of \( \csc\left( -\frac{9\pi}{4} \right) \), we can follow these steps:

### Step 1: Simplify the angle
The cosecant function is related to the sine function as:
\[
\csc(\theta) = \frac{1}{\sin(\theta)}
\]
We first simplify the angle \( -\frac{9\pi}{4} \) by expressing it as an equivalent angle between \( 0 \) and \( 2\pi \).

To do this, add multiples of \( 2\pi \) (since sine is periodic with period \( 2\pi \)):
\[
-\frac{9\pi}{4} + 2\pi = -\frac{9\pi}{4} + \frac{8\pi}{4} = -\frac{\pi}{4}
\]
So, \( \csc\left( -\frac{9\pi}{4} \right) = \csc\left( -\frac{\pi}{4} \right) \).

### Step 2: Use the identity for negative angles
The sine function is odd, meaning:
\[
\sin(-\theta) = -\sin(\theta)
\]
Therefore:
\[
\sin\left( -\frac{\pi}{4} \right) = -\sin\left( \frac{\pi}{4} \right)
\]

### Step 3: Find the value of \( \sin\left( \frac{\pi}{4} \right) \)
We know:
\[
\sin\left( \frac{\pi}{4} \right) = \frac{\sqrt{2}}{2}
\]
Thus:
\[
\sin\left( -\frac{\pi}{4} \right) = -\frac{\sqrt{2}}{2}
\]

### Step 4: Find the value of \( \csc\left( -\frac{\pi}{4} \right) \)
Now, using the definition of cosecant:
\[
\csc\left( -\frac{\pi}{4} \right) = \frac{1}{\sin\left( -\frac{\pi}{4} \right)} = \frac{1}{-\frac{\sqrt{2}}{2}} = -\frac{2}{\sqrt{2}} = -\sqrt{2}
\]

### Final Answer:
\[
\csc\left( -\frac{9\pi}{4} \right) = -\sqrt{2}
\]

Would you like more details on any step? Here are some related questions:

1. How do you convert angles beyond \( 2\pi \) to simpler forms?
2. What is the period of sine and cosecant functions?
3. Why is the sine function considered odd?
4. Can you find \( \csc(\frac{9\pi}{4}) \) without using negative angles?
5. How are cosecant and secant functions different?

**Tip:** The unit circle is a great tool for visualizing trigonometric values at different angles!

Find the exact value of cosecant left parenthesis negative StartFraction 9 pi Over 4 EndFraction right parenthesis.

Learn how to find the exact value of cosecant(-9π/4) with a step-by-step guide. This problem explores angle simplification, the periodicity of trigonometric functions, and the relationship between sine and cosecant. Suitable for students in Grades 10-12.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation