Given tan *θ* = 1 and 0 ≤ *θ* ≤ 90°, what is csc *θ*?




Given that \(\tan(\theta) = 1\) and \(0^\circ \leq \theta \leq 90^\circ\), we need to find \(\csc(\theta)\).

### Step 1: Determine \(\theta\)
Since \(\tan(\theta) = 1\), we know that:
\[
\theta = 45^\circ
\]
because \(\tan(45^\circ) = 1\).

### Step 2: Find \(\csc(\theta)\)
The cosecant function is defined as:
\[
\csc(\theta) = \frac{1}{\sin(\theta)}
\]
For \(\theta = 45^\circ\), we know:
\[
\sin(45^\circ) = \frac{\sqrt{2}}{2}
\]

### Step 3: Calculate \(\csc(45^\circ)\)
\[
\csc(45^\circ) = \frac{1}{\frac{\sqrt{2}}{2}} = \frac{2}{\sqrt{2}} = \sqrt{2}
\]

So, \(\csc(\theta) = \sqrt{2}\).

The final answer is:

\[
\csc(\theta) = \sqrt{2}
\]

Would you like to explore more about trigonometric identities, or do you have any other questions?

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**Related Questions:**
1. What is \(\sin(\theta)\) when \(\tan(\theta) = \sqrt{3}\)?
2. How can you find \(\cos(\theta)\) given \(\tan(\theta)\)?
3. What are the values of other trigonometric functions at \(\theta = 45^\circ\)?
4. How do you solve for \(\theta\) if \(\cot(\theta) = 1\)?
5. What is the relationship between \(\csc(\theta)\) and \(\sin(\theta)\)?
6. What is the value of \(\sec(\theta)\) when \(\tan(\theta) = 1\)?
7. How do you graph the function \(\csc(\theta)\)?
8. What is \(\csc(\theta)\) if \(\theta = 30^\circ\)?

**Tip:** Always check if the trigonometric function corresponds to special angles like \(30^\circ\), \(45^\circ\), or \(60^\circ\) to simplify calculations.

Learn how to calculate cosecant (csc(θ)) when given that tangent (tan(θ)) equals 1 and θ ranges from 0 to 90 degrees. Understand the step-by-step process using trigonometric identities and special angles. Suitable for students in Grades 10-12.

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