The question on your screen is asking about the secant function for an angle of $135^\circ$. Let's break it down step by step:

1. Quadrant for $135^\circ$:

   • $135^\circ$ is in the second quadrant because angles between $90^\circ$ and $180^\circ$ lie in this quadrant.

2. Reference angle:

   • The reference angle for an angle in the second quadrant is calculated as $180^\circ - \theta$.
   • For $\theta = 135^\circ$, the reference angle is: $180^\circ - 135^\circ = 45^\circ$

3. Secant in the second quadrant:

   • Secant ($\sec \theta$) is the reciprocal of the cosine ($\cos \theta$).
   • In the second quadrant, cosine is negative, so the secant will also be negative.

4. Exact value of $\sec 135^\circ$:

   • The reference angle is $45^\circ$, and $\cos 45^\circ = \frac{1}{\sqrt{2}}$.
   • Therefore, $\sec 45^\circ = \sqrt{2}$, and since $135^\circ$ is in the second quadrant, $\sec 135^\circ = -\sqrt{2}$.

Correct Choices:

• Quadrant: Second quadrant.
• Reference Angle: $45^\circ$.
• Exact Value: $-\sqrt{2}$.

Do you need more details or further clarification?

Here are 5 related questions to enhance your understanding:

1. What is the secant function in terms of sine and cosine?
2. How do you find the reference angle for any given angle?
3. Why is the cosine negative in the second quadrant?
4. Can you explain how to compute secant for angles in different quadrants?
5. What is the relationship between secant and cosecant?

Tip: To easily find the reference angle, always compare the given angle to the nearest x-axis boundary (0°, 180°, or 360°).