Huff Mccoy
09/23/2023 · Senior High School
If \( \theta=45^{\circ} \), then \( \sec (\theta)=\square \) \( \csc (\theta)=\square \) Give exact values. No derimals.allowodI
UpStudy ThothAI Solution
Tutor-Verified Answer
Step-by-step Solution
To find the exact values of \( \sec(\theta) \) and \( \csc(\theta) \) when \( \theta = 45^{\circ} \), we can use the definitions of these functions.
1. **Secant**:
\[
\sec(\theta) = \frac{1}{\cos(\theta)}
\]
For \( \theta = 45^{\circ} \):
\[
\cos(45^{\circ}) = \frac{\sqrt{2}}{2}
\]
Therefore,
\[
\sec(45^{\circ}) = \frac{1}{\cos(45^{\circ})} = \frac{1}{\frac{\sqrt{2}}{2}} = \frac{2}{\sqrt{2}} = \sqrt{2}
\]
2. **Cosecant**:
\[
\csc(\theta) = \frac{1}{\sin(\theta)}
\]
For \( \theta = 45^{\circ} \):
\[
\sin(45^{\circ}) = \frac{\sqrt{2}}{2}
\]
Therefore,
\[
\csc(45^{\circ}) = \frac{1}{\sin(45^{\circ})} = \frac{1}{\frac{\sqrt{2}}{2}} = \frac{2}{\sqrt{2}} = \sqrt{2}
\]
Thus, the exact values are:
\[
\sec(45^{\circ}) = \sqrt{2}
\]
\[
\csc(45^{\circ}) = \sqrt{2}
\]
Quick Answer
\[
\sec(45^{\circ}) = \sqrt{2}
\]
\[
\csc(45^{\circ}) = \sqrt{2}
\]
Answered by UpStudy AI and reviewed by a Professional Tutor
UpStudy ThothAI
Self-Developed and Ever-Improving
Thoth AI product is constantly being upgraded and optimized.
Covers All Major Subjects
Capable of handling homework in math, chemistry, biology, physics, and more.
Instant and Accurate
Provides immediate and precise solutions and guidance.
Try Now
Ask Tutors
Ask AI
10x
Fastest way to Get Answers & Solutions
By text
Enter your question here…
By image
Re-Upload
Submit