Find the exact value of the trig function. Don't forget to rationalize any denominators. Sec 945° mono
Find the exact value of the trig function. Don't forget to rationalize any denominators. Sec 945° mono
Trigonometry (MindTap Course List)
8th Edition
ISBN:9781305652224
Author:Charles P. McKeague, Mark D. Turner
Publisher:Charles P. McKeague, Mark D. Turner
Chapter2: Right Triangle Trigonometry
Section: Chapter Questions
Problem 5GP
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Question
![**Finding the Exact Value of the Trigonometric Function**
Ensure to rationalize any denominators.
**Problem Statement:**
Find the exact value of the trigonometric function.
Don't forget to rationalize any denominators.
**Trigonometric Function:**
\[ \sec 945^\circ \]
**Solution:**
To find the exact value of \(\sec 945^\circ\):
1. **Reduce the Angle:**
The angle given, \(945^\circ\), is more than one full circle (360°). First, we reduce \(945^\circ\) by finding the equivalent angle between \(0^\circ\) and \(360^\circ\):
\[
945^\circ - 2 \times 360^\circ = 945^\circ - 720^\circ = 225^\circ
\]
2. **Identify Reference Angle:**
The equivalent angle within one cycle (0° to 360°) is \(225^\circ\). \(225^\circ\) is in the third quadrant where both sine and cosine values are negative.
3. **Secant Function:**
The secant function is the reciprocal of the cosine function:
\[
\sec \theta = \frac{1}{\cos \theta}
\]
4. **Cosine of 225°:**
From the unit circle, the cosine of 225° (or \(\cos 225^\circ\)):
\[
\cos 225^\circ = -\frac{\sqrt{2}}{2}
\]
5. **Calculate Secant:**
Taking the reciprocal of \(\cos 225^\circ\):
\[
\sec 225^\circ = \frac{1}{\cos 225^\circ} = \frac{1}{-\frac{\sqrt{2}}{2}} = -\frac{2}{\sqrt{2}}
\]
6. **Rationalize the Denominator:**
To rationalize the denominator:
\[
-\frac{2}{\sqrt{2}} \times \frac{\sqrt{2}}{\sqrt{2}} = -\frac{2\sqrt{2}}{2} = -\sqrt{2}
\]
Thus, the exact value of \(\sec 945^\circ\) is:
\[ \boxed{-\sqrt{2}} \](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F24a6a4c4-f26c-45ad-82dd-c449c9817221%2F40482270-2a93-416f-8713-c0809d8228e3%2Fei1w2jc_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Finding the Exact Value of the Trigonometric Function**
Ensure to rationalize any denominators.
**Problem Statement:**
Find the exact value of the trigonometric function.
Don't forget to rationalize any denominators.
**Trigonometric Function:**
\[ \sec 945^\circ \]
**Solution:**
To find the exact value of \(\sec 945^\circ\):
1. **Reduce the Angle:**
The angle given, \(945^\circ\), is more than one full circle (360°). First, we reduce \(945^\circ\) by finding the equivalent angle between \(0^\circ\) and \(360^\circ\):
\[
945^\circ - 2 \times 360^\circ = 945^\circ - 720^\circ = 225^\circ
\]
2. **Identify Reference Angle:**
The equivalent angle within one cycle (0° to 360°) is \(225^\circ\). \(225^\circ\) is in the third quadrant where both sine and cosine values are negative.
3. **Secant Function:**
The secant function is the reciprocal of the cosine function:
\[
\sec \theta = \frac{1}{\cos \theta}
\]
4. **Cosine of 225°:**
From the unit circle, the cosine of 225° (or \(\cos 225^\circ\)):
\[
\cos 225^\circ = -\frac{\sqrt{2}}{2}
\]
5. **Calculate Secant:**
Taking the reciprocal of \(\cos 225^\circ\):
\[
\sec 225^\circ = \frac{1}{\cos 225^\circ} = \frac{1}{-\frac{\sqrt{2}}{2}} = -\frac{2}{\sqrt{2}}
\]
6. **Rationalize the Denominator:**
To rationalize the denominator:
\[
-\frac{2}{\sqrt{2}} \times \frac{\sqrt{2}}{\sqrt{2}} = -\frac{2\sqrt{2}}{2} = -\sqrt{2}
\]
Thus, the exact value of \(\sec 945^\circ\) is:
\[ \boxed{-\sqrt{2}} \
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