14+ Unit Circle Chart Templates - DOC, PDF - Free Printable
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Step-by-step solution for: 14+ Unit Circle Chart Templates - DOC, PDF
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Show Answer Key & Explanations
Step-by-step solution for: 14+ Unit Circle Chart Templates - DOC, PDF
To solve the problems using the unit circle and trigonometric identities, we will evaluate each trigonometric function step by step. Here's a detailed explanation for each problem:
---
- The angle \(45^\circ\) is in the first quadrant.
- On the unit circle, \(\sin(45^\circ) = \frac{\sqrt{2}}{2}\).
Answer: \(\boxed{\frac{\sqrt{2}}{2}}\)
---
- The angle \(30^\circ\) is in the first quadrant.
- On the unit circle, \(\cos(30^\circ) = \frac{\sqrt{3}}{2}\).
Answer: \(\boxed{\frac{\sqrt{3}}{2}}\)
---
- The angle \(60^\circ\) is in the first quadrant.
- On the unit circle, \(\tan(60^\circ) = \sqrt{3}\).
Answer: \(\boxed{\sqrt{3}}\)
---
- The angle \(120^\circ\) is in the second quadrant.
- \(\sec(120^\circ) = \frac{1}{\cos(120^\circ)}\).
- \(\cos(120^\circ) = -\frac{1}{2}\) (since \(\cos(180^\circ - 60^\circ) = -\cos(60^\circ)\)).
- Therefore, \(\sec(120^\circ) = \frac{1}{-\frac{1}{2}} = -2\).
Answer: \(\boxed{-2}\)
---
- The angle \(225^\circ\) is in the third quadrant.
- \(\cot(225^\circ) = \frac{1}{\tan(225^\circ)}\).
- \(\tan(225^\circ) = \tan(180^\circ + 45^\circ) = \tan(45^\circ) = 1\) (since tangent is positive in the third quadrant).
- Therefore, \(\cot(225^\circ) = \frac{1}{1} = 1\).
Answer: \(\boxed{1}\)
---
- The angle \(330^\circ\) is in the fourth quadrant.
- \(\csc(330^\circ) = \frac{1}{\sin(330^\circ)}\).
- \(\sin(330^\circ) = \sin(360^\circ - 30^\circ) = -\sin(30^\circ) = -\frac{1}{2}\).
- Therefore, \(\csc(330^\circ) = \frac{1}{-\frac{1}{2}} = -2\).
Answer: \(\boxed{-2}\)
---
- The angle \(270^\circ\) is on the negative \(y\)-axis.
- \(\cos(270^\circ) = 0\).
Answer: \(\boxed{0}\)
---
- The angle \(90^\circ\) is on the positive \(y\)-axis.
- \(\tan(90^\circ)\) is undefined because \(\cos(90^\circ) = 0\) and division by zero is undefined.
Answer: \(\boxed{\text{undefined}}\)
---
- The angle \(180^\circ\) is on the negative \(x\)-axis.
- \(\sin(180^\circ) = 0\).
Answer: \(\boxed{0}\)
---
- The angle \(-45^\circ\) is in the fourth quadrant.
- \(\csc(-45^\circ) = \frac{1}{\sin(-45^\circ)}\).
- \(\sin(-45^\circ) = -\sin(45^\circ) = -\frac{\sqrt{2}}{2}\).
- Therefore, \(\csc(-45^\circ) = \frac{1}{-\frac{\sqrt{2}}{2}} = -\sqrt{2}\).
Answer: \(\boxed{-\sqrt{2}}\)
---
- The angle \(-150^\circ\) is equivalent to \(360^\circ - 150^\circ = 210^\circ\) (in the third quadrant).
- \(\sec(-150^\circ) = \sec(210^\circ) = \frac{1}{\cos(210^\circ)}\).
- \(\cos(210^\circ) = \cos(180^\circ + 30^\circ) = -\cos(30^\circ) = -\frac{\sqrt{3}}{2}\).
- Therefore, \(\sec(-150^\circ) = \frac{1}{-\frac{\sqrt{3}}{2}} = -\frac{2}{\sqrt{3}} = -\frac{2\sqrt{3}}{3}\).
Answer: \(\boxed{-\frac{2\sqrt{3}}{3}}\)
---
- The angle \(-120^\circ\) is equivalent to \(360^\circ - 120^\circ = 240^\circ\) (in the third quadrant).
- \(\cot(-120^\circ) = \cot(240^\circ) = \frac{1}{\tan(240^\circ)}\).
- \(\tan(240^\circ) = \tan(180^\circ + 60^\circ) = \tan(60^\circ) = \sqrt{3}\).
- Therefore, \(\cot(-120^\circ) = \frac{1}{\sqrt{3}} = \frac{\sqrt{3}}{3}\).
Answer: \(\boxed{\frac{\sqrt{3}}{3}}\)
---
- The angle \(570^\circ\) is equivalent to \(570^\circ - 360^\circ = 210^\circ\) (in the third quadrant).
- \(\tan(570^\circ) = \tan(210^\circ) = \tan(180^\circ + 30^\circ) = \tan(30^\circ) = \frac{1}{\sqrt{3}} = \frac{\sqrt{3}}{3}\).
Answer: \(\boxed{\frac{\sqrt{3}}{3}}\)
---
- The angle \(495^\circ\) is equivalent to \(495^\circ - 360^\circ = 135^\circ\) (in the second quadrant).
- \(\cos(495^\circ) = \cos(135^\circ) = \cos(180^\circ - 45^\circ) = -\cos(45^\circ) = -\frac{\sqrt{2}}{2}\).
Answer: \(\boxed{-\frac{\sqrt{2}}{2}}\)
---
- The angle \(660^\circ\) is equivalent to \(660^\circ - 360^\circ = 300^\circ\) (in the fourth quadrant).
- \(\sin(660^\circ) = \sin(300^\circ) = \sin(360^\circ - 60^\circ) = -\sin(60^\circ) = -\frac{\sqrt{3}}{2}\).
Answer: \(\boxed{-\frac{\sqrt{3}}{2}}\)
---
- This is already in radians.
- Using the unit circle or trigonometric tables, \(\sin\left(\frac{\pi}{5}\right)\) is a specific value that can be approximated or left as is.
Answer: \(\boxed{\sin\left(\frac{\pi}{5}\right)}\)
---
- Convert \(\frac{\pi}{3}\) to degrees: \(\frac{\pi}{3} = 60^\circ\).
- \(\cos(60^\circ) = \frac{1}{2}\).
Answer: \(\boxed{\frac{1}{2}}\)
---
- Convert \(\frac{\pi}{4}\) to degrees: \(\frac{\pi}{4} = 45^\circ\).
- \(\tan(45^\circ) = 1\).
Answer: \(\boxed{1}\)
---
- Convert \(\frac{3\pi}{4}\) to degrees: \(\frac{3\pi}{4} = 135^\circ\) (in the second quadrant).
- \(\sec\left(\frac{3\pi}{4}\right) = \frac{1}{\cos\left(\frac{3\pi}{4}\right)}\).
- \(\cos\left(\frac{3\pi}{4}\right) = \cos(135^\circ) = -\frac{\sqrt{2}}{2}\).
- Therefore, \(\sec\left(\frac{3\pi}{4}\right) = \frac{1}{-\frac{\sqrt{2}}{2}} = -\sqrt{2}\).
Answer: \(\boxed{-\sqrt{2}}\)
---
- Convert \(\frac{5\pi}{3}\) to degrees: \(\frac{5\pi}{3} = 300^\circ\) (in the fourth quadrant).
- \(\cot\left(\frac{5\pi}{3}\right) = \frac{1}{\tan\left(\frac{5\pi}{3}\right)}\).
- \(\tan\left(\frac{5\pi}{3}\right) = \tan(300^\circ) = \tan(360^\circ - 60^\circ) = -\tan(60^\circ) = -\sqrt{3}\).
- Therefore, \(\cot\left(\frac{5\pi}{3}\right) = \frac{1}{-\sqrt{3}} = -\frac{\sqrt{3}}{3}\).
Answer: \(\boxed{-\frac{\sqrt{3}}{3}}\)
---
- Convert \(\frac{7\pi}{6}\) to degrees: \(\frac{7\pi}{6} = 210^\circ\) (in the third quadrant).
- \(\csc\left(\frac{7\pi}{6}\right) = \frac{1}{\sin\left(\frac{7\pi}{6}\right)}\).
- \(\sin\left(\frac{7\pi}{6}\right) = \sin(210^\circ) = \sin(180^\circ + 30^\circ) = -\sin(30^\circ) = -\frac{1}{2}\).
- Therefore, \(\csc\left(\frac{7\pi}{6}\right) = \frac{1}{-\frac{1}{2}} = -2\).
Answer: \(\boxed{-2}\)
---
- This is already in radians.
- Using the unit circle or trigonometric tables, \(\cos\left(\frac{\pi}{7}\right)\) is a specific value that can be approximated or left as is.
Answer: \(\boxed{\cos\left(\frac{\pi}{7}\right)}\)
---
- Convert \(\pi\) to degrees: \(\pi = 180^\circ\).
- \(\tan(180^\circ) = 0\).
Answer: \(\boxed{0}\)
---
- Convert \(\frac{3\pi}{2}\) to degrees: \(\frac{3\pi}{2} = 270^\circ\) (on the negative \(y\)-axis).
- \(\sin(270^\circ) = -1\).
Answer: \(\boxed{-1}\)
---
- Convert \(-\frac{2\pi}{3}\) to degrees: \(-\frac{2\pi}{3} = -120^\circ\).
- \(\csc\left(-\frac{2\pi}{3}\right) = \frac{1}{\sin\left(-\frac{2\pi}{3}\right)}\).
- \(\sin\left(-\frac{2\pi}{3}\right) = -\sin\left(\frac{2\pi}{3}\right) = -\sin(120^\circ) = -\frac{\sqrt{3}}{2}\).
- Therefore, \(\csc\left(-\frac{2\pi}{3}\right) = \frac{1}{-\frac{\sqrt{3}}{2}} = -\frac{2}{\sqrt{3}} = -\frac{2\sqrt{3}}{3}\).
Answer: \(\boxed{-\frac{2\sqrt{3}}{3}}\)
---
- Convert \(-\frac{5\pi}{4}\) to degrees: \(-\frac{5\pi}{4} = -225^\circ\).
- \(\sec\left(-\frac{5\pi}{4}\right) = \frac{1}{\cos\left(-\frac{5\pi}{4}\right)}\).
- \(\cos\left(-\frac{5\pi}{4}\right) = \cos(225^\circ) = \cos(180^\circ + 45^\circ) = -\cos(45^\circ) = -\frac{\sqrt{2}}{2}\).
- Therefore, \(\sec\left(-\frac{5\pi}{4}\right) = \frac{1}{-\frac{\sqrt{2}}{2}} = -\sqrt{2}\).
Answer: \(\boxed{-\sqrt{2}}\)
---
- Convert \(-\frac{11\pi}{6}\) to degrees: \(-\frac{11\pi}{6} = -330^\circ\).
- \(\cot\left(-\frac{11\pi}{6}\right) = \frac{1}{\tan\left(-\frac{11\pi}{6}\right)}\).
- \(\tan\left(-\frac{11\pi}{6}\right) = \tan(330^\circ) = \tan(360^\circ - 30^\circ) = -\tan(30^\circ) = -\frac{1}{\sqrt{3}} = -\frac{\sqrt{3}}{3}\).
- Therefore, \(\cot\left(-\frac{11\pi}{6}\right) = \frac{1}{-\frac{\sqrt{3}}{3}} = -\sqrt{3}\).
Answer: \(\boxed{-\sqrt{3}}\)
---
- Convert \(\frac{11\pi}{4}\) to degrees: \(\frac{11\pi}{4} = 495^\circ\).
- \(\tan\left(\frac{11\pi}{4}\right) = \tan(495^\circ) = \tan(210^\circ) = \tan(180^\circ + 30^\circ) = \tan(30^\circ) = \frac{1}{\sqrt{3}} = \frac{\sqrt{3}}{3}\).
Answer: \(\boxed{\frac{\sqrt{3}}{3}}\)
---
- Convert \(\frac{17\pi}{3}\) to degrees: \(\frac{17\pi}{3} = 1020^\circ\).
- Simplify the angle: \(1020^\circ - 3 \times 360^\circ = 1020^\circ - 1080^\circ = -60^\circ\).
- \(\cos\left(\frac{17\pi}{3}\right) = \cos(-60^\circ) = \cos(60^\circ) = \frac{1}{2}\).
Answer: \(\boxed{\frac{1}{2}}\)
---
- Convert \(\frac{19\pi}{6}\) to degrees: \(\frac{19\pi}{6} = 570^\circ\).
- Simplify the angle: \(570^\circ - 360^\circ = 210^\circ\).
- \(\sin\left(\frac{19\pi}{6}\right) = \sin(210^\circ) = \sin(180^\circ + 30^\circ) = -\sin(30^\circ) = -\frac{1}{2}\).
Answer: \(\boxed{-\frac{1}{2}}\)
---
\[
\boxed{
\begin{array}{l}
1. \frac{\sqrt{2}}{2} \\
2. \frac{\sqrt{3}}{2} \\
3. \sqrt{3} \\
4. -2 \\
5. 1 \\
6. -2 \\
7. 0 \\
8. \text{undefined} \\
9. 0 \\
10. -\sqrt{2} \\
11. -\frac{2\sqrt{3}}{3} \\
12. \frac{\sqrt{3}}{3} \\
13. \frac{\sqrt{3}}{3} \\
14. -\frac{\sqrt{2}}{2} \\
15. -\frac{\sqrt{3}}{2} \\
16. \sin\left(\frac{\pi}{5}\right) \\
17. \frac{1}{2} \\
18. 1 \\
19. -\sqrt{2} \\
20. -\frac{\sqrt{3}}{3} \\
21. -2 \\
22. \cos\left(\frac{\pi}{7}\right) \\
23. 0 \\
24. -1 \\
25. -\frac{2\sqrt{3}}{3} \\
26. -\sqrt{2} \\
27. -\sqrt{3} \\
28. \frac{\sqrt{3}}{3} \\
29. \frac{1}{2} \\
30. -\frac{1}{2}
\end{array}
}
\]
---
1. \(\sin(45^\circ)\)
- The angle \(45^\circ\) is in the first quadrant.
- On the unit circle, \(\sin(45^\circ) = \frac{\sqrt{2}}{2}\).
Answer: \(\boxed{\frac{\sqrt{2}}{2}}\)
---
2. \(\cos(30^\circ)\)
- The angle \(30^\circ\) is in the first quadrant.
- On the unit circle, \(\cos(30^\circ) = \frac{\sqrt{3}}{2}\).
Answer: \(\boxed{\frac{\sqrt{3}}{2}}\)
---
3. \(\tan(60^\circ)\)
- The angle \(60^\circ\) is in the first quadrant.
- On the unit circle, \(\tan(60^\circ) = \sqrt{3}\).
Answer: \(\boxed{\sqrt{3}}\)
---
4. \(\sec(120^\circ)\)
- The angle \(120^\circ\) is in the second quadrant.
- \(\sec(120^\circ) = \frac{1}{\cos(120^\circ)}\).
- \(\cos(120^\circ) = -\frac{1}{2}\) (since \(\cos(180^\circ - 60^\circ) = -\cos(60^\circ)\)).
- Therefore, \(\sec(120^\circ) = \frac{1}{-\frac{1}{2}} = -2\).
Answer: \(\boxed{-2}\)
---
5. \(\cot(225^\circ)\)
- The angle \(225^\circ\) is in the third quadrant.
- \(\cot(225^\circ) = \frac{1}{\tan(225^\circ)}\).
- \(\tan(225^\circ) = \tan(180^\circ + 45^\circ) = \tan(45^\circ) = 1\) (since tangent is positive in the third quadrant).
- Therefore, \(\cot(225^\circ) = \frac{1}{1} = 1\).
Answer: \(\boxed{1}\)
---
6. \(\csc(330^\circ)\)
- The angle \(330^\circ\) is in the fourth quadrant.
- \(\csc(330^\circ) = \frac{1}{\sin(330^\circ)}\).
- \(\sin(330^\circ) = \sin(360^\circ - 30^\circ) = -\sin(30^\circ) = -\frac{1}{2}\).
- Therefore, \(\csc(330^\circ) = \frac{1}{-\frac{1}{2}} = -2\).
Answer: \(\boxed{-2}\)
---
7. \(\cos(270^\circ)\)
- The angle \(270^\circ\) is on the negative \(y\)-axis.
- \(\cos(270^\circ) = 0\).
Answer: \(\boxed{0}\)
---
8. \(\tan(90^\circ)\)
- The angle \(90^\circ\) is on the positive \(y\)-axis.
- \(\tan(90^\circ)\) is undefined because \(\cos(90^\circ) = 0\) and division by zero is undefined.
Answer: \(\boxed{\text{undefined}}\)
---
9. \(\sin(180^\circ)\)
- The angle \(180^\circ\) is on the negative \(x\)-axis.
- \(\sin(180^\circ) = 0\).
Answer: \(\boxed{0}\)
---
10. \(\csc(-45^\circ)\)
- The angle \(-45^\circ\) is in the fourth quadrant.
- \(\csc(-45^\circ) = \frac{1}{\sin(-45^\circ)}\).
- \(\sin(-45^\circ) = -\sin(45^\circ) = -\frac{\sqrt{2}}{2}\).
- Therefore, \(\csc(-45^\circ) = \frac{1}{-\frac{\sqrt{2}}{2}} = -\sqrt{2}\).
Answer: \(\boxed{-\sqrt{2}}\)
---
11. \(\sec(-150^\circ)\)
- The angle \(-150^\circ\) is equivalent to \(360^\circ - 150^\circ = 210^\circ\) (in the third quadrant).
- \(\sec(-150^\circ) = \sec(210^\circ) = \frac{1}{\cos(210^\circ)}\).
- \(\cos(210^\circ) = \cos(180^\circ + 30^\circ) = -\cos(30^\circ) = -\frac{\sqrt{3}}{2}\).
- Therefore, \(\sec(-150^\circ) = \frac{1}{-\frac{\sqrt{3}}{2}} = -\frac{2}{\sqrt{3}} = -\frac{2\sqrt{3}}{3}\).
Answer: \(\boxed{-\frac{2\sqrt{3}}{3}}\)
---
12. \(\cot(-120^\circ)\)
- The angle \(-120^\circ\) is equivalent to \(360^\circ - 120^\circ = 240^\circ\) (in the third quadrant).
- \(\cot(-120^\circ) = \cot(240^\circ) = \frac{1}{\tan(240^\circ)}\).
- \(\tan(240^\circ) = \tan(180^\circ + 60^\circ) = \tan(60^\circ) = \sqrt{3}\).
- Therefore, \(\cot(-120^\circ) = \frac{1}{\sqrt{3}} = \frac{\sqrt{3}}{3}\).
Answer: \(\boxed{\frac{\sqrt{3}}{3}}\)
---
13. \(\tan(570^\circ)\)
- The angle \(570^\circ\) is equivalent to \(570^\circ - 360^\circ = 210^\circ\) (in the third quadrant).
- \(\tan(570^\circ) = \tan(210^\circ) = \tan(180^\circ + 30^\circ) = \tan(30^\circ) = \frac{1}{\sqrt{3}} = \frac{\sqrt{3}}{3}\).
Answer: \(\boxed{\frac{\sqrt{3}}{3}}\)
---
14. \(\cos(495^\circ)\)
- The angle \(495^\circ\) is equivalent to \(495^\circ - 360^\circ = 135^\circ\) (in the second quadrant).
- \(\cos(495^\circ) = \cos(135^\circ) = \cos(180^\circ - 45^\circ) = -\cos(45^\circ) = -\frac{\sqrt{2}}{2}\).
Answer: \(\boxed{-\frac{\sqrt{2}}{2}}\)
---
15. \(\sin(660^\circ)\)
- The angle \(660^\circ\) is equivalent to \(660^\circ - 360^\circ = 300^\circ\) (in the fourth quadrant).
- \(\sin(660^\circ) = \sin(300^\circ) = \sin(360^\circ - 60^\circ) = -\sin(60^\circ) = -\frac{\sqrt{3}}{2}\).
Answer: \(\boxed{-\frac{\sqrt{3}}{2}}\)
---
16. \(\sin\left(\frac{\pi}{5}\right)\)
- This is already in radians.
- Using the unit circle or trigonometric tables, \(\sin\left(\frac{\pi}{5}\right)\) is a specific value that can be approximated or left as is.
Answer: \(\boxed{\sin\left(\frac{\pi}{5}\right)}\)
---
17. \(\cos\left(\frac{\pi}{3}\right)\)
- Convert \(\frac{\pi}{3}\) to degrees: \(\frac{\pi}{3} = 60^\circ\).
- \(\cos(60^\circ) = \frac{1}{2}\).
Answer: \(\boxed{\frac{1}{2}}\)
---
18. \(\tan\left(\frac{\pi}{4}\right)\)
- Convert \(\frac{\pi}{4}\) to degrees: \(\frac{\pi}{4} = 45^\circ\).
- \(\tan(45^\circ) = 1\).
Answer: \(\boxed{1}\)
---
19. \(\sec\left(\frac{3\pi}{4}\right)\)
- Convert \(\frac{3\pi}{4}\) to degrees: \(\frac{3\pi}{4} = 135^\circ\) (in the second quadrant).
- \(\sec\left(\frac{3\pi}{4}\right) = \frac{1}{\cos\left(\frac{3\pi}{4}\right)}\).
- \(\cos\left(\frac{3\pi}{4}\right) = \cos(135^\circ) = -\frac{\sqrt{2}}{2}\).
- Therefore, \(\sec\left(\frac{3\pi}{4}\right) = \frac{1}{-\frac{\sqrt{2}}{2}} = -\sqrt{2}\).
Answer: \(\boxed{-\sqrt{2}}\)
---
20. \(\cot\left(\frac{5\pi}{3}\right)\)
- Convert \(\frac{5\pi}{3}\) to degrees: \(\frac{5\pi}{3} = 300^\circ\) (in the fourth quadrant).
- \(\cot\left(\frac{5\pi}{3}\right) = \frac{1}{\tan\left(\frac{5\pi}{3}\right)}\).
- \(\tan\left(\frac{5\pi}{3}\right) = \tan(300^\circ) = \tan(360^\circ - 60^\circ) = -\tan(60^\circ) = -\sqrt{3}\).
- Therefore, \(\cot\left(\frac{5\pi}{3}\right) = \frac{1}{-\sqrt{3}} = -\frac{\sqrt{3}}{3}\).
Answer: \(\boxed{-\frac{\sqrt{3}}{3}}\)
---
21. \(\csc\left(\frac{7\pi}{6}\right)\)
- Convert \(\frac{7\pi}{6}\) to degrees: \(\frac{7\pi}{6} = 210^\circ\) (in the third quadrant).
- \(\csc\left(\frac{7\pi}{6}\right) = \frac{1}{\sin\left(\frac{7\pi}{6}\right)}\).
- \(\sin\left(\frac{7\pi}{6}\right) = \sin(210^\circ) = \sin(180^\circ + 30^\circ) = -\sin(30^\circ) = -\frac{1}{2}\).
- Therefore, \(\csc\left(\frac{7\pi}{6}\right) = \frac{1}{-\frac{1}{2}} = -2\).
Answer: \(\boxed{-2}\)
---
22. \(\cos\left(\frac{\pi}{7}\right)\)
- This is already in radians.
- Using the unit circle or trigonometric tables, \(\cos\left(\frac{\pi}{7}\right)\) is a specific value that can be approximated or left as is.
Answer: \(\boxed{\cos\left(\frac{\pi}{7}\right)}\)
---
23. \(\tan(\pi)\)
- Convert \(\pi\) to degrees: \(\pi = 180^\circ\).
- \(\tan(180^\circ) = 0\).
Answer: \(\boxed{0}\)
---
24. \(\sin\left(\frac{3\pi}{2}\right)\)
- Convert \(\frac{3\pi}{2}\) to degrees: \(\frac{3\pi}{2} = 270^\circ\) (on the negative \(y\)-axis).
- \(\sin(270^\circ) = -1\).
Answer: \(\boxed{-1}\)
---
25. \(\csc\left(-\frac{2\pi}{3}\right)\)
- Convert \(-\frac{2\pi}{3}\) to degrees: \(-\frac{2\pi}{3} = -120^\circ\).
- \(\csc\left(-\frac{2\pi}{3}\right) = \frac{1}{\sin\left(-\frac{2\pi}{3}\right)}\).
- \(\sin\left(-\frac{2\pi}{3}\right) = -\sin\left(\frac{2\pi}{3}\right) = -\sin(120^\circ) = -\frac{\sqrt{3}}{2}\).
- Therefore, \(\csc\left(-\frac{2\pi}{3}\right) = \frac{1}{-\frac{\sqrt{3}}{2}} = -\frac{2}{\sqrt{3}} = -\frac{2\sqrt{3}}{3}\).
Answer: \(\boxed{-\frac{2\sqrt{3}}{3}}\)
---
26. \(\sec\left(-\frac{5\pi}{4}\right)\)
- Convert \(-\frac{5\pi}{4}\) to degrees: \(-\frac{5\pi}{4} = -225^\circ\).
- \(\sec\left(-\frac{5\pi}{4}\right) = \frac{1}{\cos\left(-\frac{5\pi}{4}\right)}\).
- \(\cos\left(-\frac{5\pi}{4}\right) = \cos(225^\circ) = \cos(180^\circ + 45^\circ) = -\cos(45^\circ) = -\frac{\sqrt{2}}{2}\).
- Therefore, \(\sec\left(-\frac{5\pi}{4}\right) = \frac{1}{-\frac{\sqrt{2}}{2}} = -\sqrt{2}\).
Answer: \(\boxed{-\sqrt{2}}\)
---
27. \(\cot\left(-\frac{11\pi}{6}\right)\)
- Convert \(-\frac{11\pi}{6}\) to degrees: \(-\frac{11\pi}{6} = -330^\circ\).
- \(\cot\left(-\frac{11\pi}{6}\right) = \frac{1}{\tan\left(-\frac{11\pi}{6}\right)}\).
- \(\tan\left(-\frac{11\pi}{6}\right) = \tan(330^\circ) = \tan(360^\circ - 30^\circ) = -\tan(30^\circ) = -\frac{1}{\sqrt{3}} = -\frac{\sqrt{3}}{3}\).
- Therefore, \(\cot\left(-\frac{11\pi}{6}\right) = \frac{1}{-\frac{\sqrt{3}}{3}} = -\sqrt{3}\).
Answer: \(\boxed{-\sqrt{3}}\)
---
28. \(\tan\left(\frac{11\pi}{4}\right)\)
- Convert \(\frac{11\pi}{4}\) to degrees: \(\frac{11\pi}{4} = 495^\circ\).
- \(\tan\left(\frac{11\pi}{4}\right) = \tan(495^\circ) = \tan(210^\circ) = \tan(180^\circ + 30^\circ) = \tan(30^\circ) = \frac{1}{\sqrt{3}} = \frac{\sqrt{3}}{3}\).
Answer: \(\boxed{\frac{\sqrt{3}}{3}}\)
---
29. \(\cos\left(\frac{17\pi}{3}\right)\)
- Convert \(\frac{17\pi}{3}\) to degrees: \(\frac{17\pi}{3} = 1020^\circ\).
- Simplify the angle: \(1020^\circ - 3 \times 360^\circ = 1020^\circ - 1080^\circ = -60^\circ\).
- \(\cos\left(\frac{17\pi}{3}\right) = \cos(-60^\circ) = \cos(60^\circ) = \frac{1}{2}\).
Answer: \(\boxed{\frac{1}{2}}\)
---
30. \(\sin\left(\frac{19\pi}{6}\right)\)
- Convert \(\frac{19\pi}{6}\) to degrees: \(\frac{19\pi}{6} = 570^\circ\).
- Simplify the angle: \(570^\circ - 360^\circ = 210^\circ\).
- \(\sin\left(\frac{19\pi}{6}\right) = \sin(210^\circ) = \sin(180^\circ + 30^\circ) = -\sin(30^\circ) = -\frac{1}{2}\).
Answer: \(\boxed{-\frac{1}{2}}\)
---
Final Answer:
\[
\boxed{
\begin{array}{l}
1. \frac{\sqrt{2}}{2} \\
2. \frac{\sqrt{3}}{2} \\
3. \sqrt{3} \\
4. -2 \\
5. 1 \\
6. -2 \\
7. 0 \\
8. \text{undefined} \\
9. 0 \\
10. -\sqrt{2} \\
11. -\frac{2\sqrt{3}}{3} \\
12. \frac{\sqrt{3}}{3} \\
13. \frac{\sqrt{3}}{3} \\
14. -\frac{\sqrt{2}}{2} \\
15. -\frac{\sqrt{3}}{2} \\
16. \sin\left(\frac{\pi}{5}\right) \\
17. \frac{1}{2} \\
18. 1 \\
19. -\sqrt{2} \\
20. -\frac{\sqrt{3}}{3} \\
21. -2 \\
22. \cos\left(\frac{\pi}{7}\right) \\
23. 0 \\
24. -1 \\
25. -\frac{2\sqrt{3}}{3} \\
26. -\sqrt{2} \\
27. -\sqrt{3} \\
28. \frac{\sqrt{3}}{3} \\
29. \frac{1}{2} \\
30. -\frac{1}{2}
\end{array}
}
\]
Parent Tip: Review the logic above to help your child master the concept of unit circle printable quiz.
