International Baccalaureate Diploma Program (IBDP) - Analysis and Approaches (AAHL) || Unit Circle and Trigonometric Ratios Beyond Acute Angles

Unit Circle and Trigonometric Ratios Beyond Acute Angles - Worksheet

Name: ___________________

Date: ___________________

Instructions:

• Answer the following questions.
• Show all your work, including any necessary diagrams.
• Ensure you clearly explain your steps where applicable.

Questions

1. Evaluate the following expressions using the unit circle:
   • a) \( \sin\left(\frac{5\pi}{4}\right) \)
   • b) \( \cos\left(\frac{7\pi}{3}\right) \)
2. Given \( \theta = \frac{5\pi}{6} \), find the exact values of \( \sin(\theta) \), \( \cos(\theta) \), and \( \tan(\theta) \) using the unit circle.
3. Find the exact value of \( \csc\left(\frac{3\pi}{2}\right) \).
4. The point \( P \) lies on the unit circle at an angle of \( \theta = \frac{4\pi}{3} \). What are the coordinates of \( P \)?
5. Prove the identity \( 1 + \tan^2(\theta) = \sec^2(\theta) \) using the unit circle.
6. For \( \theta = \frac{5\pi}{3} \), determine the value of \( \sin(\theta) \), \( \cos(\theta) \), and \( \tan(\theta) \). Then, verify that \( \sin(\theta) \) and \( \cos(\theta) \) satisfy the Pythagorean identity \( \sin^2(\theta) + \cos^2(\theta) = 1 \).
7. Given \( \tan(\theta) = -\frac{1}{\sqrt{3}} \), and \( \theta \) is in the fourth quadrant, find the exact values of \( \sin(\theta) \) and \( \cos(\theta) \).
8. A point \( P \) on the unit circle forms an angle of \( \theta = \frac{7\pi}{6} \). Determine the reference angle for \( \theta \), and find the exact values of \( \sin(\theta) \), \( \cos(\theta) \), and \( \tan(\theta) \).
9. A pendulum swings back and forth following the sine function \( y(t) = 5 \sin\left( \frac{\pi}{2} t \right) \), where \( t \) is in seconds. Find the period of the pendulum’s motion and the amplitude of the sine wave.
10. Using the unit circle, determine the exact value of \( \tan\left(\frac{11\pi}{6}\right) \) and explain your reasoning.

Bonus Question (Optional)

1. Show that \( \cos(\theta + \pi) = -\cos(\theta) \) using the unit circle, and use this identity to find \( \cos\left( \frac{5\pi}{3} \right) \).

End of Worksheet