Aurasma Unit Circle Worksheet | Download Free PDF | Trigonometry ... - Free Printable

Let's solve each problem step-by-step using knowledge of the unit circle and trigonometric identities. We'll use the standard values for common angles (in radians) and recall:

- $ \sin\theta = y $, $ \cos\theta = x $ on the unit circle
- $ \tan\theta = \frac{\sin\theta}{\cos\theta} $
- $ \sec\theta = \frac{1}{\cos\theta} $, $ \csc\theta = \frac{1}{\sin\theta} $, $ \cot\theta = \frac{1}{\tan\theta} $
- Angles are in radians, so we convert or recognize them as standard angles.

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🔷 Set I: Basic Sine and Cosine

1. $ \sin\frac{5\pi}{6} $

- $ \frac{5\pi}{6} $ is in Quadrant II.
- Reference angle: $ \pi - \frac{5\pi}{6} = \frac{\pi}{6} $
- $ \sin\frac{\pi}{6} = \frac{1}{2} $, and sine is positive in QII.
- ✔ $ \boxed{\frac{1}{2}} $

2. $ \cos\frac{7\pi}{6} $

- $ \frac{7\pi}{6} $ is in Quadrant III.
- Reference angle: $ \frac{7\pi}{6} - \pi = \frac{\pi}{6} $
- $ \cos\frac{\pi}{6} = \frac{\sqrt{3}}{2} $, but cosine is negative in QIII.
- ✔ $ \boxed{-\frac{\sqrt{3}}{2}} $

3. $ \sin\frac{3\pi}{4} $

- $ \frac{3\pi}{4} $ is in QII.
- Reference angle: $ \pi - \frac{3\pi}{4} = \frac{\pi}{4} $
- $ \sin\frac{\pi}{4} = \frac{\sqrt{2}}{2} $, positive in QII.
- ✔ $ \boxed{\frac{\sqrt{2}}{2}} $

4. $ \cos\frac{11\pi}{6} $

- $ \frac{11\pi}{6} $ is in QIV.
- Reference angle: $ 2\pi - \frac{11\pi}{6} = \frac{\pi}{6} $
- $ \cos\frac{\pi}{6} = \frac{\sqrt{3}}{2} $, cosine is positive in QIV.
- ✔ $ \boxed{\frac{\sqrt{3}}{2}} $

5. $ \sin\frac{5\pi}{4} $

- $ \frac{5\pi}{4} $ is in QIII.
- Reference angle: $ \frac{5\pi}{4} - \pi = \frac{\pi}{4} $
- $ \sin\frac{\pi}{4} = \frac{\sqrt{2}}{2} $, but sine is negative in QIII.
- ✔ $ \boxed{-\frac{\sqrt{2}}{2}} $

6. $ \cos\frac{4\pi}{3} $

- $ \frac{4\pi}{3} $ is in QIII.
- Reference angle: $ \frac{4\pi}{3} - \pi = \frac{\pi}{3} $
- $ \cos\frac{\pi}{3} = \frac{1}{2} $, negative in QIII.
- ✔ $ \boxed{-\frac{1}{2}} $

7. $ \cos\frac{5\pi}{4} $

- $ \frac{5\pi}{4} $ is in QIII.
- Reference angle: $ \frac{\pi}{4} $
- $ \cos\frac{\pi}{4} = \frac{\sqrt{2}}{2} $, negative in QIII.
- ✔ $ \boxed{-\frac{\sqrt{2}}{2}} $

8. $ \sin\frac{2\pi}{3} $

- $ \frac{2\pi}{3} $ is in QII.
- Reference angle: $ \pi - \frac{2\pi}{3} = \frac{\pi}{3} $
- $ \sin\frac{\pi}{3} = \frac{\sqrt{3}}{2} $, positive in QII.
- ✔ $ \boxed{\frac{\sqrt{3}}{2}} $

9. $ \cos\frac{5\pi}{6} $

- $ \frac{5\pi}{6} $ is in QII.
- Reference angle: $ \frac{\pi}{6} $
- $ \cos\frac{\pi}{6} = \frac{\sqrt{3}}{2} $, negative in QII.
- ✔ $ \boxed{-\frac{\sqrt{3}}{2}} $

10. $ \sin\frac{3\pi}{2} $

- This is straight down on the unit circle.
- $ \sin\frac{3\pi}{2} = -1 $
- ✔ $ \boxed{-1} $

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🔷 Set II: Reciprocal and Tangent Functions

1. $ \sec\frac{2\pi}{3} $

- $ \frac{2\pi}{3} $ is in QII.
- $ \cos\frac{2\pi}{3} = -\frac{1}{2} $
- So $ \sec = \frac{1}{\cos} = \frac{1}{-1/2} = -2 $
- ✔ $ \boxed{-2} $

2. $ \csc\frac{7\pi}{4} $

- $ \frac{7\pi}{4} $ is in QIV.
- $ \sin\frac{7\pi}{4} = -\frac{\sqrt{2}}{2} $
- $ \csc = \frac{1}{\sin} = \frac{1}{-\sqrt{2}/2} = -\frac{2}{\sqrt{2}} = -\sqrt{2} $
- ✔ $ \boxed{-\sqrt{2}} $

3. $ \tan\frac{\pi}{2} $

- $ \tan\theta = \frac{\sin}{\cos} $
- $ \sin\frac{\pi}{2} = 1 $, $ \cos\frac{\pi}{2} = 0 $
- Division by zero → undefined
- ✔ $ \boxed{\text{DNE}} $

4. $ \cot\frac{5\pi}{6} $

- $ \frac{5\pi}{6} $ is in QII.
- $ \tan\frac{5\pi}{6} = \frac{\sin}{\cos} = \frac{1/2}{-\sqrt{3}/2} = -\frac{1}{\sqrt{3}} $
- So $ \cot = \frac{1}{\tan} = -\sqrt{3} $
- ✔ $ \boxed{-\sqrt{3}} $

5. $ \sec\frac{3\pi}{4} $

- $ \frac{3\pi}{4} $ is in QII.
- $ \cos\frac{3\pi}{4} = -\frac{\sqrt{2}}{2} $
- $ \sec = \frac{1}{\cos} = \frac{1}{-\sqrt{2}/2} = -\frac{2}{\sqrt{2}} = -\sqrt{2} $
- ✔ $ \boxed{-\sqrt{2}} $

6. $ \csc\frac{11\pi}{6} $

- $ \frac{11\pi}{6} $ is in QIV.
- $ \sin\frac{11\pi}{6} = -\frac{1}{2} $
- $ \csc = \frac{1}{\sin} = \frac{1}{-1/2} = -2 $
- ✔ $ \boxed{-2} $

7. $ \cot\frac{4\pi}{3} $

- $ \frac{4\pi}{3} $ is in QIII.
- $ \tan\frac{4\pi}{3} = \frac{\sin}{\cos} = \frac{-\sqrt{3}/2}{-1/2} = \sqrt{3} $
- So $ \cot = \frac{1}{\tan} = \frac{1}{\sqrt{3}} = \frac{\sqrt{3}}{3} $
- ✔ $ \boxed{\frac{\sqrt{3}}{3}} $

8. $ \sec\frac{7\pi}{6} $

- $ \frac{7\pi}{6} $ is in QIII.
- $ \cos\frac{7\pi}{6} = -\frac{\sqrt{3}}{2} $
- $ \sec = \frac{1}{\cos} = \frac{1}{-\sqrt{3}/2} = -\frac{2}{\sqrt{3}} = -\frac{2\sqrt{3}}{3} $
- ✔ $ \boxed{-\frac{2\sqrt{3}}{3}} $

9. $ \cot\frac{5\pi}{3} $

- $ \frac{5\pi}{3} $ is in QIV.
- $ \tan\frac{5\pi}{3} = \frac{\sin}{\cos} = \frac{-\sqrt{3}/2}{1/2} = -\sqrt{3} $
- So $ \cot = \frac{1}{\tan} = -\frac{1}{\sqrt{3}} = -\frac{\sqrt{3}}{3} $
- ✔ $ \boxed{-\frac{\sqrt{3}}{3}} $

10. $ \sec\pi $

- $ \cos\pi = -1 $
- $ \sec\pi = \frac{1}{-1} = -1 $
- ✔ $ \boxed{-1} $

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🔷 Set III: More Trig Values

1. $ \cot\frac{11\pi}{6} $

- $ \frac{11\pi}{6} $ is in QIV.
- $ \tan\frac{11\pi}{6} = \frac{\sin}{\cos} = \frac{-1/2}{\sqrt{3}/2} = -\frac{1}{\sqrt{3}} $
- $ \cot = \frac{1}{\tan} = -\sqrt{3} $
- ✔ $ \boxed{-\sqrt{3}} $

2. $ \tan\pi $

- $ \sin\pi = 0 $, $ \cos\pi = -1 $
- $ \tan\pi = \frac{0}{-1} = 0 $
- ✔ $ \boxed{0} $

3. $ \tan\frac{4\pi}{3} $

- $ \frac{4\pi}{3} $ is in QIII.
- $ \sin = -\frac{\sqrt{3}}{2}, \cos = -\frac{1}{2} $
- $ \tan = \frac{-\sqrt{3}/2}{-1/2} = \sqrt{3} $
- ✔ $ \boxed{\sqrt{3}} $

4. $ \cot\frac{\pi}{2} $

- $ \tan\frac{\pi}{2} $ is undefined → $ \cot = \frac{1}{\tan} $ → DNE
- ✔ $ \boxed{\text{DNE}} $

5. $ \sec\frac{5\pi}{3} $

- $ \frac{5\pi}{3} $ is in QIV.
- $ \cos\frac{5\pi}{3} = \frac{1}{2} $
- $ \sec = \frac{1}{1/2} = 2 $
- ✔ $ \boxed{2} $

6. $ \csc\frac{7\pi}{4} $

- $ \sin\frac{7\pi}{4} = -\frac{\sqrt{2}}{2} $
- $ \csc = \frac{1}{-\sqrt{2}/2} = -\frac{2}{\sqrt{2}} = -\sqrt{2} $
- ✔ $ \boxed{-\sqrt{2}} $

7. $ \cot\frac{2\pi}{3} $

- $ \frac{2\pi}{3} $ is in QII.
- $ \tan\frac{2\pi}{3} = \frac{\sin}{\cos} = \frac{\sqrt{3}/2}{-1/2} = -\sqrt{3} $
- $ \cot = \frac{1}{-\sqrt{3}} = -\frac{\sqrt{3}}{3} $
- ✔ $ \boxed{-\frac{\sqrt{3}}{3}} $

8. $ \sec\frac{11\pi}{6} $

- $ \cos\frac{11\pi}{6} = \frac{\sqrt{3}}{2} $
- $ \sec = \frac{1}{\sqrt{3}/2} = \frac{2}{\sqrt{3}} = \frac{2\sqrt{3}}{3} $
- ✔ $ \boxed{\frac{2\sqrt{3}}{3}} $

9. $ \csc 0 $

- $ \sin 0 = 0 $
- $ \csc = \frac{1}{0} $ → undefined
- ✔ $ \boxed{\text{DNE}} $

10. $ \cot\frac{5\pi}{4} $

- $ \frac{5\pi}{4} $ is in QIII.
- $ \tan\frac{5\pi}{4} = \frac{\sin}{\cos} = \frac{-\sqrt{2}/2}{-\sqrt{2}/2} = 1 $
- $ \cot = \frac{1}{1} = 1 $
- ✔ $ \boxed{1} $

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🔷 Set IV: Final Set

1. $ \sin 2\pi $

- $ \sin 2\pi = 0 $
- ✔ $ \boxed{0} $

2. $ \cos\frac{3\pi}{2} $

- $ \cos\frac{3\pi}{2} = 0 $
- ✔ $ \boxed{0} $

3. $ \tan\frac{\pi}{6} $

- $ \tan\frac{\pi}{6} = \frac{\sin}{\cos} = \frac{1/2}{\sqrt{3}/2} = \frac{1}{\sqrt{3}} = \frac{\sqrt{3}}{3} $
- ✔ $ \boxed{\frac{\sqrt{3}}{3}} $

4. $ \cot\frac{7\pi}{4} $

- $ \frac{7\pi}{4} $ is in QIV.
- $ \tan\frac{7\pi}{4} = \frac{-\sqrt{2}/2}{\sqrt{2}/2} = -1 $
- $ \cot = \frac{1}{-1} = -1 $
- ✔ $ \boxed{-1} $

5. $ \sec\frac{5\pi}{6} $

- $ \cos\frac{5\pi}{6} = -\frac{\sqrt{3}}{2} $
- $ \sec = \frac{1}{-\sqrt{3}/2} = -\frac{2}{\sqrt{3}} = -\frac{2\sqrt{3}}{3} $
- ✔ $ \boxed{-\frac{2\sqrt{3}}{3}} $

6. $ \csc\frac{5\pi}{3} $

- $ \sin\frac{5\pi}{3} = -\frac{\sqrt{3}}{2} $
- $ \csc = \frac{1}{-\sqrt{3}/2} = -\frac{2}{\sqrt{3}} = -\frac{2\sqrt{3}}{3} $
- ✔ $ \boxed{-\frac{2\sqrt{3}}{3}} $

7. $ \cot\frac{4\pi}{3} $

- Already did in Set II: $ \frac{\sqrt{3}}{3} $
- ✔ $ \boxed{\frac{\sqrt{3}}{3}} $

8. $ \sec\pi $

- $ \cos\pi = -1 $ → $ \sec\pi = -1 $
- ✔ $ \boxed{-1} $

9. $ \sin\frac{7\pi}{6} $

- $ \frac{7\pi}{6} $ is in QIII.
- $ \sin = -\frac{1}{2} $
- ✔ $ \boxed{-\frac{1}{2}} $

10. $ \cos\frac{2\pi}{3} $

- $ \cos\frac{2\pi}{3} = -\frac{1}{2} $
- ✔ $ \boxed{-\frac{1}{2}} $

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✔ Final Answers Summary

#### Set I
1. $ \frac{1}{2} $
2. $ -\frac{\sqrt{3}}{2} $
3. $ \frac{\sqrt{2}}{2} $
4. $ \frac{\sqrt{3}}{2} $
5. $ -\frac{\sqrt{2}}{2} $
6. $ -\frac{1}{2} $
7. $ -\frac{\sqrt{2}}{2} $
8. $ \frac{\sqrt{3}}{2} $
9. $ -\frac{\sqrt{3}}{2} $
10. $ -1 $

#### Set II
1. $ -2 $
2. $ -\sqrt{2} $
3. DNE
4. $ -\sqrt{3} $
5. $ -\sqrt{2} $
6. $ -2 $
7. $ \frac{\sqrt{3}}{3} $
8. $ -\frac{2\sqrt{3}}{3} $
9. $ -\frac{\sqrt{3}}{3} $
10. $ -1 $

#### Set III
1. $ -\sqrt{3} $
2. $ 0 $
3. $ \sqrt{3} $
4. DNE
5. $ 2 $
6. $ -\sqrt{2} $
7. $ -\frac{\sqrt{3}}{3} $
8. $ \frac{2\sqrt{3}}{3} $
9. DNE
10. $ 1 $

#### Set IV
1. $ 0 $
2. $ 0 $
3. $ \frac{\sqrt{3}}{3} $
4. $ -1 $
5. $ -\frac{2\sqrt{3}}{3} $
6. $ -\frac{2\sqrt{3}}{3} $
7. $ \frac{\sqrt{3}}{3} $
8. $ -1 $
9. $ -\frac{1}{2} $
10. $ -\frac{1}{2} $

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✔ All answers are simplified, exact, and rationalized where needed. Let me know if you'd like this formatted as a printable answer key!