Question

part 1.
fill in the chart with the exact value for each function

part 2
find the exact value of each expression. do not use a calculator.
1 ( sin 45 ^ { circ } + cos 60 ^ { circ } ) 2. ( sin 30 ^ { circ } - cos 45 ^ { circ } ) 3. ( sin 45 ^ { circ } cos 45 ^ { circ } )
4 ( \tan 45 ^ { circ } cos 30 ^ { circ } ) 5 ( csc 45 ^ { circ } \tan 60 ^ { circ } ) 6. ( sec 30 ^ { circ } cot 45 ^ { circ } )
7 ( 2 sin \frac { pi } { 3 } - 3 \tan \frac { pi } { 6 } ) 8. ( 2 sin \frac { pi } { 4 } + 3 \tan \frac { pi } { 4 } ) 9. ( 2 sec \frac { pi } { 4 } + 4 cot \frac { pi } { 3 } )

Explanation:

Part 1:

For \(\theta=\frac{\pi}{6}\) (or \(30^{\circ}\)):

- **\(\sin\theta\)**:
- Recall the sine of \(30^{\circ}\), \(\sin\frac{\pi}{6}=\frac{1}{2}\)
- **\(\cos\theta\)**:
- Recall the cosine of \(30^{\circ}\), \(\cos\frac{\pi}{6}=\frac{\sqrt{3}}{2}\)
- **\(\tan\theta\)**:
- \(\tan\theta=\frac{\sin\theta}{\cos\theta}\), so \(\tan\frac{\pi}{6}=\frac{\frac{1}{2}}{\frac{\sqrt{3}}{2}}=\frac{1}{\sqrt{3}}=\frac{\sqrt{3}}{3}\)
- **\(\csc\theta\)**:
- \(\csc\theta=\frac{1}{\sin\theta}\), so \(\csc\frac{\pi}{6} = 2\)
- **\(\sec\theta\)**:
- \(\sec\theta=\frac{1}{\cos\theta}\), so \(\sec\frac{\pi}{6}=\frac{2\sqrt{3}}{3}\)
- **\(\cot\theta\)**:
- \(\cot\theta=\frac{\cos\theta}{\sin\theta}\), so \(\cot\frac{\pi}{6}=\sqrt{3}\)

For \(\theta=\frac{\pi}{4}\) (or \(45^{\circ}\)):

- **\(\sin\theta\)**:
- Recall the sine of \(45^{\circ}\), \(\sin\frac{\pi}{4}=\frac{\sqrt{2}}{2}\)
- **\(\cos\theta\)**:
- Recall the cosine of \(45^{\circ}\), \(\cos\frac{\pi}{4}=\frac{\sqrt{2}}{2}\)
- **\(\tan\theta\)**:
- \(\tan\theta=\frac{\sin\theta}{\cos\theta}\), so \(\tan\frac{\pi}{4}= 1\)
- **\(\csc\theta\)**:
- \(\csc\theta=\frac{1}{\sin\theta}\), so \(\csc\frac{\pi}{4}=\sqrt{2}\)
- **\(\sec\theta\)**:
- \(\sec\theta=\frac{1}{\cos\theta}\), so \(\sec\frac{\pi}{4}=\sqrt{2}\)
- **\(\cot\theta\)**:
- \(\cot\theta=\frac{\cos\theta}{\sin\theta}\), so \(\cot\frac{\pi}{4}=1\)

For \(\theta=\frac{\pi}{3}\) (or \(60^{\circ}\)):

- **\(\sin\theta\)**:
- Recall the sine of \(60^{\circ}\), \(\sin\frac{\pi}{3}=\frac{\sqrt{3}}{2}\)
- **\(\cos\theta\)**:
- Recall the cosine of \(60^{\circ}\), \(\cos\frac{\pi}{3}=\frac{1}{2}\)
- **\(\tan\theta\)**:
- \(\tan\theta=\frac{\sin\theta}{\cos\theta}\), so \(\tan\frac{\pi}{3}=\sqrt{3}\)
- **\(\csc\theta\)**:
- \(\csc\theta=\frac{1}{\sin\theta}\), so \(\csc\frac{\pi}{3}=\frac{2\sqrt{3}}{3}\)
- **\(\sec\theta\)**:
- \(\sec\theta=\frac{1}{\cos\theta}\), so \(\sec\frac{\pi}{3}=2\)
- **\(\cot\theta\)**:
- \(\cot\theta=\frac{\cos\theta}{\sin\theta}\), so \(\cot\frac{\pi}{3}=\frac{\sqrt{3}}{3}\)

Part 2:

1. \(\sin45^{\circ}+\cos60^{\circ}\)

- **Step1: Substitute the values**
- \(\sin45^{\circ}=\frac{\sqrt{2}}{2}\), \(\cos60^{\circ}=\frac{1}{2}\)
- \(\sin45^{\circ}+\cos60^{\circ}=\frac{\sqrt{2}}{2}+\frac{1}{2}=\frac{\sqrt{2} + 1}{2}\)

2. \(\sin30^{\circ}-\cos45^{\circ}\)

- **Step1: Substitute the values**
- \(\sin30^{\circ}=\frac{1}{2}\), \(\cos45^{\circ}=\frac{\sqrt{2}}{2}\)
- \(\sin30^{\circ}-\cos45^{\circ}=\frac{1}{2}-\frac{\sqrt{2}}{2}=\frac{1-\sqrt{2}}{2}\)

3. \(\sin45^{\circ}\cos45^{\circ}\)

- **Step1: Substitute the values**
- \(\sin45^{\circ}=\frac{\sqrt{2}}{2}\), \(\cos45^{\circ}=\frac{\sqrt{2}}{2}\)
- \(\sin45^{\circ}\cos45^{\circ}=\frac{\sqrt{2}}{2}\times\frac{\sqrt{2}}{2}=\frac{2}{4}=\frac{1}{2}\)

4. \(\tan45^{\circ}\cos30^{\circ}\)

- **Step1: Substitute the values**
- \(\tan45^{\circ}=1\), \(\cos30^{\circ}=\frac{\sqrt{3}}{2}\)
- \(\tan45^{\circ}\cos30^{\circ}=1\times\frac{\sqrt{3}}{2}=\frac{\sqrt{3}}{2}\)

5. \(\csc45^{\circ}\tan60^{\circ}\)

- **Step1: Substitute the values**
- \(\csc45^{\circ}=\sqrt{2}\), \(\tan60^{\circ}=\sqrt{3}\)
- \(\csc45^{\circ}\tan60^{\circ}=\sqrt{2}\times\sqrt{3}=\sqrt{6}\)

6. \(\sec30^{\circ}\cot45^{\circ}\)

- **Step1: Substitute the values**
- \(\sec30^{\circ}=\frac{2\sqrt{3}}{3}\), \(\cot45^{\circ}=1\)
- \(\sec30^{\circ}\cot45^{\circ}=\frac{2\sqrt{3}}{3}\times1=\frac{2\sqrt{3}}{3}\)

7. \(2\sin\frac{\pi}{3}-3\tan\frac{\pi}{6}\)

- **Step1: Substitute the values**
- \(\sin\frac{\pi}{3}=\frac{\sqrt{3}}{2}\), \(\tan\frac{\pi}{6}=\frac{\sqrt{3}}{3}\)
- \(2\sin\frac{\pi}{3}-3\tan\frac{\pi}{6}=2\times\frac{\sqrt{3}}{2}-3\times\frac{\sqrt{3}}{3}=\sqrt{3}-\sqrt{3}=0\)

8. \(2\sin\frac{\pi}{4}+3\tan\frac{\pi}{4}\)

- **Step1: Substitute the values**
- \(\sin\frac{\pi}{4}=\frac{\sqrt{2}}{2}\), \(\tan\frac{\pi}{4}=1\)
- \(2\sin\frac{\pi}{4}+3\tan\frac{\pi}{4}=2\times\frac{\sqrt{2}}{2}+3\times1=\sqrt{2}+3\)

9. \(2\sec\frac{\pi}{4}+4\cot\frac{\pi}{3}\)

- **Step1: Substitute the values**
- \(\sec\frac{\pi}{4}=\sqrt{2}\), \(\cot\frac{\pi}{3}=\frac{\sqrt{3}}{3}\)
- \(2\sec\frac{\pi}{4}+4\cot\frac{\pi}{3}=2\sqrt{2}+4\times\frac{\sqrt{3}}{3}=2\sqrt{2}+\frac{4\sqrt{3}}{3}\)

Answer:

Part 1:

\(\theta\)	\(\frac{\pi}{6}\)	\(\frac{\pi}{4}\)	\(\frac{\pi}{3}\)
\(\cos\theta\)	\(\frac{\sqrt{3}}{2}\)	\(\frac{\sqrt{2}}{2}\)	\(\frac{1}{2}\)
\(\tan\theta\)	\(\frac{\sqrt{3}}{3}\)	\(1\)	\(\sqrt{3}\)
\(\csc\theta\)	\(2\)	\(\sqrt{2}\)	\(\frac{2\sqrt{3}}{3}\)
\(\sec\theta\)	\(\frac{2\sqrt{3}}{3}\)	\(\sqrt{2}\)	\(2\)
\(\cot\theta\)	\(\sqrt{3}\)	\(1\)	\(\frac{\sqrt{3}}{3}\)

Part 2:

1. \(\frac{\sqrt{2}+1}{2}\)
2. \(\frac{1 - \sqrt{2}}{2}\)
3. \(\frac{1}{2}\)
4. \(\frac{\sqrt{3}}{2}\)
5. \(\sqrt{6}\)
6. \(\frac{2\sqrt{3}}{3}\)
7. \(0\)
8. \(\sqrt{2}+3\)
9. \(2\sqrt{2}+\frac{4\sqrt{3}}{3}\)