part 1.\nfill in the chart with the exact value for each function\n\n\npart 2\nfind the exact value of each…

part 1.\nfill in the chart with the exact value for each function\n\n\npart 2\nfind the exact value of each expression. do not use a calculator.\n1 ( sin 45 ^ { circ } + cos 60 ^ { circ } ) 2. ( sin 30 ^ { circ } - cos 45 ^ { circ } ) 3. ( sin 45 ^ { circ } cos 45 ^ { circ } )\n4 ( \tan 45 ^ { circ } cos 30 ^ { circ } ) 5 ( csc 45 ^ { circ } \tan 60 ^ { circ } ) 6. ( sec 30 ^ { circ } cot 45 ^ { circ } )\n7 ( 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 } )

part 1.\nfill in the chart with the exact value for each function\n\n\npart 2\nfind the exact value of each expression. do not use a calculator.\n1 ( sin 45 ^ { circ } + cos 60 ^ { circ } ) 2. ( sin 30 ^ { circ } - cos 45 ^ { circ } ) 3. ( sin 45 ^ { circ } cos 45 ^ { circ } )\n4 ( \tan 45 ^ { circ } cos 30 ^ { circ } ) 5 ( csc 45 ^ { circ } \tan 60 ^ { circ } ) 6. ( sec 30 ^ { circ } cot 45 ^ { circ } )\n7 ( 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 } )

Answer

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})
(\sin\theta) (\frac{1}{2}) (\frac{\sqrt{2}}{2}) (\frac{\sqrt{3}}{2})
(\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})