Answer :
Given the values [tex]\(\sin \left(\frac{3 \pi}{16}\right) \approx 0.5556\)[/tex] and [tex]\(\cos \left(\frac{3 \pi}{16}\right) \approx 0.8315\)[/tex], we will calculate the other trigonometric functions for [tex]\(\frac{3\pi}{16}\)[/tex] and [tex]\(\frac{5\pi}{16}\)[/tex].
### a) Trigonometric functions for [tex]\(\frac{3\pi}{16}\)[/tex]:
1. Tangent:
[tex]\[\tan \left(\frac{3 \pi}{16}\right) = \frac{\sin \left(\frac{3 \pi}{16}\right)}{\cos \left(\frac{3 \pi}{16}\right)} \approx \frac{0.5556}{0.8315} \approx 0.6682.\][/tex]
2. Cosecant (reciprocal of sine):
[tex]\[\csc \left(\frac{3 \pi}{16}\right) = \frac{1}{\sin \left(\frac{3 \pi}{16}\right)} \approx \frac{1}{0.5556} \approx 1.7999.\][/tex]
3. Secant (reciprocal of cosine):
[tex]\[\sec \left(\frac{3 \pi}{16}\right) = \frac{1}{\cos \left(\frac{3 \pi}{16}\right)} \approx \frac{1}{0.8315} \approx 1.2026.\][/tex]
4. Cotangent (reciprocal of tangent):
[tex]\[\cot \left(\frac{3 \pi}{16}\right) = \frac{1}{\tan \left(\frac{3 \pi}{16}\right)} \approx \frac{1}{0.6682} \approx 1.4966.\][/tex]
Hence, the trigonometric functions for [tex]\(\frac{3 \pi}{16}\)[/tex] are:
- [tex]\(\tan \left(\frac{3 \pi}{16}\right) \approx 0.6682\)[/tex],
- [tex]\(\csc \left(\frac{3 \pi}{16}\right) \approx 1.7999\)[/tex],
- [tex]\(\sec \left(\frac{3 \pi}{16}\right) \approx 1.2026\)[/tex],
- [tex]\(\cot \left(\frac{3 \pi}{16}\right) \approx 1.4966\)[/tex].
### b) Trigonometric functions for [tex]\(\frac{5\pi}{16}\)[/tex] (using complementary angle [tex]\(\frac{5\pi}{16} = \frac{\pi}{2} - \frac{3\pi}{16}\)[/tex]):
1. Sine:
[tex]\[\sin \left(\frac{5 \pi}{16}\right) = \cos \left(\frac{3 \pi}{16}\right) \approx 0.8315.\][/tex]
2. Cosine:
[tex]\[\cos \left(\frac{5 \pi}{16}\right) = \sin \left(\frac{3 \pi}{16}\right) \approx 0.5556.\][/tex]
3. Tangent:
[tex]\[\tan \left(\frac{5 \pi}{16}\right) = \frac{\sin \left(\frac{5 \pi}{16}\right)}{\cos \left(\frac{5 \pi}{16}\right)} \approx \frac{0.8315}{0.5556} \approx 1.4966.\][/tex]
4. Cosecant (reciprocal of sine):
[tex]\[\csc \left(\frac{5 \pi}{16}\right) = \frac{1}{\sin \left(\frac{5 \pi}{16}\right)} \approx \frac{1}{0.8315} \approx 1.2026.\][/tex]
5. Secant (reciprocal of cosine):
[tex]\[\sec \left(\frac{5 \pi}{16}\right) = \frac{1}{\cos \left(\frac{5 \pi}{16}\right)} \approx \frac{1}{0.5556} \approx 1.7999.\][/tex]
6. Cotangent (reciprocal of tangent):
[tex]\[\cot \left(\frac{5 \pi}{16}\right) = \frac{1}{\tan \left(\frac{5 \pi}{16}\right)} \approx \frac{1}{1.4966} \approx 0.6682.\][/tex]
Hence, the trigonometric functions for [tex]\(\frac{5\pi}{16}\)[/tex] are:
- [tex]\(\sin \left(\frac{5 \pi}{16}\right) \approx 0.8315\)[/tex],
- [tex]\(\cos \left(\frac{5 \pi}{16}\right) \approx 0.5556\)[/tex],
- [tex]\(\tan \left(\frac{5 \pi}{16}\right) \approx 1.4966\)[/tex],
- [tex]\(\csc \left(\frac{5 \pi}{16}\right) \approx 1.2026\)[/tex],
- [tex]\(\sec \left(\frac{5 \pi}{16}\right) \approx 1.7999\)[/tex],
- [tex]\(\cot \left(\frac{5 \pi}{16}\right) \approx 0.6682\)[/tex].
In summary, based on the given values:
- [tex]\(\csc \left(\frac{3 \pi}{16}\right) = 1.7999\)[/tex]
- [tex]\(\sec \left(\frac{3 \pi}{16}\right) = 1.2026\)[/tex]
- [tex]\(\cot \left(\frac{3 \pi}{16}\right) = 1.4966\)[/tex]
- [tex]\(\sin \left(\frac{5 \pi}{16}\right) = 0.8315\)[/tex]
- [tex]\(\cos \left(\frac{5 \pi}{16}\right) = 0.5556\)[/tex]
- [tex]\(\tan \left(\frac{5 \pi}{16}\right) = 1.4966\)[/tex]
- [tex]\(\csc \left(\frac{5 \pi}{16}\right) = 1.2026\)[/tex]
- [tex]\(\sec \left(\frac{5 \pi}{16}\right) = 1.7999\)[/tex]
- [tex]\(\cot \left(\frac{5 \pi}{16}\right) = 0.6682\)[/tex].
### a) Trigonometric functions for [tex]\(\frac{3\pi}{16}\)[/tex]:
1. Tangent:
[tex]\[\tan \left(\frac{3 \pi}{16}\right) = \frac{\sin \left(\frac{3 \pi}{16}\right)}{\cos \left(\frac{3 \pi}{16}\right)} \approx \frac{0.5556}{0.8315} \approx 0.6682.\][/tex]
2. Cosecant (reciprocal of sine):
[tex]\[\csc \left(\frac{3 \pi}{16}\right) = \frac{1}{\sin \left(\frac{3 \pi}{16}\right)} \approx \frac{1}{0.5556} \approx 1.7999.\][/tex]
3. Secant (reciprocal of cosine):
[tex]\[\sec \left(\frac{3 \pi}{16}\right) = \frac{1}{\cos \left(\frac{3 \pi}{16}\right)} \approx \frac{1}{0.8315} \approx 1.2026.\][/tex]
4. Cotangent (reciprocal of tangent):
[tex]\[\cot \left(\frac{3 \pi}{16}\right) = \frac{1}{\tan \left(\frac{3 \pi}{16}\right)} \approx \frac{1}{0.6682} \approx 1.4966.\][/tex]
Hence, the trigonometric functions for [tex]\(\frac{3 \pi}{16}\)[/tex] are:
- [tex]\(\tan \left(\frac{3 \pi}{16}\right) \approx 0.6682\)[/tex],
- [tex]\(\csc \left(\frac{3 \pi}{16}\right) \approx 1.7999\)[/tex],
- [tex]\(\sec \left(\frac{3 \pi}{16}\right) \approx 1.2026\)[/tex],
- [tex]\(\cot \left(\frac{3 \pi}{16}\right) \approx 1.4966\)[/tex].
### b) Trigonometric functions for [tex]\(\frac{5\pi}{16}\)[/tex] (using complementary angle [tex]\(\frac{5\pi}{16} = \frac{\pi}{2} - \frac{3\pi}{16}\)[/tex]):
1. Sine:
[tex]\[\sin \left(\frac{5 \pi}{16}\right) = \cos \left(\frac{3 \pi}{16}\right) \approx 0.8315.\][/tex]
2. Cosine:
[tex]\[\cos \left(\frac{5 \pi}{16}\right) = \sin \left(\frac{3 \pi}{16}\right) \approx 0.5556.\][/tex]
3. Tangent:
[tex]\[\tan \left(\frac{5 \pi}{16}\right) = \frac{\sin \left(\frac{5 \pi}{16}\right)}{\cos \left(\frac{5 \pi}{16}\right)} \approx \frac{0.8315}{0.5556} \approx 1.4966.\][/tex]
4. Cosecant (reciprocal of sine):
[tex]\[\csc \left(\frac{5 \pi}{16}\right) = \frac{1}{\sin \left(\frac{5 \pi}{16}\right)} \approx \frac{1}{0.8315} \approx 1.2026.\][/tex]
5. Secant (reciprocal of cosine):
[tex]\[\sec \left(\frac{5 \pi}{16}\right) = \frac{1}{\cos \left(\frac{5 \pi}{16}\right)} \approx \frac{1}{0.5556} \approx 1.7999.\][/tex]
6. Cotangent (reciprocal of tangent):
[tex]\[\cot \left(\frac{5 \pi}{16}\right) = \frac{1}{\tan \left(\frac{5 \pi}{16}\right)} \approx \frac{1}{1.4966} \approx 0.6682.\][/tex]
Hence, the trigonometric functions for [tex]\(\frac{5\pi}{16}\)[/tex] are:
- [tex]\(\sin \left(\frac{5 \pi}{16}\right) \approx 0.8315\)[/tex],
- [tex]\(\cos \left(\frac{5 \pi}{16}\right) \approx 0.5556\)[/tex],
- [tex]\(\tan \left(\frac{5 \pi}{16}\right) \approx 1.4966\)[/tex],
- [tex]\(\csc \left(\frac{5 \pi}{16}\right) \approx 1.2026\)[/tex],
- [tex]\(\sec \left(\frac{5 \pi}{16}\right) \approx 1.7999\)[/tex],
- [tex]\(\cot \left(\frac{5 \pi}{16}\right) \approx 0.6682\)[/tex].
In summary, based on the given values:
- [tex]\(\csc \left(\frac{3 \pi}{16}\right) = 1.7999\)[/tex]
- [tex]\(\sec \left(\frac{3 \pi}{16}\right) = 1.2026\)[/tex]
- [tex]\(\cot \left(\frac{3 \pi}{16}\right) = 1.4966\)[/tex]
- [tex]\(\sin \left(\frac{5 \pi}{16}\right) = 0.8315\)[/tex]
- [tex]\(\cos \left(\frac{5 \pi}{16}\right) = 0.5556\)[/tex]
- [tex]\(\tan \left(\frac{5 \pi}{16}\right) = 1.4966\)[/tex]
- [tex]\(\csc \left(\frac{5 \pi}{16}\right) = 1.2026\)[/tex]
- [tex]\(\sec \left(\frac{5 \pi}{16}\right) = 1.7999\)[/tex]
- [tex]\(\cot \left(\frac{5 \pi}{16}\right) = 0.6682\)[/tex].
