Answer:
Part 1)
![cos(A)=(√(65))/(9)](https://img.qammunity.org/2020/formulas/mathematics/middle-school/4ks4rgpgj32d6qj1g8xzbyas5k803ocuwa.png)
Part 2)
![tan(A)=4(√(65))/(65)](https://img.qammunity.org/2020/formulas/mathematics/middle-school/q38693kyjikafmzaaduagab3528shw3xr2.png)
Part 3)
![cot(A)=(√(65))/(4)](https://img.qammunity.org/2020/formulas/mathematics/middle-school/zjtw2muwqo0z2ns6ki8pasl3u2n71wjt9p.png)
Part 4)
![sec(A)=9(√(65))/(65)](https://img.qammunity.org/2020/formulas/mathematics/middle-school/su8jggu3uf1t223u8o57utbgohof8kdens.png)
Part 5)
![csc(A)=(9)/(4)](https://img.qammunity.org/2020/formulas/mathematics/middle-school/dnsrwtoqy071910uw42js796oodn0y4my0.png)
Explanation:
Let
A------> the angle
we have that
![sin(A)=(4)/(9)](https://img.qammunity.org/2020/formulas/mathematics/middle-school/a8odxafcsdahgusuca8hl34n5p1e10d2ay.png)
step 1
Find the cos(A)
we know that
![cos^(2)(A)+sin^(2)(A)=1](https://img.qammunity.org/2020/formulas/mathematics/middle-school/tbixm0xnrc2e2zstetpkobxo3djn855wvf.png)
substitute the value of sin(A)
![cos^(2)(A)+((4)/(9))^(2)=1](https://img.qammunity.org/2020/formulas/mathematics/middle-school/12a53nhai71mfo1pcjogrq861r2x1s4ksi.png)
![cos^(2)(A)=1-((4)/(9))^(2)](https://img.qammunity.org/2020/formulas/mathematics/middle-school/q15eqohv36a7ivvezt2bkpvild71ksrqxr.png)
![cos^(2)(A)=1-((16)/(81))](https://img.qammunity.org/2020/formulas/mathematics/middle-school/h0yju8qd0g68e22xm5jkxwnpttc9on072t.png)
![cos^(2)(A)=((65)/(81))](https://img.qammunity.org/2020/formulas/mathematics/middle-school/1hpxjh9b5mqvudu7xx2x3k2658zpvy2y7t.png)
![cos(A)=(√(65))/(9)](https://img.qammunity.org/2020/formulas/mathematics/middle-school/4ks4rgpgj32d6qj1g8xzbyas5k803ocuwa.png)
step 2
Find the tan(A)
we know that
![tan(A)=(sin(A))/(cos(A))](https://img.qammunity.org/2020/formulas/mathematics/middle-school/b9n2jhmirgq7nc0zqfha5qtixtwnwvt3s8.png)
substitute the values
![tan(A)=((4)/(9))/((√(65))/(9))](https://img.qammunity.org/2020/formulas/mathematics/middle-school/mfdlbrtkbf6y834opu2kopepl0iec3htqg.png)
![tan(A)=(4)/(√(65))](https://img.qammunity.org/2020/formulas/mathematics/middle-school/a843hucgedyxzyikfllx7lhk7tqcc8ntjx.png)
![tan(A)=4(√(65))/(65)](https://img.qammunity.org/2020/formulas/mathematics/middle-school/q38693kyjikafmzaaduagab3528shw3xr2.png)
step 3
Find the cot(A)
we know that
![cot(A)=(1)/(tan(A))](https://img.qammunity.org/2020/formulas/mathematics/middle-school/zgjd6subiege6cz2wvi69xi9qx038er9ed.png)
we have
![tan(A)=(4)/(√(65))](https://img.qammunity.org/2020/formulas/mathematics/middle-school/a843hucgedyxzyikfllx7lhk7tqcc8ntjx.png)
substitute
![cot(A)=(1)/((4)/(√(65)))](https://img.qammunity.org/2020/formulas/mathematics/middle-school/2lwv8bthmrs7pdx5lsyj8ftv2addphjisv.png)
![cot(A)=(√(65))/(4)](https://img.qammunity.org/2020/formulas/mathematics/middle-school/zjtw2muwqo0z2ns6ki8pasl3u2n71wjt9p.png)
step 4
Find the sec(A)
we know that
![sec(A)=(1)/(cos(A))](https://img.qammunity.org/2020/formulas/mathematics/middle-school/7fdqhv6vc7we163d93w9z94hs6c2y9fy8o.png)
we have
![cos(A)=(√(65))/(9)](https://img.qammunity.org/2020/formulas/mathematics/middle-school/4ks4rgpgj32d6qj1g8xzbyas5k803ocuwa.png)
substitute
![sec(A)=(1)/((√(65))/(9))](https://img.qammunity.org/2020/formulas/mathematics/middle-school/jwfzwkw8j444idde723hy7xlab047aom9v.png)
![sec(A)=(9)/(√(65))](https://img.qammunity.org/2020/formulas/mathematics/middle-school/u2dl6ux9wircgiuiq8ooe8nvbezhinyf0i.png)
![sec(A)=9(√(65))/(65)](https://img.qammunity.org/2020/formulas/mathematics/middle-school/su8jggu3uf1t223u8o57utbgohof8kdens.png)
step 5
Find the csc(A)
we know that
![csc(A)=(1)/(sin(A))](https://img.qammunity.org/2020/formulas/mathematics/middle-school/bix3ld5a8g457xuxf2a3klb9xe28gs8hwh.png)
we have
![sin(A)=(4)/(9)](https://img.qammunity.org/2020/formulas/mathematics/middle-school/a8odxafcsdahgusuca8hl34n5p1e10d2ay.png)
substitute the value
![csc(A)=(1)/((4)/(9))](https://img.qammunity.org/2020/formulas/mathematics/middle-school/lji77uii2t2ik4vcho3o7he96dcu3lcpbo.png)
![csc(A)=(9)/(4)](https://img.qammunity.org/2020/formulas/mathematics/middle-school/dnsrwtoqy071910uw42js796oodn0y4my0.png)