Answer:
Part 1) [tex]cos(A)=\frac{\sqrt{65}}{9}[/tex]
Part 2) [tex]tan(A)=4\frac{\sqrt{65}}{65}[/tex]
Part 3) [tex]cot(A)=\frac{\sqrt{65}}{4}[/tex]
Part 4) [tex]sec(A)=9\frac{\sqrt{65}}{65}[/tex]
Part 5) [tex]csc(A)=\frac{9}{4}[/tex]
Step-by-step explanation:
Let
A------> the angle
we have that
[tex]sin(A)=\frac{4}{9}[/tex]
step 1
Find the cos(A)
we know that
[tex]cos^{2}(A)+sin^{2}(A)=1[/tex]
substitute the value of sin(A)
[tex]cos^{2}(A)+(\frac{4}{9})^{2}=1[/tex]
[tex]cos^{2}(A)=1-(\frac{4}{9})^{2}[/tex]
[tex]cos^{2}(A)=1-(\frac{16}{81})[/tex]
[tex]cos^{2}(A)=(\frac{65}{81})[/tex]
[tex]cos(A)=\frac{\sqrt{65}}{9}[/tex]
step 2
Find the tan(A)
we know that
[tex]tan(A)=\frac{sin(A)}{cos(A)}[/tex]
substitute the values
[tex]tan(A)=\frac{\frac{4}{9}}{\frac{\sqrt{65}}{9}}[/tex]
[tex]tan(A)=\frac{4}{\sqrt{65}}[/tex]
[tex]tan(A)=4\frac{\sqrt{65}}{65}[/tex]
step 3
Find the cot(A)
we know that
[tex]cot(A)=\frac{1}{tan(A)}[/tex]
we have
[tex]tan(A)=\frac{4}{\sqrt{65}}[/tex]
substitute
[tex]cot(A)=\frac{1}{\frac{4}{\sqrt{65}}}[/tex]
[tex]cot(A)=\frac{\sqrt{65}}{4}[/tex]
step 4
Find the sec(A)
we know that
[tex]sec(A)=\frac{1}{cos(A)}[/tex]
we have
[tex]cos(A)=\frac{\sqrt{65}}{9}[/tex]
substitute
[tex]sec(A)=\frac{1}{\frac{\sqrt{65}}{9}}[/tex]
[tex]sec(A)=\frac{9}{\sqrt{65}}[/tex]
[tex]sec(A)=9\frac{\sqrt{65}}{65}[/tex]
step 5
Find the csc(A)
we know that
[tex]csc(A)=\frac{1}{sin(A)}[/tex]
we have
[tex]sin(A)=\frac{4}{9}[/tex]
substitute the value
[tex]csc(A)=\frac{1}{\frac{4}{9}}[/tex]
[tex]csc(A)=\frac{9}{4}[/tex]
