Answer:
[tex]csc(\theta)=\frac{5}{4}[/tex]
[tex]cos(\theta)=\frac{3}{5}[/tex]
[tex]sec(\theta)=\frac{5}{3}[/tex]
[tex]tan(\theta)=\frac{4}{3}[/tex]
[tex]cot(\theta)=\frac{3}{4}[/tex]
Step-by-step explanation:
step 1
Find the [tex]csc(\theta)[/tex]
we know that
[tex]csc(\theta)=\frac{1}{sin(\theta)}[/tex]
we have
[tex]sin(\theta)=\frac{4}{5}[/tex]
therefore
[tex]csc(\theta)=\frac{5}{4}[/tex]
step 2
Find the [tex]cos(\theta)[/tex]
we know that
[tex]sin^2(\theta)+cos^2(\theta)=1[/tex]
we have
[tex]sin(\theta)=\frac{4}{5}[/tex]
substitute
[tex](\frac{4}{5})^2+cos^2(\theta)=1[/tex]
[tex]cos^2(\theta)=1-(\frac{4}{5})^2[/tex]
[tex]cos^2(\theta)=1-(\frac{16}{25})[/tex]
[tex]cos^2(\theta)=(\frac{9}{25})[/tex]
square root both sides
[tex]cos(\theta)=\pm(\frac{3}{5})[/tex]
Remember that the angle theta is in quadrant I
so
The value of cosine of angle theta is positive
[tex]cos(\theta)=\frac{3}{5}[/tex]
step 3
Find the [tex]sec(\theta)[/tex]
we know that
[tex]sec(\theta)=\frac{1}{cos(\theta)}[/tex]
we have
[tex]cos(\theta)=\frac{3}{5}[/tex]
therefore
[tex]sec(\theta)=\frac{5}{3}[/tex]
step 4
Find the value of [tex]tan(\theta)[/tex]
we know that
[tex]tan(\theta)=\frac{sin(\theta)}{cos(\theta)}[/tex]
we have
[tex]sin(\theta)=\frac{4}{5}[/tex]
[tex]cos(\theta)=\frac{3}{5}[/tex]
substitute the values
[tex]tan(\theta)=\frac{4}{3}[/tex]
step 5
Find the value of [tex]cot(\theta)[/tex]
we know that
[tex]cot(\theta)=\frac{1}{tan(\theta)}[/tex]
we have
[tex]tan(\theta)=\frac{4}{3}[/tex]
therefore
[tex]cot(\theta)=\frac{3}{4}[/tex]
