Question

In a right triangle, angle is an acute angle and sin angle = 4/9. Evaluate the other five trigonometric functions of angle.

Answer 1

Part 1)
`cos(A)=(√(65))/(9)`

Part 2)
`tan(A)=4(√(65))/(65)`

Part 3)
`cot(A)=(√(65))/(4)`

Part 4)
`sec(A)=9(√(65))/(65)`

Part 5)
`csc(A)=(9)/(4)`

Explanation:

Let

A------> the angle

we have that

`sin(A)=(4)/(9)`

step 1

Find the cos(A)

we know that

`cos^(2)(A)+sin^(2)(A)=1`

substitute the value of sin(A)

`cos^(2)(A)+((4)/(9))^(2)=1`

`cos^(2)(A)=1-((4)/(9))^(2)`

`cos^(2)(A)=1-((16)/(81))`

`cos^(2)(A)=((65)/(81))`

`cos(A)=(√(65))/(9)`

step 2

Find the tan(A)

we know that

`tan(A)=(sin(A))/(cos(A))`

substitute the values

`tan(A)=((4)/(9))/((√(65))/(9))`

`tan(A)=(4)/(√(65))`

`tan(A)=4(√(65))/(65)`

step 3

Find the cot(A)

we know that

`cot(A)=(1)/(tan(A))`

we have

`tan(A)=(4)/(√(65))`

substitute

`cot(A)=(1)/((4)/(√(65)))`

`cot(A)=(√(65))/(4)`

step 4

Find the sec(A)

we know that

`sec(A)=(1)/(cos(A))`

we have

`cos(A)=(√(65))/(9)`

substitute

`sec(A)=(1)/((√(65))/(9))`

`sec(A)=(9)/(√(65))`

`sec(A)=9(√(65))/(65)`

step 5

Find the csc(A)

we know that

`csc(A)=(1)/(sin(A))`

we have

`sin(A)=(4)/(9)`

substitute the value

`csc(A)=(1)/((4)/(9))`

`csc(A)=(9)/(4)`