Question

Suppose 0 is an angle in the standard position whose terminal side is in Quadrant 1 and sin 0= 84/85. Find the exact values of the five remaining trigonometric functions of 0

Answer 1

Part 1)
`cos(\theta)=((13)/(85))`

Part 2)
`tan(\theta)=(84)/(13)`

Part 3)
`cot(\theta)=(13)/(84)`

Part 4)
`sec(\theta)=(85)/(13)`

Part 5)
`csc(\theta)=(85)/(84)`

Explanation:

we know that

Angle
`\theta` lie on Quadrant I

so

`sin(\theta)` is positive

`cos(\theta)` is positive

`tan(\theta)` is positive

`cot(\theta)` is positive

`sec(\theta)` is positive

`csc(\theta)` is positive

step 1

Find the value of
`cos(\theta)`

we have

`sin(\theta)=(84)/(85)`

we know that

`sin^2(\theta)+cos^2(\theta)=1`

substitute

`((84)/(85))^2+cos^2(\theta)=1`

`((7,056)/(7,225))+cos^2(\theta)=1`

`cos^2(\theta)=1-((7,056)/(7,225))`

`cos^2(\theta)=((169)/(7,225))`

`cos(\theta)=((13)/(85))`

step 2

Find the value of
`tan(\theta)`

we know that

`tan(\theta)=(sin(\theta))/(cos(\theta))`

we have

`sin(\theta)=(84)/(85)`

`cos(\theta)=((13)/(85))`

substitute

`tan(\theta)=((84/85))/((13/85))`

`tan(\theta)=(84)/(13)`

step 3

Find the value of
`cot(\theta)`

we know that

`cot(\theta)=(1)/(tan(\theta))`

we have

`tan(\theta)=(84)/(13)`

therefore

`cot(\theta)=(13)/(84)`

step 4

Find the value of
`sec(\theta)`

we know that

`sec(\theta)=(1)/(cos(\theta))`

we have

`cos(\theta)=((13)/(85))`

therefore

`sec(\theta)=(85)/(13)`

step 5

Find the value of
`csc(\theta)`

we know that

`csc(\theta)=(1)/(sin(\theta))`

we have

`sin(\theta)=(84)/(85)`

therefore

`csc(\theta)=(85)/(84)`