Question

The angle θ is in the first quadrant of the unit circle and tan(θ)=
`(8)/(15)` .Since tan(θ)=
`(sin(θ))/(cos(θ))` , does this mean that sin(θ)=8 and cos(θ)=15? If not, clearly explain and give the exact values of sin(θ) and cos(θ).

Answer 1

`\sin(\theta)=0.47` and
`\cos(\theta)=0.88`.

Explanation:

Given:

Angle lies in first quadrant.

`\tan\theta=(8)/(15)`

Since,
`\tan\theta=(\cos(\theta)/(\sin(\theta))` this does not mean that
`\sin(\theta)=8` and
`\cos(\theta)=15`, because it is the ratio of the
`\sin(\theta)` and
`\cos(\theta)` which is not necessarily the exact values of them. And also
`\sin(\theta)` and
`\cos(\theta)` values lie between
`-1 \ to\ 1` and so it cannot be
`=8\ or\ 15`.

In order to find the exact values, we need to find the exact angle.

`\tan\theta=(8)/(15)`

So,

`\theta=\tan^(-1)((8)/(15))`

`\theta= 28.07\°` As the angle lies in 1st quadrant.

Using the angle
`\theta` we can find the exact values for
`\sin(\theta)` and
`\cos(\theta)`.

`sin(28.07\°)=0.47`

`cos(28.07\°)=0.88`