Question

Question #10e48

Answer 1

Write the given as two equations:

`cos(theta) = x" [1]"`

`theta=tan^-1(4/3)" [2]"`

and we want to find the value of x.

Use equation [2] to find the value of `tan(theta)` by applying the tangent function to both sides:

`tan(theta)=tan(tan^-1(4/3))" [2.1]"`

The tangent of its inverse reduces to `4/3` on the right:

`tan(theta) = 4/3" [2.2]"`

We can use the identity

`1 + tan^2(theta) = sec^2(theta)`

to give us a relationship between `tan(theta)` and `cos(theta)`:

`1 + (4/3)^2 = sec^2(theta)`

`9/9+16/9 = sec^2(theta)`

`25/9= sec^2(theta)`

We know that `sec^2(theta) = 1/cos^2(theta)`:

`25/9= 1/cos^2(theta)`

`cos^2(theta) = 9/25`

`cos(theta) = +-3/5`

We do not know whether we are in the 1st or 3rd quadrant, therefore, we must leave the `+-` as is.

Answer 2

`+-3/5`

Explanation:

We know that,

`53 approx tan^(-1) (4/3)` Using a calculator...

`=> tan53 approx 4/3`

So, `cos(tan^(-1)(4/3) ) approx cos(tan^(-1)(tan(53))`

`cos(npi+ 53)=+-3/5`, `n in Z`