Question

Given that \tan\theta =\frac{12}{35}tanθ= 35 12 ​ and that angle \thetaθ terminates in quadrant \text{III}III, then what is the value of \cos\thetacosθ?

Answer 1

`cos\theta = -(12)/(37)`

Explanation:

Given

`tan\theta = (35)/(12)`

Quadrant: 3rd

Required

Determine
`cos\theta`

This question will be solved using the Pythagoras theorem

`Hyp^2 = Adj^2 + Opp^2`

The tangent of an angle is calculated as thus;

`tan\theta = (Opp)/(Adj)`

Comparing

`tan\theta = (35)/(12)` to
`tan\theta = (Opp)/(Adj)`

We can conclude that

`Opp = 35`
`Adj = 12`

Substitute these values in the Pythagoras formula

`Hyp^2 = 35^2 + 12^2`

`Hyp^2 = 1225 + 144`

`Hyp^2 = 1369`

Square root of both sides

`Hyp = √(1369)`

`Hyp = 37`

SInce
`\theta` is in the 3rd quadrant, then

`cos\theta = -(Adj)/(Hyp)`

Where
`Adj = 12` and
`Hyp = 37`

`cos\theta = -(12)/(37)`