Question

Find the value of tan( π + θ) if θ terminates in Quadrant III and sinθ = -5/13

Answer 1

`\tan(\pi + \theta)= (5)/(12)`

Step-by-step explanation

We first obtain

`\cos( \theta)`

using the Pythagorean Identity.

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

`\implies \: \cos ^(2) ( \theta) + ( - (5)/(13) )^(2) = 1`

`\implies \: \cos ^(2) ( \theta) + (25)/(169)= 1`

`\implies \: \cos ^(2) ( \theta) = 1 - (25)/(169)`

`\implies \: \cos ^(2) ( \theta) = (144)/(169)`

`\implies \: \cos ( \theta) = \pm \: \sqrt{(144)/(169) }`

`\implies \: \cos ( \theta) = \pm \: (12)/(13)`

In the third quadrant, the cosine ratio is negative.

`\implies \: \cos ( \theta) = - \: (12)/(13)`

The tangent function has a period of π and
`\pi + \theta` is in the third quadrant.

This implies that:

`\tan(\pi + \theta)= \tan( \theta)`

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

`\tan(\pi + \theta)= ( - ( 5)/(13) )/( - (12)/(13) )`

This gives us:

`\tan(\pi + \theta)= (5)/(12)`