Question

If cos ⁡θ=−8/17, and 180°<θ<270°, what is tan ⁡θ?

Answer 1

15/8.

Explanation:

This angle is in the third quadrant in which the cosine is negative and the tangent is positive.

The opposite side in the triangle formed = √(17)^2 - (-8)^2)

= √(225)

= 15

So tan⁡ θ = 15/8

= 15/8.

Answer 2

tanΘ =
`(15)/(8)`

Explanation:

Using the trigonometric identities

tan x =
`(sinx)/(cosx)`

sin²x + cos²x = 1 ⇒ sin x = ±
`√(1-cos^2x)`

Since 180° < Θ < 270° then sinΘ < 0 and tanΘ > 0

sinΘ = -
`√(1-(-8/17)^2)` = -
`\sqrt{1-(64)/(289) }` = -
`\sqrt{(225)/(289) }` = -
`(15)/(17)`

Hence

tanΘ =
`(-(15)/(17) )/((-8)/(17) )` = -
`(15)/(17)` × -
`(17)/(8)` =
`(15)/(8)`