Question

Find the exact value of the trigonometric function given that sin u = − 3/5 and cos v = − 8/17? (Both u and v are in Quadrant III.)

cot(v − u)

Answer 1

`sin (u + v) = 84/85`

Explanation:

Use trig identity:
sin (u + v) = sin u.cos v + sin v.cos u
We know: `sin u = -3/5` and `cos v = -8/17`. Find sin v and cos u.
`sin^2 v = 1 - cos^2 v = 1 - 64/289 = 225/289` --> `sin v = +- 15/17`
Since v is in Q.3, then, sin v is negative. Take the negative value.
`cos^2 u = 1 - sin^2 u = 1 - 9/25 = 16/25` --> `cos u = +- 4/5`.
Take the negative value, because u is in Q. 3.
`sin (u + v) = (-3/5)(-8/17) + (-4/5)(-15/17) =`
`= (24 + 60)/85 = 84/85`

Answer 2

`77/36`

Explanation:

Trignometric function required to be evaluated is cot (v-u). Before evaluation, there is need to find the value of cos u and sin v, because these would be required in calculations.

Given sin u= -3/5. cos u would be `-sqrt (1-sin^2 u)= - sqrt(1-9/25)= -4/5`. Negative sign is to used for the square root, because u is in IIIrd quadrant, where cos u would also be negative

Likewise `sin v= -sqrt (1-cos^2 v)= -sqrt (1-64/289)= - sqrt 225/289= -15/17`

`cot(v-u)= (cos (v-u))/(sin (v-u))`

= `(cosv cosu + sin v sin u)/(sin v cos u- cosv sin u)`

=`( -8/17 * -4/5 + -15/17 *-3/5)`

`/(-15/17 * -4/5) - (-8/17 * -3/5)`

= `77/85/36/85=77/36`