# point (-6,-8) lies on the terminal terminal arm of an angle `theta` in standard position. Determine the exact value of sin`theta` plus tan`theta`(sin+tan) ` `Sin `theta` = Y/R Cos `theta` =...

point (-6,-8) lies on the terminal terminal arm of an angle `theta` in standard position. Determine the exact value of sin`theta` plus tan`theta`(sin+tan) ` `

Sin `theta` = Y/R

Cos `theta` = X/R

Tan `theta` = Y/X

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You need to evaluate sin theta such that:

`sin theta = y/R`

You may evaluate ` R` using the pythagorean theorem such that:

`R = sqrt(x^2 + y^2) => R = sqrt((-6)^2 + (-8)^2)`

`R = sqrt(36 + 64) => R = 10`

Substituting -8 for y and 10 for R yields:

`sin theta = x/R => sin theta = -8/10 => sin theta = -4/5 => theta = sin^(-1)(-4/5)`

Since sin theta and cos theta are both negative yields that `theta` is an angle in quadrant 3.

`theta = 180^o + 53^o => theta = 233^o`

`cos theta = -6/10 => cos theta = -3/5 => theta = cos^(-1)(-3/5)`

`tan theta = (sin theta)/(cos theta) => tan theta = (-4/5)/(-3/5)`

Reducing duplicate factors yields:

`tan theta = 4/3`

You need to evaluate the sum `sin theta + tan theta` such that:

`sin theta + tan theta = -4/5 + 4/3 => sin theta + tan theta = (-12 + 20)/15 => sin theta + tan theta = 8/15`

**Hence, evaluating the sum sin theta + tan theta, under he given conditions, yields `sin theta + tan theta = 8/15` .**