I am looking for help with this question from my grade 11 advanced functions course:
csc θ = -17/15 where 270° ≤ θ ≤ 360° and
cot β = - 3/4 where 90° ≤ β ≤ 180°.
Find the exact value of sin(θ + β). Show all the work.
This what I have done:
cscθ = -17/15 where 270° ≤ θ ≤ 360°
sinθ = 1/cscθ
sinθ = 1/(-17/15)
sinθ = 1/1 / (-17/15)
sinθ = 1/1 * (-15/17)
sinθ = -15/17
θ = sin^-1 (-15/17)
** I don't know what to do from here because if I solve for θ completely I won't have an EXACT value **
cot β = - 3/4 where 90° ≤ β ≤ 180°
tanβ = 1/cotβ
tanβ = 1/(-3/4)
tanβ = 1/1 / (-3/4)
tanβ = 1/1 * (-4/3)
tanβ = -4/3
β = tan^-1 (-4/3)
** I don't know what to do from here because if I solve for β completely I won't have an EXACT value **
From this point I'm not sure how to find an exact value for sin(θ + β) because I do not have values for θ or β
For starters, you should know that $\displaystyle \displaystyle \sin{(\theta \pm \beta)} \equiv \sin{(\theta)}\cos{(\beta)} \pm \cos{(\theta)}\sin{(\beta)}$.
You don't need to solve for $\displaystyle \displaystyle \theta$ or $\displaystyle \displaystyle \beta$, you only need to solve for $\displaystyle \displaystyle \sin{(\theta)}, \cos{(\theta)}, \sin{(\beta)}$ and $\displaystyle \displaystyle \cos{(\beta)}$. You will need to make use of the Pythagorean Identity.
I knew about the sum and difference formulas and that I would have to use the sum formula for sin in order to solve the final step, but I'm not sure what you mean by pythagorean identities. I'm not sure how to solve for sin(\theta), cos(\theta), sin(\beta), or cos(\beta).
From that given you know that $\displaystyle \theta\in IV$.
So $\displaystyle \sin \left( \theta \right) = \frac{{ - 15}}{{17}}\;\& \,\cos \left( \theta \right) = \sqrt {1 - \left( {\frac{15}{17}} \right)^2 }=\frac{8}{{17}}$
Moreover, $\displaystyle \beta\in II$ so $\displaystyle \sin \left( \beta \right) = \frac{4}{5}\;\& \,\cos \left( \beta \right) = \frac{{ - 3}}{5}$.