Question

Evaluate the expression under the given conditions.
sin(θ + ϕ);
sin(θ) = 12/13, θ in Quadrant I,
cos(ϕ) = −(2(sqrt 5))/5, ϕ in Quadrant II
Thanks

Answer 1

Sketch a triangle in the matching quadrant for each case
sin(θ) = 12/13, you should recognize the 5-12-13 right-angled triangle, so
r = 13, y = 12, and x = 5
so cosθ = 5/13

(If not use x^2 + y^2 = r^2 to find the missing value, use the sign to match the quadrant properties)

cos(ϕ) = −(2(sqrt 5))/5 in quad II
x = -2√5, r = 5,
x^2 + y^2 = r^2
20 + y^2 = 25 ----> y = √5 in II
then sinϕ = √5/5

so sin(θ + ϕ) = sinθcosϕ + cosθsinϕ
you have all 4 values, sub them in and simplify.
Make sure you keep the radicals for exact answers.