evaluate the expression under the given conditions.

Question

cos(θ − ϕ); cos(θ) = 5/13, θ in Quadrant IV, tan(ϕ) = − square root15,ϕ in Quadrant II

Anyone know how to do this I have never been this confused in my life

Answer 1

start with : cos(θ) = 5/13, θ in Quadrant IV
you should recognize the 5-12-13 right-angled triangle
and since cosØ = adjacent/hypotenuse
x = 5, r = 13 , y = -12, since Ø is in IV
and sinØ = -12/13

also tan(ϕ) = −√15 = -√15/1 = y/x and ϕ is in II,
y = √15 , x = -1
r^2 = x^2 + y^2 = 15+1 = 16
r = 4
sinϕ = √15/4 , cosϕ = -1/4

you must know that:
cos(θ − ϕ) = cosθcosϕ + sinθsinϕ
= (5/13)(-1/4) + (-12/13)(√15/4)
= -5/52 - 12√15/52
= (-5-12√15)/52

To do these type of problems you must know your basic trig definitions as well as the CAST rule.