Function h(t) = sin (t) cos(pi/6)+sin(pi/6)cos(t) represents height of a wave in an ocean, where h, is height in meters, and t, is time in seconds. Determine all values of time when height of this wave is 1/√2 meters.

Answer 1

you might recognize
sin(A+B) = sinAcosB + cosAsinB
we have that in
h(t) = sin (t) cos(pi/6)+sin(pi/6)cos(t)
= sin(t + π/6)

so you want sin(t+π/6) = 1/√2
I know that sin π/4 = 1/√2
so t + π/6 = π/4 or t+π/6 = π-π/4
t = π/4 - π/6 = π/12

or

t+π/6 = π-π/4 = 3π/4
t = 3π/4 - π/6 = 7π/12

of course this will repeat every 2π seconds for each answer
so t = π/12, 7π/12, 25π/12, 31π/12, .....

general solution: (π/12 + 2kπ), (7π/12 + 2kπ) , where k is an integer