Number of solutions of sin^2θ+3cosθ=3 in [-π,π] ?

1 Answer
Jun 23, 2018

Answer:

One answer: t = 0

Explanation:

Replace in the equation #sin^2 t# by #(1 - cos^2 t)#:, then, solve the quadratic equation for cos t:
#1 - cos^2 t + 3cos t - 3 = 0#
#- cos^2 t + 3cos t - 2 = 0#
Sine a + b + c = 0, use shortcut. The 2 real roots are:
cos x = 1 and #cos x = c/a = 2# (rejected as > 1)
cos x = 1.
Unit circle gives --> t = 0, and #t = 2pi#
Inside the interval #(-pi, pi)#, there is only one answer:
t = 0
Check
t = 0 --> sin^2 t = 0 --> 3cos t = 3
sin^2 t + 3cos t = 0 + 3 = 3. Proved.