If $4 cos θ + 3 sin θ = x$ and $4 sin θ − 3 cos θ = y$, find the value of $x^ 2 + y^2$.
Answer & explanation
Correct answer: option 3
4 cos θ + 3 sin θ = x and 4 sin θ − 3 cos θ = y
On squaring both the equations and adding ,
( 4 cos θ + 3 sin θ)² +( 4 sin θ − 3 cos θ )² = x² + y²
16 cos² θ + 9 sin² θ + 24 cos θ . sin θ + 16 sin² θ + 9 cos² θ . 24 cos θ . sin θ = x² + y²
16 cos² θ + 9 sin² θ + 16 sin² θ + 9 cos² θ = x² + y²
{ using sin² θ + cos² θ = 1 }
16 ( cos² θ + sin² θ) + 9 ( sin² θ + cos² θ) = x² + y²
x² + y² = 16 + 9
x² + y² = 25