Introduction: Graphical method
Graphs of 5sinθ + 2cosθ and sqrt(29)sin(θ+21.8º)
The graph of sqrt(29)sin(θ+21.8º) is an approximation, and thus is slightly out of phase with the graph of 5 sin θ + 2 cos θ. However, when drawn, the 2 graphs are almost the same, which reflects that the functions 5sinθ + 2cosθ and sqrt(29)sin(θ+21.8º) are identical.
Graph of 5sinθ and graph of 2cosθ
Finding Ymin and Ymax of the functions from the graph:
The Ymax and Ymin values of 5sinθ + 2cosθ and sqrt(29)sin(θ+21.8º) are not the Ymax and Ymin values of 5sinθ and 2cosθ added together.
Graph of 5sinθ + 2cosθ plotted with the graphs of 5sinθ and 2cosθ
Solving 5sinθ + 2cosθ =0:
Solving 5sinθ + 2cosθ by expressing in the form Rsin(θ+α), to find the ymin and ymax: