Quite a number of students have difficulty understanding R formula.
Hopefully this graph will help.
3 sin(x)+4 cos (x) expressed in R formula is 5 sin(x+0.927)
So the two graphs appear as 1 graph.
Max value of 3 sin(x)+4 cos (x) = 5. Occurs when sin(x+0.927)=1
Min value of 3 sin(x)+4 cos (x) = -5. Occurs when sin(x+0.927)=-1
Students can try their hands in R formula at https://www.desmos.com/calculator
Quite a number of Amath students have difficulty sketching the transformation of trigonometric graph.
This is what is required in latest Amath syllabus 4047
Students can play around with transformation of graphs at https://www.desmos.com/calculator
cos (2x) moves twice as fast as cos x, so there are twice as many cycles within the same time frame.
cos (0.5x) moves half as fast as cos x, so there are half as many cycles within the same time frame.
-sin(x) is a reflection of sin(x) about the X-axis. 2 sin (x) a vertical stretch of factor 2 parallel to the Y axis.
sin(x)+2 is a vertical translation of sin(x) by 2 units upwards parallel to the Y axis.
sin(x)-2 is a vertical translation of sin(x) by 2 units downwards parallel to the Y axis.
The above shows the sequence of transforming sin(x) to 3 sin(2x)+2.
Explore transformation of graph at https://www.desmos.com/calculator. This will help in learning the transformation of trigo graphs.