Hi, can someone please explain how to solve the problem below.
Find the maximum value of 3sinx + cos2x.
Your help is very much appreciated.
There are a few ways. If you know calculus, take the derivative and find the critical points.
If you don't know calculus. You can also graph the equation either by hand with several test test points or with a graphing utilitiy.
If you prefer to not graph it, you can do some critical thinking. For example, on the interval from x=0 to x = pi/2, you have 3sinx going from 0 to 3. On this same interval you have cos2x going from 1 to -1. Thus you can determine which one changes value faster. Then use that information to figure out at what angle the fastest growing function stop growing.
$\displaystyle y = 3\sin{x} + \cos(2x)$
$\displaystyle y = 3\sin{x} + 1 - 2\sin^2{x}$
$\displaystyle y = -2\sin^2{x} + 3\sin{x} + 1$
y is quadratic in $\displaystyle sin{x}$ ...
maximum value of y occurs at $\displaystyle \sin{x} = \left(\frac{-b}{2a}\right)$ ... $\displaystyle \sin{x} = \frac{3}{4}$
$\displaystyle y_{max} = 3\left(\frac{3}{4}\right) + 1 - 2\left(\frac{3}{4}\right)^2 = \frac{17}{8}$
the max value can also be found using the 1st and 2nd derivative tests taught in calculus
If that were "3 cos(x)+ 2sin(x)" you could use r sin(x+ a)= r sin(a)cos(x)+ r cos(a)sin(x) and look for a and r so that r cos(a)= 3, r, sin(a)= 2 and $\displaystyle r^2cos^2(a)+ r^2 sin^2(a)= r^2= 3^2+ 2^2$. (I say this because I misread the problem and started to do it that way!)
But with "3 cos(x)+ sin(2x)" I see no method simpler than taking the first derivative and setting it equal to 0- a "Calculus" rather than "Trigonometry" method.