Question

Please solve q 11?

Answer 1

Find the minimum value of `4 cos theta + 3 sin theta.`

The linear combination is a phase shifted and scaled sine wave, the scale determined by the magnitude of the coefficients in polar form, `\sqrt{3^2+4^2}=5,` so a minimum of `-5`.

Explanation:

Find the minimum value of `4 cos theta + 3 sin theta`

The linear combination of sine and cosine of the same angle is a phase shift and a scaling. We recognize the Pythagorean Triple `3^2+4^2=5^2.`

Let `phi` be the angle such that `cos phi=4/5` and `sin phi = 3/5`. The angle `phi` is the principal value of `arctan(3/4)` but that doesn't really matter to us. What matters to us is we can rewrite our constants: `4 = 5 cos phi` and `3 = 5 sin phi`. So

`4 cos theta + 3 sin theta`

`= 5 (cos phi cos theta + sin phi sin theta)`

`= 5 cos(theta - phi)`

so has a minimum of `-5`.

Answer 2

`-5` is the required minimum value.

Explanation:

Divide the equation `3sinx+4cosx` by `sqrt(a^2+b^2)` to reduce it to the form `sin(x+-alpha) or cos(x+-alpha)` where `a` and `b`
are the coefficients of `sinx` and `cosx` respectively.

`rarr3sinx+4cosx`

`=5[sinx*(3/5)+cosx*(4/5)]`

Let `cosalpha=3/5` then `sinalpha=4/5`

Now, `3sinx+4cosx`

`=5[sinx*cosalpha+cosx*sinalpha]`

`=5sin(x+alpha)=5sin(x+alpha)`

The value of `5sin(x+alpha)` will be minimum when `sin(x+alpha`) is minimum and the minimum value of `sin(x+alpha)` is `-1`.

So, the minimum value of `5sin(x+alpha)=-5`