Question

Find the range of the trigonometric function $12\sin \theta -9{{\sin }^{2}}\theta $.

Answer 1

Hint: You know that the trigonometric sine and cosine functions are periodic functions. The range of the trigonometric sine and cosine function is $\left[ -1,1 \right]$, where -1 is the minimum value and 1 is the maximum value.

Complete step-by-step answer:

Let the given function be $f(\theta )=12\sin \theta -9{{\sin }^{2}}\theta $

Taking -1 is common on the right side, we get

$f(\theta )=-\left( 9{{\sin }^{2}}\theta -12\sin \right)$

Now the term on the right side can be converted into complete square by adding and subtracting 4, we get

$f(\theta )=-\left( 9{{\sin }^{2}}\theta -12\sin +4-4 \right)$

Rearranging the term on the right side, we get

$f(\theta )=-\left[ \left( 9{{\sin }^{2}}\theta -12\sin +4 \right)-4 \right]$

$f(\theta )=-\left[ {{\left( 3\sin \theta -2 \right)}^{2}}-4 \right]$

Open the bracket on the right side, we get

 \[f(\theta )=4-{{\left( 3\sin \theta -2 \right)}^{2}}\]

Minimum value of function $f(\theta )$ occurs when \[{{\left( 3\sin \theta -2 \right)}^{2}}\] is minimum.

Minimum value of function $f(\theta )=4-{{\left( 3\times (-1)-2 \right)}^{2}}=4-{{(-3-2)}^{2}}=4-25=-21$

Maximum value of function $f(\theta )$ occurs when \[{{\left( 3\sin \theta -2 \right)}^{2}}\] is maximum.

Maximum value of function $f(\theta )=4-{{(0)}^{2}}=4$

Hence the range of the given function is $\left[ -21,4 \right]$.


Note: You might get confused about the difference between the domain of a function and the range of a function. Domain is the independent variable and range is the dependent variable. On the other hand, range is defined as a set of all probable output values. Domain is what is put into a function, whereas range is what is the result of the function with the domain value.