Animated Solution for Mathematics - Trigonometry: The maximum value of the expression $\frac{1}{{\sin }^2\theta +3\sin \theta \cos \theta +5{\cos }^2\theta }$ is ____.

The expression we are analyzing is:

To maximize this fraction, we must minimize the denominator. Let us define the denominator as:

The presence of squared terms and products makes $D(\theta )$ difficult to handle directly. We utilize double-angle identities to transform the expression:

By substituting these identities, we convert the quadratic terms into a linear combination of $\sin 2\theta$ and $\cos 2\theta$.

Substituting the identities into $D(\theta )$, we obtain:

Distributing the fractions yields:

Combining the constant terms and the trigonometric coefficients, we simplify the expression to:

We have arrived at the form $a\cos x+b\sin x$, where $a=2$ and $b=1.5$. The range of this trigonometric component is $[-\sqrt{a^2+b^2},\sqrt{a^2+b^2}]$.

Calculating the amplitude:

The expression $2\cos 2\theta +1.5\sin 2\theta$ oscillates between $-2.5$ and $2.5$. To minimize $D(\theta )$, we select the minimum value of this oscillation, which is $-2.5$.

Thus, the minimum denominator is:

The maximum value of the original expression is: