Animated Solution for Mathematics - Trigonometry: The number of roots of the equation, $(81{)}^{{\sin }^{2}x}+(81{)}^{{\cos }^{2}x}=30$ in the interval $[0,\pi ]$ is equal to :

The given equation is $81^{{\sin }^{2}x}+81^{{\cos }^{2}x}=30$. At first glance, this appears to be a complex mix of exponential growth and periodic oscillation.

However, we can utilize the fundamental trigonometric identity ${\sin }^{2}x+{\cos }^{2}x=1$. This allows us to express the equation in terms of a single trigonometric function: ${\cos }^{2}x=1-{\sin }^{2}x$.

Substituting this into the original expression, we obtain:

Using the laws of exponents, we recognize that $81^{1-{\sin }^{2}x}$ can be rewritten as $\frac{81}{81^{{\sin }^{2}x}}$. Let us introduce a substitution variable $t=81^{{\sin }^{2}x}$.

The equation now transforms into a standard algebraic form:

Multiplying the entire equation by $t$ yields the quadratic equation:

To solve $t^2-30t+81=0$, we factor the quadratic by finding two numbers that multiply to $81$ and add to $-30$. These numbers are $-3$ and $-27$.

The factored form is $(t-3)(t-27)=0$, which gives us two possible values for $t$:

Now, we substitute back $t=81^{{\sin }^{2}x}$ to solve for $x$.

Case 1: $81^{{\sin }^{2}x}=3$ Since $81=3^4$, we have $(3^4{)}^{{\sin }^{2}x}=3^1$. This implies $4{\sin }^{2}x=1$, or:

Case 2: $81^{{\sin }^{2}x}=27$ Since $81=3^4$ and $27=3^3$, we have $(3^4{)}^{{\sin }^{2}x}=3^3$. This implies $4{\sin }^{2}x=3$, or:

We are restricted to the interval $[0,\pi ]$. In this interval, the sine function is always non-negative, so we discard the negative roots.

For $\sin x=\frac{1}{2}$, the solutions are $x=\frac{\pi }{6}$ and $x=\frac{5\pi }{6}$.

For $\sin x=\frac{\sqrt{3}}{2}$, the solutions are $x=\frac{\pi }{3}$ and $x=\frac{2\pi }{3}$.

Counting these values, we find that there are exactly four roots in the given interval.