Animated Solution for Mathematics - Trigonometry: If $\sin x=-\frac{3}{5}$, where $\pi <x<\frac{3\pi }{2}$, then $80({\tan }^{2}x-\cos x)$ is equal to

We are given that $\sin x=-\frac{3}{5}$ and the angle $x$ lies in the third quadrant, defined by the interval $\pi <x<\frac{3\pi }{2}$. In the third quadrant, both the $x$-coordinate (base) and the $y$-coordinate (perpendicular) are negative, while the hypotenuse remains positive.

This quadrant constraint is critical, as it dictates the signs of all trigonometric functions. Failing to account for the negative base in this quadrant is a common pitfall.

Our Compass in the Dark

We construct a right-angled triangle where $\sin x=\frac{\text{Perpendicular}}{\text{Hypotenuse}}=-\frac{3}{5}$. This identifies the perpendicular as $-3$ and the hypotenuse as $5$.

Using the Pythagorean theorem, $b^2+p^2=h^2$, we calculate the base $b$:

Since we are in the third quadrant, the base must be negative. Therefore, $b=-4$.

With the sides determined, we find the required trigonometric ratios:

Substitution and Simplification

We now evaluate the expression $80({\tan }^{2}x-\cos x)$ by substituting our derived values:

First, we compute the square and handle the double negative:

The expression simplifies to:

To add the fractions, we use a common denominator of $80$:

Multiplying the terms, the $80$ in the numerator and denominator cancel out perfectly:

The final result of the expression is $109$.