Question

Find a numerical value of one trigonometric function of x if tan^2x-sin^2x/sin^2x=5

Answer 1

To find the numerical values of $x$, we are going to solve the trigonometric equation $tan^2(x)- \frac{sin^2(x)}{sin^2(x)}= 5$ for $x$.

Step 1. Simplify, $\frac{sin^2(x)}{sin^2(x)}=1$:
$tan^2(x)- \frac{sin^2(x)}{sin^2(x)}= 5$
$tan^2(x)- 1= 5$

Step 2. Add 1 to both sides of the equation:
$tan^2(x)- 1+1= 5+1$
$tan^2(x)=6$

Step 3. Take square root to both sides of the equation:
$\sqrt{tan^2(x)}=(+/-) \sqrt{6}$
$tan(x)= \sqrt{6} ,- \sqrt{6}$

Step 4. Solve for $x$:
$x=arctan( \sqrt{6} )+k \pi ,x=arctan(- \sqrt{6} )+k \pi$
$x=1.1832+k \pi ,x=-1.1832+k \pi$

We can conclude that the numerical values of $x$ that satisfy the equation are $x=1.1832+k \pi$ and $x=-1.1832+k \pi$