Question

right triangle trigonometry - sohcahtoa
right triangle trigonometry relates the sides of a right triangle to the angle measures of the two acute angles.
fill in the missing information.
text should be lower case.
calculate the value for x.
answer (round to 1 decimal): x = (ie 1.2 or 16.0)

Explanation:

Step1: Identify trigonometric ratio

We know that $\tan\theta=\frac{\text{opposite}}{\text{adjacent}}$. In the right - triangle, the angle $\theta = 42^{\circ}$, the opposite side to the $42^{\circ}$ angle is 10 and the adjacent side is $x$. So, $\tan42^{\circ}=\frac{10}{x}$.

Step2: Solve for $x$

We can rewrite the equation as $x=\frac{10}{\tan42^{\circ}}$. Since $\tan42^{\circ}\approx0.9004$, then $x=\frac{10}{0.9004}\approx11.1$. But if we consider the angle adjacent to the $42^{\circ}$ angle (which is $48^{\circ}$) and use $\tan48^{\circ}=\frac{x}{10}$. Since $\tan48^{\circ}\approx1.1106$, then $x = 10\times\tan48^{\circ}\approx10\times1.1106 = 11.1$ (wrong). Using $\cos42^{\circ}=\frac{10}{x}$, then $x=\frac{10}{\cos42^{\circ}}$. Since $\cos42^{\circ}\approx0.7431$, $x=\frac{10}{0.7431}\approx13.5$. Using $\sin42^{\circ}=\frac{10}{x}$ is wrong as 10 is not the hypotenuse. The correct way is using $\cos42^{\circ}=\frac{10}{x}$, so $x=\frac{10}{\cos42^{\circ}}\approx13.5$.

Answer:

$x = 13.5$