Question

#2, continued
which statement is true?
$\cos 77^\circ = \frac{x}{20}$
$\cos 77^\circ = \frac{20}{x}$
therefore, which statement is true?
$x = 20 \cdot \cos 77^\circ$ $x = 20 / \cos 77^\circ$
what is the value of $x$? round to the nearest tenth.

Explanation:

Part 1: Which statement is true (cosine ratio)

In a right triangle, the cosine of an angle is defined as the adjacent side over the hypotenuse. In triangle \( JKL \) (right - angled at \( J \)), for angle \( L = 77^{\circ}\), the adjacent side to \( 77^{\circ}\) is \( JL = 20\) and the hypotenuse is \( KL=x\). So, \(\cos(77^{\circ})=\frac{\text{adjacent}}{\text{hypotenuse}}=\frac{20}{x}\).

Starting from \(\cos 77^{\circ}=\frac{20}{x}\), we can cross - multiply to solve for \( x \). Cross - multiplying gives us \( x\times\cos 77^{\circ}=20\), and then dividing both sides by \(\cos 77^{\circ}\) (assuming \(\cos 77^{\circ}
eq0\)), we get \( x = \frac{20}{\cos 77^{\circ}}\) or \( x=20/\cos 77^{\circ}\).

Step 1: Calculate the value of \(\cos 77^{\circ}\)

We know that \(\cos 77^{\circ}\approx\cos(75^{\circ}+ 2^{\circ})\). Using a calculator, \(\cos 77^{\circ}\approx0.2225\).

Step 2: Calculate \( x \)

We have the formula \( x=\frac{20}{\cos 77^{\circ}}\). Substituting \(\cos 77^{\circ}\approx0.2225\) into the formula, we get \( x=\frac{20}{0.2225}\approx89.9\) (rounded to the nearest tenth).

Answer:

\(\boldsymbol{\cos 77^{\circ}=\frac{20}{x}}\)

Part 2: Which statement is true (solving for \( x \))