Find z. Write your answer in simplest radical form.
Understand the Problem
The question asks to find the length of side 'z' in a right triangle, where one angle is 60 degrees, another is 30 degrees, and the side opposite the 30-degree angle has a length of $2\sqrt{6}$ yards. We can use trigonometric ratios (sine, cosine, tangent) to solve for 'z', expressing the answer in simplest radical form.
Answer
$4\sqrt{6}$
Answer for screen readers
$4\sqrt{6}$
Steps to Solve
- Identify the relevant trigonometric ratio
We are given the side opposite the $30^\circ$ angle and we want to find the hypotenuse (side $z$). The trigonometric ratio that relates the opposite side and the hypotenuse is the sine function.
- Set up the trigonometric equation
We can set up the following equation: $$ \sin(30^\circ) = \frac{\text{opposite}}{\text{hypotenuse}} = \frac{2\sqrt{6}}{z} $$
- Solve for $z$
We know that $\sin(30^\circ) = \frac{1}{2}$. Substituting this into the equation gives us: $$ \frac{1}{2} = \frac{2\sqrt{6}}{z} $$ To solve for $z$, we can cross-multiply: $$ z = 2 \cdot 2\sqrt{6} $$ $$ z = 4\sqrt{6} $$
$4\sqrt{6}$
More Information
The length of side $z$ in the right triangle is $4\sqrt{6}$ yards, which is in simplest radical form.
Tips
A common mistake is using the wrong trigonometric ratio (e.g., cosine or tangent instead of sine) or incorrectly identifying the opposite and adjacent sides relative to the given angle. Another mistake is not knowing the value of $\sin(30^\circ)$.
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