Find the values of x and y.
Understand the Problem
The question asks for the values of x and y in a right triangle defined by the given dimensions and angle. We will use trigonometric ratios to find these values.
Answer
$x = 18$, $y = 6\sqrt{3}$
Answer for screen readers
The values are: $x = 18$ $y = 6\sqrt{3}$
Steps to Solve
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Identify the given values The hypotenuse of the triangle is given as $12\sqrt{3}$ and the angle opposite to $y$ is $30^\circ$.
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Use the sine function to find y For a right triangle, the sine of an angle is the ratio of the opposite side to the hypotenuse. Thus, we can write: $$ \sin(30^\circ) = \frac{y}{12\sqrt{3}} $$
Substituting the known value, we have: $$ \frac{1}{2} = \frac{y}{12\sqrt{3}} $$
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Solve for y Multiply both sides by $12\sqrt{3}$: $$ y = 12\sqrt{3} \cdot \frac{1}{2} = 6\sqrt{3} $$
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Use the cosine function to find x Now, we can find $x$ using the cosine function. The formula for cosine is: $$ \cos(30^\circ) = \frac{x}{12\sqrt{3}} $$
Substituting the known value of $\cos(30^\circ) = \frac{\sqrt{3}}{2}$ gives: $$ \frac{\sqrt{3}}{2} = \frac{x}{12\sqrt{3}} $$
- Solve for x Multiply both sides by $12\sqrt{3}$: $$ x = 12\sqrt{3} \cdot \frac{\sqrt{3}}{2} = 12 \cdot \frac{3}{2} = 18 $$
The values are: $x = 18$ $y = 6\sqrt{3}$
More Information
In a right triangle, the sine and cosine functions relate the angles to the lengths of the sides, allowing us to determine any unknown side lengths when given one side and an angle.
Tips
- Confusing sine and cosine: Ensure to use the correct function based on the angle's position relative to the sides.
- Not simplifying the answers properly: Always check if the result can be simplified further.
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