Question
\( \sin (x)=\frac{\sqrt{3}}{2} \) \( x= \) is an acute angle. Find the value of \( x \) in degrees. Submit
Ask by Reyes Tyler. in the United States
Feb 03,2025
Real Tutor Solution
Tutor-Verified Answer
Answer
\( x = 60^\circ \)
Solution
To find the acute angle \( x \) such that \( \sin(x) = \frac{\sqrt{3}}{2} \), follow these steps:
1. **Understand the Sine Values:**
The sine of an angle in a right triangle is defined as the ratio of the length of the opposite side to the hypotenuse. Certain angles have well-known sine values. For example:
\[
\sin(30^\circ) = \frac{1}{2}, \quad \sin(45^\circ) = \frac{\sqrt{2}}{2}, \quad \sin(60^\circ) = \frac{\sqrt{3}}{2}
\]
2. **Identify the Angle:**
Comparing the given sine value with the standard sine values:
\[
\sin(x) = \frac{\sqrt{3}}{2} \implies x = 60^\circ
\]
Since the problem specifies that \( x \) is an **acute angle** (an angle less than \( 90^\circ \)), \( 60^\circ \) is the valid solution.
3. **Conclusion:**
\[
x = 60^\circ
\]
So, the acute angle \( x \) satisfying \( \sin(x) = \frac{\sqrt{3}}{2} \) is **\( 60^\circ \)**.
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Simplify this solution Beyond the Answer
To solve the equation \( \sin (x) = \frac{\sqrt{3}}{2} \), we recall that for acute angles, the sine function equals \( \frac{\sqrt{3}}{2} \) at \( 60^\circ \). So, the value of \( x \) is \( 60^\circ \). Moreover, remember that \( \sin(60^\circ) = \frac{\sqrt{3}}{2} \) is fundamental in trigonometry, often seen when dealing with equilateral triangles, where each angle measures \( 60^\circ \). So, next time you're working with triangles, keep that angle in mind—it’s a sine saver!
