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An angle in standard position measures \( \frac{\pi}{2} \) radians, and \( P(0,1) \) is on the terminal side of the angle. What is the value of the cosine of this angle? -1 1 undefined

Ask by Davies Turner. in the United States
Mar 11,2025

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The cosine of \( \frac{\pi}{2} \) radians is \( 0 \).

Solución

To find the cosine of the angle measuring \( \frac{\pi}{2} \) radians, we can analyze the position of the point \( P(0,1) \) on the terminal side of the angle. 1. **Understanding the angle**: An angle of \( \frac{\pi}{2} \) radians corresponds to 90 degrees. In standard position, this angle starts from the positive x-axis and rotates counterclockwise to the positive y-axis. 2. **Identifying the coordinates**: The point \( P(0,1) \) lies on the positive y-axis, which is indeed the terminal side of the angle \( \frac{\pi}{2} \). 3. **Calculating cosine**: The cosine of an angle in the unit circle is defined as the x-coordinate of the point on the terminal side of the angle. For the point \( P(0,1) \): - The x-coordinate is \( 0 \). Thus, the cosine of the angle \( \frac{\pi}{2} \) radians is: \[ \cos\left(\frac{\pi}{2}\right) = 0 \] Since the options provided are -1, 1, and undefined, the correct answer is not listed. However, the cosine of \( \frac{\pi}{2} \) is indeed \( 0 \).

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The cosine of an angle in standard position is defined as the x-coordinate of the point on the terminal side of the angle where it intersects the unit circle. For the angle \( \frac{\pi}{2} \) radians, which lies along the positive y-axis, the cosine value corresponds to the x-coordinate of the point \( P(0,1) \). Since the x-coordinate is \( 0 \), the cosine of this angle is \( 0 \). Although none of the provided options included \( 0 \), it's an important takeaway that the cosine of \( \frac{\pi}{2} \) is not \( -1 \), \( 1 \), or undefined; rather, it is \( 0 \). This highlights the relationship between angles and their corresponding coordinates on the unit circle! Remember, angles and their trigonometric functions can often lead to unexpected results, so always look at the unit circle to double-check your work! Encountering these \( \pi/2 \) radian angles is quite common, especially in calculus and geometry, where they pop up in many problem-solving scenarios!

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