Explain how to derive the sine and cosine values of standard angles using the unit circle.
Answer 1
Lucas Brown
To derive the sine and cosine values of standard angles (0°, 30°, 45°, 60°, and 90°) using the unit circle, follow these steps:
1. Draw the unit circle centered at the origin with a radius of 1.
2. Mark the standard angles on the unit circle. For example, angle 30° (or π/6) will be marked from the positive x-axis moving counter-clockwise.
3. For each angle, drop a perpendicular from the point on the unit circle to the x-axis. This forms a right triangle.
4. Use the definitions of sine and cosine: sine is the y-coordinate of the point, and cosine is the x-coordinate.
5. By using the properties of special triangles (such as 30°-60°-90° and 45°-45°-90° triangles), one can determine the exact coordinates of each point. For example, for 45° (or π/4), the coordinates are (√2/2, √2/2), so cos(45°) = √2/2 and sin(45°) = √2/2.
Answer 2
Christopher Garcia
To derive the sine and cosine values of standard angles using the unit circle:
1. Draw the unit circle with a radius of 1.
2. Mark angles 0°, 30°, 45°, 60°, and 90° on the circumference.
3. Drop perpendiculars from the points to the x-axis, forming right triangles.
4. The x-coordinate gives the cosine of the angle, and the y-coordinate gives the sine.
5. Using special triangles, find the sine and cosine values. For example, at 30° (or π/6), the coordinates are (√3/2, 1/2), thus cos(30°) = √3/2 and sin(30°) = 1/2.
Answer 3
John Anderson
To derive sine and cosine values of standard angles using the unit circle:
1. Draw the unit circle with radius 1 and mark standard angles.
2. Drop perpendiculars to the x-axis to form right triangles.
3. Coordinates of points give sine (y) and cosine (x) values.
4. Example: At 60° (or π/3), coordinates are (1/2, √3/2), so cos(60°) = 1/2 and sin(60°) = √3/2.
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