Designing a Circular Ruler?

📌 CONCEPT: Inverse trigonometric functions are used to determine the angle or side length of a right triangle when the other is known. The circular ruler problem involves using these functions to find the new scale of numbers on a straight line. The inverse trigonometric function used here is the arcsine function, which gives the angle whose sine is a given value.

📐 RULE / FORMULA: The new scale of numbers on the straight line is given by the formula y = r ∗ sin(x), where r is the radius of the circular ruler and x is the angle in degrees. Since the circular ruler is folded into a straight line, the new scale of numbers will be the absolute value of the sine of the angle.

💡 WORKED EXAMPLE: Suppose the circular ruler has a radius of 5 cm and the angle marked on it is 30°. Using the formula y = r ∗ sin(x), we get y = 5 ∗ sin(30°) = 5 ∗ 0.5 = 2.5 cm. Therefore, the new scale of numbers on the straight line is 2.5 cm.

⚠️ COMMON MISTAKE: Students often forget to take the absolute value of the sine function, which can lead to incorrect answers. Additionally, they may not use the correct formula for the new scale of numbers, resulting in an incorrect solution.