Why The Limit Of Sin(Theta)/Theta As Theta Approaches 0 Is Equal To 1: A Visual Explanation Using The Unit Circle

lim theta->0 sin(theta) / theta = ___________________

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The limit of sin(theta)/theta as theta approaches 0 is equal to 1.

To see why, consider the unit circle in the xy-plane, centered at the origin. Label the x-coordinate of a point on the circle as cos(theta) and the y-coordinate as sin(theta).

Now, envision a line segment that starts at the point (1, 0) on the circle (corresponding to theta = 0) and ends at the point (cos(theta), sin(theta)) for some small positive theta. This line segment will clearly lie entirely in the first quadrant and will make an angle of theta with the positive x-axis.

Using the distance formula, the length of this line segment is the square root of [(cos(theta) – 1)^2 + sin^2(theta)]. However, since theta is small, we can make the approximation that cos(theta) is close to 1 and that sin(theta) is close to theta (as measured in radians). Therefore, we can say that the length of the line segment is approximately equal to the square root of [(cos(theta) – 1)^2 + theta^2].

Now, draw a line from the point (cos(theta), sin(theta)) to the x-axis, forming a right triangle. The hypotenuse of the triangle is the line segment we just found, and the adjacent side is (1 – cos(theta)). Therefore, using the Pythagorean theorem, the length of the opposite side is approximately equal to the square root of [((1 – cos(theta))^2) + theta^2].

Dividing the opposite side by the hypotenuse of the triangle (i.e. dividing the square root expression above by the square root expression from earlier), we get:

sin(theta) / [sqrt((1 – cos(theta))^2 + theta^2)].

However, as theta approaches 0, we know that cos(theta) approaches 1 and that sin(theta) approaches theta, so the above expression simplifies to:

theta / [sqrt((1 – 1)^2 + theta^2)] = theta / theta = 1.

Therefore, the limit of sin(theta)/theta as theta approaches 0 is equal to 1, and we can write:

lim theta->0 sin(theta)/theta = 1.

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