University:
York University
Course:
MATH 1013 | Applied Calculus
Academic year:

2025
Views:

483

Pages:

4
Author:

Noya Ali

Limits of Trigonometric Functions Consider the graphs of y = sin 𝜃 and y = cos 𝜃 Inspection of the graphs of y = sin 𝜃 and y = cos 𝜃 gives us the first two limits. lim sin 𝜃 = 0 lim cos 𝜃 = 1 𝜃→0− 𝜃→0− lim sin 𝜃 = 0 lim cos 𝜃 = 1 𝜃→0+ 𝜃→0+ Therefore, lim cos 𝜃 = 1 lim sin 𝜃 = 0 𝜃→0 𝜃→0 One of the important limits involving trigonometric functions is lim 𝜃→0 We examine the value of sin 𝜃 𝜃 sin 𝜃 𝜃 𝜃 Approaching 0 from the Left sin 𝜃 𝜃 𝜃 0.3 - 0.3 0.2 - 0.2 0.1 - 0.1 0.05 - 0.05 0.02 - 0.02 0.01 - 0.01 sin 𝜃 𝜃 𝜃 for values of 𝜃 close to 0. Approaching 0 from the Right The trend of the values of sin 𝜃 in the tables suggests that lim 𝜃→0 sin 𝜃 𝜃 = ___________________ Proof: In the diagram, point A is on the unit circle x2 + y2 = 1. 𝜋 Point A determines angle 𝜃, 0 < 𝜃 < 2 . The perpendicular drawn from point A meets the x-axis at B. Another important limit involving trigonometric functions is lim 𝜃→0 Proof: cos 𝜃−1 𝜃 Examples: a) lim 𝜃→0 sin2𝜃 sin7𝜃 b) lim 𝑠𝑖𝑛4𝜃 𝜃 𝜃→0 sin𝜃 c) lim 𝜋−𝜃 𝜃→𝜋 Summary:: sin 𝜃 lim 𝜃 = 𝜃→0 lim 𝜃→0 cos 𝜃−1 𝜃 = lim 𝜃→0 lim sin 𝑚𝜃 𝑛𝜃 sin 𝑚𝜃 𝜃→0 𝑠𝑖𝑛 𝑛𝜃 = =