Essential Limits

For the four essential trigonometric limits it is essential that we use radian measure. Then for we get diagram

PROOF:

In the diagram the length of the line segment AD is and the length of the line segment BD is . Since these are the legs of a right triangle both are shorter than the hypotenuse AB, which is in turn shorter than the arc t.

Application of the squeeze theorem to these inequalities gives

The other basic limit needed to calculate the derivative of the sine function comes from the inequality

PROOF:

For t in this range we are looking at the first quadrant. In the diagram the coordinates of the points are given by

The triangle OAB is contained in the sector of the circle OAB which is in turn contained in the triangle OAC. The areas of the two triangles are given by the formula area = base height. The area of the sector of the circle is given by , taking the fraction of the unit circle's area corresponding to the fraction of the circumference given by the arc t. Thus their areas give the inequalities

Multiplying by 2 gives the desired inequalities.

Dividing all terms of the inequality by and then taking reciprocals gives

Applying the squeeze theorem to this inequality gives

PROOF:

Multiply both numerator and denominator by to get

Symmetry relations can be used to obtain these same limits from the right, thus allowing us to replace with .