Calculate the area of the sectorabd in terms of angle theta

Question

Goal: To move that lim_θ→0 sin (θ) / θ = 1. This limit is essential to proving that the derivative of sin(x) is cos(x).

Rules: You may discuss with other students to work out the answers. llowever you may NOT copy other students' work. You must complete the quiz alone.

The above diagram shows two triangles ΔABD, ΔHED, and a sector ABD. The sectorABD is part of a circle centered at A with radius 1. So IABI = IADI = 1.

1. Let L₁ = |BC|, and L₂=|ED|.

Calculate L₁ and L₂ in terms of the angle θ.

2. Using L₁ and L₂, calculate the areas of ΔHED and ΔABD.

3. Calculate the area of the sectorABD in terms of angle θ.

4. Take the formulas you got in 2) and 3) and just by looking at the diagram, put the three areas in order of increasing size.

5. Simplify the inequalities to show that 1 ≥ sin x / x ≥ cos x.

6. Use the Squeeze Theorem to get the final result.