Maths with David

    IBDP. AA. HL. Trigonometric Limits

    Investigation: \frac{ sin \theta }{ \theta }

    1a.) Graph y = \frac{sin x}{x} for \frac{ - \pi }{2} \leq x \leq \frac{ \pi }{2} (you can use technology to help you)

    1b.) Discuss the behaviour of the graph as x approaches zero.

    2. We will now prove the result lim_{ \theta \to 0 } \frac{sin \theta}{ \theta } = 1

    1. Show that f( \theta ) = \frac{ sin \theta }{ \theta } is an even function. What does this mean graphically?
    2. Suppose f ( \theta ) is an even function for which lim_{ \theta \to 0^{+} } f( \theta ) = A .
    3. Explain why:
      • lim_{ \theta \to 0^{-} } f( \theta ) = A
      • lim_{ \theta \to 0 } f( \theta ) = A
    4. Since \frac{ sin \theta }{ \theta } is even, we need only examine \frac{ sin \theta }{ \theta } for positive \theta .
      • A circle of radius r contains n congruent isosceles triangles as shown at the end of this question.
      • Use the diagram to show that lim_{ n \to \infty } \frac{n}{2} r^2 sin \frac{2 \pi}{n} = \pi r^2
      • Hence show that:
        • lim_{ n \to \infty } \frac{sin (2 \pi / n )}{ 2 \pi / n } = 1
        • lim_{ \theta \to 0} \frac{ sin \theta}{ \theta } = 1

    From the above investigation, we found that if \theta is in radians then lim_{ \theta \to 0 } \frac{ sin \theta }{ \theta } = 1

    Worked Example

    Find lim_{ \theta \to 0} \frac{ sin 3 \theta}{ \theta }

    Exercise

    Answers

    (1.) (a.) 2, (b.) 1, (c.) 1, (d.) 0, (e.) 4, (f.) 1/2, (g.) 3pi, (h.) 7/4, (i.) 1/2

    (2.) (a.) Does not exist, (b.) 0, (c.) 1/2